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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. In Kaihuang 4 the calendar of Zhang Bin was adopted; in Kaihuang 17 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. In Tianjian 3 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." In Tianjian 8, 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 Tianjian 8 through the seventh month of Tianjian 9. 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 Tianjian 9, Zu Chongzhi's Jiazi Origin Calendar was adopted and the new year's calendar was promulgated. By Datong 10 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 Tianbao 1 (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." In Wuping 7 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 Daxiang 1 (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 Kaihuang 4 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 Kaihuang 4 (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 year 3, second month, day jisi: 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, year 12, third month, day gengwu: 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
Year 15, 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, day dingsi: 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。
Second: verify the changing discrepancy in celestial degrees.
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, year 5, first month, day renzi, new-moon dawn, winter solstice. Calculated by the Jiazi Origin Calendar, the match is exact. The History of Song records that in Yuanjia 10, 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 Yuanjia 13 through Yuanjia 20; 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，
Day 15: 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，
Day 26: 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，
Day 18: 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，
Day 29: 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，
Day 10: 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，
Day 3: longest shadow;
89
即是今曆冬至日。
This matches the present calendar's winter solstice.
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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 Kaihuang 14, 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 Kaihuang 17, 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, day renzi, new-moon dawn, winter solstice. The Zuo Commentary records Duke Xi's fifth year, first month, day xinhai, 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. Yuanjia 12, 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. Yuanjia 13, 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. Yuanjia 15, eleventh month, dingmao-day new moon; eighteenth day, jiashen, winter solstice; longest shadow. Both calendars agree on jiashen winter solstice. Yuanjia 16, 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. Yuanjia 17, eleventh month, yiyou-day new moon; tenth day, jiawu, winter solstice; longest shadow. Zhang Bin's calendar: jiawu winter solstice. Zhang Zhouxuan's calendar: yiwu winter solstice—one day late. Yuanjia 18, 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. Yuanjia 19, 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 Tianhe 1 (bingxu) through Sui's Kaihuang 15 (yimao), fourteen winter and summer solstice shadow records are available. Zhang Bin's calendar matched ten and missed four—three one day early, one one day late. Zhang Zhouxuan's calendar matched five and missed nine—eight one day late, one one day early. Tianhe 2, 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. Tianhe 3, 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. Jiande 1, 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. Jiande 2, 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. Year 3, 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. Year 6, 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 Year 1, eleventh month, jia-wu-day new moon;, day 5, wu-xu, winter solstice, longest shadow. Both calendars agree on wuxu winter solstice. KaihuangYear 4, 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. Year 5, 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. Year 7, fifth month, yi-hai-day new moon;, day 9, gui-wei, Summer Solstice, shortest shadow. Summer Solstice,, Summer Solstice. 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. Year 11, 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. Kaihuang 14, 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. Jiande 4, 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. In Kaihuang 4, 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 Jiande 4: 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 Kaihuang 17: 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 Kaihuang 4, 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. Kaihuang 5, 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. Kaihuang 6, 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. Kaihuang 10, 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. Kaihuang 12, 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. Kaihuang 13, 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. Kaihuang 14, 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. Kaihuang 15, 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. Kaihuang 16, 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 Kaihuang 17 set the winter solstice at the fifth degree of the Xu lodge. Later its imprecision became apparent; by Daye 4, 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 Daye 4 (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 by month divisor intercalary remainder, cycle yearsmall remainder,, divide by qi divisor to obtain one,, count from new moon beyond tally, Winter Solstice. 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 according to month length,, beyond tally,. 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. Itsentering qi, divide by sixteen to obtain one, 1,.
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。
Small fraction, day fraction, innersmall fraction,, submergence fraction. Nonesmall fraction, day fraction,. 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。
Accumulated days,,,, remainder is, beyond tally.
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。
Small remainder,, small fraction,.
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。
Decrease-increase, decrease-increaseanomaly, 1, yielding fixed. 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
Horn 12°, Neck 9°, Root 15°, Room 5°, Heart 5°, Tail 18°, Winnowing Basket 11°.
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
Dipper 26°, Ox 8°, Girl 12°, Emptiness 10°, Rooftop 17°, Encampment 16°, Wall 9°.
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
Legs 16°, Bond 12°, Stomach 14°, Hairy Head 11°, Net 16°, Turtle Beak 2°, Three Stars 9°.
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
Well 33°, Ghost 4°, Willow 15°, Star 7°, Extended Net 18°, Wings 18°, Axletree 17°.
146
南方七宿百一十二度
Southern seven lodges: 112°.
147
推日度術：
Method for computing 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。
,,,, degree divisorand1, yielding accumulated,. ,,, beyond tally, Winter Solstice. Winter Solstice,, 1, degree divisor,,,. 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
Small remaindercycle year,,,. 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 degree at full moon hour:
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。
Small remainder, day divisor,,,.
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。
,, day fraction,.
168
火終日，七百七十九，日分，三萬九千四百六十六。
,, day fraction,.
169
土終日，三百七十八，日分，三千八百四十七。
,, day fraction,.
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。
,, day fraction,. 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。
,, day fraction,. Morning appearance and disappearance: 63 days, fractions the same. Evening appearance and disappearance: 52 days。.
172
求星見術：
Method for finding star 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 , partswhen full degree divisorcarry 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: mean appearanceatSpring Equinox, Great Cold, mean appearance,, yielding fixed. Start of Autumn, Cold Dew,,. 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 appearanceday fraction. Winter Solstice, Great Cold,. 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.
177
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 appearanceatRain Water, Great Cold. AtStart of Summerafter, Start of Autumn, add to appearance day fraction; when full remove as before. Rain Water through Start of Summer, uniformly add 29day. Lesser Snow, End of Heat. Winter Solsticeafter, Great Cold, when full remove as before and subtract. Lesser Snow through Winter Solstice, uniformly subtract 25day. At first appearance and disappearance, each is seventeen degrees from the sun.
178
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 appearanceatEnd of Heat, Great Heat. White Dewafter, Frost Descent, add to appearance day fraction; when full remove as before. From End of Heat through White Dew add 9 days uniformly. Lesser Cold, Frost Descent, Lesser ColdStart of Spring, Start of Spring, 7, 1, Grain Rain3, Summer Solstice1, 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: mean appearance, atStart of Spring, Lesser ColdLesser Fullnessafter, Summer Solstice, add to appearance day fraction; when full remove as beforeStart of SpringLesser Fullness. Start of Autumn, Lesser Heat, Lesser SnowWinter Solstice, when full remove as before and subtract, Start of AutumnLesser Snow. Mean appearance,, Lesser Snow. Pure Brightnessafter, Grain in Ear, when full remove as before and subtract, Pure Brightness. End of Heat, Summer Solstice. Cold Dewafter, Great Snow. 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.
180
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. Start of AutumnEnd of Heat,,,. When absent, it is likewise not visible. Frost DescentStart of Winter,,,. 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。
：, hassmall fraction,, small fraction, degree divisor. Its Fast motionSlow motion, 1 daily motion parts, each its partsFast motionSlow motion thereof . Stationary, then subtract thereof , disappearancenot °, directremove 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。
Rain Water, Lesser Cold, Lesser Fullnessafter, Great Heat. Divide by 3; subtract the result from days to obtain fixed days. Rain WaterLesser Fullness, yielding fixed. All preceding entries give the day and degree counts of the prior fast-motion phase. Winter Solstice, according todecrease-increase thereof, yielding fixed. Degree divisor, 1, namely uniform motion 1day fraction, small fraction. Great ColdStart of Autumn, remainder uniform motion. End of HeatWhite Dew,, 6. White DewCold Dew, daily motion, daily motion,. 1,, uniform motion 1day fraction, day fraction. , slow motion60fraction, slow motion. Daily motion 20,600, daily decreasing slow , 60daily motion 24 and 35,640fast remove , slow add 4,264, 60daily motion 30,. 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. ,, daily motion, daily increasing fast, daily motion,, Start of AutumnAutumn Equinox,, daily motion,. 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. Start of Summer, Lesser Heat, daily motion,,. Lesser HeatStart of Autumn,,. 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: first appearance,,, then stationary. Then direct,,,, daily motion. 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; Lesser HeatStart of Winter, yielding fixed. 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 partsas daily motion parts. Uniform motion,, 15 daily motion. Lesser Cold, 1, Rain Water,. Uniformly through 10 days after spring equinox reduce by 1; through Lesser Fullness, again 15 days travel 15°. 1, End of Heat,. Frost Descentafter, 1, Winter Solstice15 daily motion, daily motion. When prior direct-slow reduces degrees, compute the reduction and add to this degree for fixed degree. 1 daily motion,, degree divisor, 1, uniform motion. Morning disappearance in the east. First appearance,,, daily motion. Summer Solstice, Lesser Fullness,. Lesser Heatafter, Start of Autumn,, Summer SolsticeLesser Heat, yielding fixed. White DewPure Brightness,, daily decreasing slow. Pure BrightnessWhite Dew, uniform motion, uniform motion,,,, daily motion. Uniform motion,, 15 daily motion. 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: first appearance,. Direct, Slow motion, daily motion 660 parts, 4 daily motion 1°. From Great Cold through Rain Water, this slow-motion phase is not required. Uniform motion,, 10 daily motion. Great Cold, 1,,. Fast motion, daily motion 1°38376 parts, 10 daily motion 19°, prior none Slow motion, subtractthis 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°. Uniform motion,, 10 daily motion. Great Heat, 1,,. 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。
Accumulated months,,, remainder is.
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. AtGrain Rainafter, Grain in Ear,, Grain Rain. When full, remove the conjunction common; the remainder is the fixed remainder. ItsLesser ColdSpring Equinox, Start of SummerGrain in Ear,,. 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 Saturn 10+ days from appearance, Mars 40+, Venus morning disappearance 22+. When only one luminary is involved, do not add the qi difference. White Dew, Lesser Heat. AtStart of Winter, Great Snow,. White DewStart of Winter,,, remainder is. 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。
Small remainder,,, beyond tally,. 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。
Small remainder,,,, 24,. In the third month of spring, inner path, node distance seven hours or above—add 24. ,, beyond tally,. The remainder is the time remainder. Set aside, when full,, or above. 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,, 3, 2, 1,, 12. Multiply by the difference rate; divide by 14 to obtain one—this is the hour difference. Half half and half half , Multiply by add remainder. Half half and half 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. Spring Equinox,, Autumn Equinox,,. 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, stardisappearance, 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, noteclipse. Awakening of Insects through Grain Rain, 13, hour of occurrence, noteclipse. End of HeatFrost Descent,,,.
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, 1eclipsefraction. ,,, 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。
AtAutumn Equinox, Summer Solstice,, remainder is. ,, remainder is. Yielding fixed. ,, yielding fixed. AtAwakening of Insects, Summer Solstice500. Autumn Equinox through Awakening of Insects, uniformly subtract 184000,, as before. Great ColdLesser Fullness,,. ,,, eclipsealready. 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.
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。
Entering qi,, entering qi, divide by fifteen to obtain one, decrease-increasethat whichentering qi, yielding fixed.
224
隋書總目
General Table of Contents of the Book of Sui.
225
隋書隋書總目隋書卷一十八志第十三
Book of Sui — General Table of Contents — Book of Sui, Volume 18, Treatise 13.
226
律曆下
Treatises on Measures and the Calendar, Part Three.
227
開皇二十年，袁充奏日長影短，高祖因以曆事付皇太子，遣更研詳著日長之候。 太子征天下曆算之士，咸集於東宮。 劉焯以太子新立，複增修其書，名曰《皇極曆》，駁正胄玄之短。 太子頗嘉之，未獲考驗。 焯為太學博士，負其精博，志解胄玄之印，官不滿意，又稱疾罷歸。 至仁壽四年，焯言胄玄之誤于皇太子：
In Kaihuang 20, Yuan Chong memorialized that days were long and shadows short; Emperor Gaozu entrusted calendrical affairs to the Crown Prince and ordered further study of the signs of lengthening days. The Crown Prince summoned calendrical and computational experts from across the realm; all assembled at the Eastern Palace. Liu Chuo, as the Crown Prince had just been installed, further revised his work, titling it the Supreme Pole Calendar, to refute Zhang Zhouxuan's errors. The Crown Prince greatly approved it, but verification had not yet been obtained. Liu Chuo, as Imperial Academy Erudite, confident in his expertise, sought to supplant Zhang Zhouxuan; dissatisfied with his office, he pleaded illness and retired. By Renshou 4, Liu Chuo reported Zhang Zhouxuan's errors to the Crown Prince.
228
其一曰，張胄玄所上見行曆，日月交食，星度見留，雖未盡善，得其大較，官至五品，誠無所愧。 但因人成事，非其實錄，就而討論，違舛甚眾。
1, Zhang Zhouxuan, eclipse, stationary,,, through 5,. But it was achieved through others' work, not his own record; on examination, the discrepancies were very numerous.
229
其二曰，胄玄弦望晦朔，違古且疏，氣節閏候，乖天爽命。 時不從子半，晨前別為後日。 日躔莫悟緩急，月逡妄為兩種，月度之轉，輒遺盈縮，交會之際，意造氣差。 七曜之行，不循其道，月星之度，行無出入，應黃反赤，當近更遠，虧食乖准，陰陽無法。 星端不協，珠璧不同，盈縮失倫，行度愆序。 去極晷漏，應有而無，食分先後，彌為煩碎。 測今不審，考古莫通，立術之疏，不可紀極。 今隨事糾駁，凡五百三十六條。
Second: Zhang Zhouxuan's first quarter, full moon, last quarter, and new moon violate antiquity and are coarse; seasonal nodes, intercalation, and pentads deviate from Heaven's clear mandate. Hours do not begin from midnight; the morning before is separately counted as the following day. Solar motion fails to grasp fast and slow; lunar anomaly is arbitrarily made into two kinds; monthly rotation omits expansion-contraction; at conjunction crossings, qi difference is fabricated. 7,,,,,, obscurationeclipse,. ,, anomaly,. Polar distance and gnomon graduations should exist but do not; eclipse fraction sequencing is excessively tedious. Present measurements are unverified, ancient comparisons fail—the method's flaws are beyond reckoning. Now item by item refuted—536 entries in all.
230
其三曰，胄玄以開皇五年，與李文琮于張賓曆行之後，本州貢舉，即齎所造曆擬以上應。 其曆在鄉陽流布，散寫甚多，今所見行，與焯前曆不異。 玄前擬獻，年將六十，非是忽迫倉卒始為，何故至京未幾，即變同焯曆，與舊懸殊？ 焯作于前，玄獻於後，舍己從人，異同暗會。 且孝孫因焯，胄玄後附孝孫，曆術之文，又皆是孝孫所作，則元本偷竊，事甚分明。 恐胄玄推諱，故依前曆為駁，凡七十五條，並前曆本俱上。
, Multiply by , , , Multiply by. His calendar circulated widely with many copies; the version in circulation matches Liu Chuo's earlier calendar. Zhang Zhouxuan had prepared to submit his work at nearly sixty—this was not hastily composed; why, shortly after reaching the capital, did he change to match Liu Chuo's calendar, diverging from his earlier system; ？. Liu Chuo composed first, Zhang Zhouxuan submitted later—abandoning his own work to follow another, their differences secretly aligned. Moreover Xiaosun derived from Liu Chuo; Zhang Zhouxuan later attached to Xiaosun—the calendrical text was all Xiaosun's work; the original theft is quite clear. Zhang Zhouxuan,, in general75items,.
231
其四曰，玄為史官，自奏虧食，前後所上，多與曆違，今算其乖舛有一十三事。 又前與太史令劉暉等校其疏密五十四事，雲五十三條新。 計後為曆應密於舊，見用算推，更疏於本。 今糾發並前，凡四十四條。
Fourth: as historiographer Zhang Zhouxuan himself reported eclipses; his submissions mostly contradicted the calendar—now calculated, his errors number thirteen. He had previously joined Grand Astrologer Liu Hui and others in comparing fifty-four points of accuracy, claiming fifty-three favored the new method. By later calculation the calendar should exceed the old in precision; yet in practical computation it proved coarser. Now refutations including prior entries—44 items in all.
232
其五曰，胄玄於曆，未為精通。 然孝孫初造，皆有意，徵天推步，事必出生，不是空文，徒為臆斷。
Fifth: Zhang Zhouxuan was not thoroughly versed in calendrical science. Xiaosun,,,,,.
233
其六曰，焯以開皇三年，奉敕修造，顧循記注，自許精微，秦漢以來，無所與讓。 尋聖人之跡，悟曩哲之心，測七曜之行，得三光之度，正諸氣朔，成一曆象，會通今古，符允經傳，稽於庶類，信而有徵。 胄玄所違，焯法皆合，胄玄所闕，今則盡有，隱括始終，謂為總備。
Sixth: Liu Chuo, by imperial order in Kaihuang 3, undertook revision, following recorded precedents, claiming precision unmatched since Qin and Han. ,, 7, 3,, 1, conjunction common,,,. Where Zhang Zhouxuan deviated, Liu Chuo's method agrees; what Zhang Zhouxuan lacked, the present system fully provides—embracing beginning and end, claiming completeness.
234
仍上啟曰：「自木鐸寢聲，緒言成燼，群生蕩析，諸夏沸騰，曲技雲浮，疇官雨絕，曆紀廢壞，千百年矣。 焯以庸鄙，謬荷甄擢，專精藝業，耽玩數象，自力群儒之下，冀睹聖人之意。 開皇之初，奉敕修撰，性不諧物，功不克終，猶被胄玄竊為己法，未能盡妙，協時多爽，屍官亂日，實玷皇猷。 請征胄玄答，驗其長短。」
He also memorialized: "Since learning was silenced and texts burned, the people were scattered and the realm in turmoil; minor arts proliferated while calendrical officers vanished—the calendar lay in ruins for a thousand years. Liu Chuo, humble and unworthy, was mistakenly elevated; devoted to calendrical arts, immersed in numerology, striving from below the ranks of scholars to glimpse the sage's intent. At the start of Kaihuang, commissioned to compile—his nature ill-suited to others, his work unfinished—yet Zhang Zhouxuan stole it as his own method; failing to achieve perfection, frequently missing the seasons, holding office in name only while corrupting the calendar—a true stain on imperial policy. Request summoning Zhang Zhouxuan to answer and verify the merits and flaws; . 」.
235
焯又造曆家同異，名曰《稽極》。 大業元年，著作郎王邵、諸葛潁二人，因入侍宴，言劉焯善曆，推步精審，證引陽明。 帝曰：「知之久矣。」 仍下其書與胄玄參校。 胄玄駁難雲：「焯曆有歲率、月率，而立定朔，月有三大、三小。 案歲率、月率者，平朔之章歲、章月也。 以平朔之率而求定朔，值三小者，猶以減三五為十四； 值三大者，增三五為十六也。 校其理實，並非十五之正。 故張衡及何承天創有此意，為難者執數以校其率，率皆自敗，故不克成。 今焯為定朔，則須除其平率，然後為可。」 互相駁難，是非不決，焯又罷歸。
Liu Chuo also compiled calendrical comparisons, titled Investigation of the Pole. Daye 1: Wang Shao and Zhuge Ying, attending a banquet, praised Liu Chuo's calendrical precision with clear evidence. The emperor said: "I have known this for a long time. He then ordered Liu Chuo's book sent to Zhang Zhouxuan for joint verification. Zhang Zhouxuan objected: "Liu Chuo's calendar has year rate and month rate, yet establishes fixed new moons, producing months with three long and three short. And , new moon cycle year and cycle month. , 3, 514. 3, 56. , 15. , as Multiply by , ,. Yielding fixed new moon,,. They traded refutations until no decision was reached, and Liu Chuo again retired.
236
四年，駕幸汾陽宮，太史奏曰：「日食無效。」 帝召焯，欲行其曆。 袁允方幸於帝，左右胄玄，共排焯曆，又會焯死，曆竟不行。 術士鹹稱其妙，故錄其術雲。 甲子元，距大隋仁壽四年甲子積一百萬八千八百四十算。
4year,, Grand Astrologer: 「dayeclipse. The emperor summoned Liu Chuo, wishing to implement his calendar. , Zhang Zhouxuan,,,. Calendrical experts praised its subtlety, and its method is recorded here. Jiazi origin: 1008840.
237
歲率，六百七十六。
Year rate: 676.
238
月率，八千三百六十一。
Month rate: 8361.
239
朔日法，千二百四十二。
New-moon day divisor: 242.
240
朔實，三萬六千六百七十七。
New-moon dividend: 36677.
241
旬周，六十。
Ten-day circuit: 60.
242
朔辰，百三半。
New-moon chronogram: 3 and a half.
243
日干元，五十二。
Day stem origin: 52.
244
日限，十一。
Day limit: 11.
245
盈泛，十六。
Expansion general: 16.
246
虧總，十七。
Obscuration, 17.
247
推經朔術：
Method for computing mean new moon:
248
置入元距所求年，月率乘之，如歲率而一，為積月，不滿為閏衰。 朔實乘積月，滿朔日法得一，為積日，不滿為朔餘。 旬周去積日，不盡為日，即所求年天正經朔日及餘。
,, 1, yielding accumulated,. Accumulated months, day divisor1, yielding accumulated,. Accumulated days,, thenthe year sought.
249
求上下弦、望：加經朔日七、餘四百七十五小，即上弦經日及餘。 又加得望、下弦及後月朔。 就徑求望者，加日十四、餘九百五十半； 下弦加日二十二、餘百八十三大； 後月朔加日二十九，餘六百五十九。 每月加閏衰二十大，即各其月閏衰也。
To find first quarter and full moon: add 7 mean-new-moon days and 475¾ remainder for the first-quarter mean day and remainder. Also add to obtain full moon, last quarter, and the following month's new moon. Full moon , add 14 and remainder 950half. Quarter add 22 and remainder. 29, remainder659. Each month add 20 greater intercalary-decline units to obtain that month's intercalary decline.
250
凡月建子為天正，建醜為地正，建寅為人正。 即以人正為正月，統求所起，本於天正。 若建歲曆從正月始，氣、候、月、星，所值節度，雖有前卻，並亦隨之。 其前地正為十二月，天正為十一月，並諸氣度皆屬往年。 其日之初，亦從星起，晨前多少，俱歸昨日。 若氣在夜半之後，量影以後日為正。 諸因加者，各以其餘減法，殘者為全餘。 若所因之餘滿全餘以上，皆增全一而加之，減其全餘； 即因餘少於全餘者，不增全加，皆得所求。 分度亦爾。 凡日不全為餘，積以成餘者曰秒； 度不全為分，積以成分者曰篾； 其有不成秒曰麽，不成篾曰么。 其分、餘、秒、篾，皆一為小，二為半，三為大，四為全，加滿全者從一。 其三分者，一為少，二為太。 若加者，秒篾成法，從分餘。 分餘滿法從日度一，日度有所滿，則從去之。 而日命以日辰者，滿旬周則亦除； 命有連分、餘、秒、篾者，亦隨全而從去。 其日度雖滿，而分秒不滿者，未可從去，仍依本數。 若減者，秒篾不足，減分餘一，加法而減之； 分餘不足減者，加所從去或前日度乃減之。 即其名有總，而日度全及分餘共者，須相加除，當皆連全及分餘共加除之。 若須相乘，有分餘者，母必通全內子，乘訖報除。 或分餘相並，母不同者，子乘而並之。 母相乘為法，其並，滿法從一為全，此即齊同之也。 既除為分餘而有不成，若例有秒篾，法乘而又法除，得秒篾數。 已為秒篾及正有分餘，而所不成不復須者，須過半從一，無半棄之。 若分餘其母不等，須變相通，以彼所法之母乘此分餘，而此母除之，得彼所須之子。 所有秒篾者，亦法乘，不滿此母，又除而得其數。 麽么亦然。 其所除去而有不盡全，則謂之不盡，亦曰不如。 其不成全，全乃為不滿分、餘、秒、篾，更曰不成。 凡以數相減，而有小及半、太須相加減，同于分餘法者，皆以其母三四除其氣度日法，以半及太、大本率二三乘之，少、小即須因所除之數隨其分餘而加減焉。 秋分後春分前為盈泛，春分後秋分前為虧總，須取其數。 泛總為名，指用其時，春分為主，虧日分後，盈日分前。 凡所不見，皆放於此。
The month establishing zi is the celestial first month; chou the terrestrial first month; yin the human first month. Multiply by as , ,. Carry to , and and and , , ,. 12month, 11month,. , alsocarry to,,. If the seasonal node falls after midnight, measure the shadow using the following day as the reference. In additive operations, subtract each remainder from the divisor; the residue is the full remainder. When full, 1,. Remainder less remainder , add , obtain. Degree fractions follow the same rule. Fractional parts of a day are remainders; accumulated remainders are called seconds. Fractional parts of a degree are fractions; accumulated fractions are called bamboo tallies. Incomplete seconds are called micro-units; incomplete bamboo tallies are called minima. And remainder and second and bamboo tally , 1as , 2as half , 3as , 4as , add when full carry to 1. Its3, 1, 2. ,, carry to. When full divisorcarry to 1, when full, thencarry to remove. , when full. Count and remainder and second and bamboo tally , carry to remove. When full, when full, carry to,. ,, 1,. , carry to. When terms share totals, full day-degrees, and fractions, add or subtract all full values and shared fractions together. For multiplication with fractions, convert denominators to include full values; after multiplying, divide back. To combine fractions with different denominators, cross-multiply numerators and combine. ,, when full divisorcarry to 1,. Divide as remainder , second bamboo tally , multiply divide , obtain second bamboo tally. ,, carry to 1,. When denominators differ, convert to common terms: multiply each fraction by the other's denominator, then divide to obtain the required numerator. ,, notwhen full,. The smallest fractional units follow the same rule. Divide remove , ,. , as when full and remainder and second and bamboo tally ,. Multiply by subtract , half and greater add subtract , remainder , Multiply by 4divide day divisor, Multiply by half greater and 3multiply , less and divide remainder add subtract. Autumn EquinoxafterSpring Equinox, Spring EquinoxafterAutumn Equinox,. ,, Spring Equinox, day fractionafter, day fraction. All unlisted cases follow this pattern.
251
氣日法，四萬六千六百四十四。
Qi day divisor: 46644.
252
歲數，千七百三萬六千四百六十六半。
Year number: 7036466 and a half.
253
度准，三百三十八。
Degree standard: 338.
254
約率，九。
Approximation rate: 9.
255
氣辰，三千八百八十七。
Qi chronogram: 3887.
256
餘通，八百九十七。
Remainder common: 897.
257
秒法，四十八。
Second divisor: 48.
258
麽法，五。
Micro-divisor: 5.
259
推氣術：
Method for computing seasonal nodes:
260
半閏衰乘朔實，又度准乘朔餘，加之，如約率而一，所得滿氣日法為去經朔日，不滿為氣餘。 以去經朔日，即天正月冬至恆日定餘，乃加夜數之半者，減日一，滿者因前，皆為定日。 命日甲子算外，即定冬至日。 其餘如半氣辰千九百四十三半以下者，為氣加子半後也； 過以上，先加此數，乃氣辰而一，命以辰算外，即氣所在辰。 十二辰外，為子初以後餘也。 又十二乘辰餘：
,,, 1, day divisor,. , Winter Solstice,, 1, when full, allyielding fixed day. Beyond tally, Winter Solstice. 943or below,. ,, 1, beyond tally,. Beyond the twelve chronograms lies the remainder after the start of zi. Also multiply the chronogram remainder by 12:
261
四為小太，亦曰少； 五為半步； 六為半；
Four parts is "lesser-greater," also called "lesser." 5 equals half a step; 6 equals half;
262
七為半太； 八為大少，亦曰太； 九為太；
7 equals half plus greater; Eight parts is "greater-lesser," also called "greater." 9 equals greater;
263
十為大太； 十一為窮辰少。
10 equals great plus greater; 11 equals exhausted chronogram less.
264
其又不成法者，半以上為進，以下為退。 退以配前為強，進以配後為弱。 即初不成一而有退者，謂之沾辰； 初成十一而有進者，謂之窮辰。 未旦其名有重者，則於間可以加之，命辰通用其餘，辨日分辰而判諸日。 因別亦皆准此。 因冬至有減日者，還加之。 每加日十五、餘萬一百九十二、秒三十七，即各次氣恆日及餘。 諸月齊其閏衰，如求冬至法，亦即其月中氣恆日去經朔數。 其求後月節氣恆日，如次之求前節者減之。
,, or below. Retrograde Multiply by as strong , Multiply by as weak. When the initial value does not reach one and retreats, this is called a stained chronogram; When eleven is initially formed with a carry, it is called an exhausted chronogram. ,,, day fraction. All separate categories follow this same standard. Winter Solstice subtract , add. Add 15 and remainder 192 and second 37, remainder. , Winter Solsticedivisor,. To find the next month's node constant day, subtract as for the prior node.
265
推每日遲速數術：
Method for computing daily slow-fast numbers:
266
見求所在氣陟降率，並後氣率半之，以日限乘而泛總除，得氣末率。 又日限乘二率相減之殘，泛總除，為總差。 其總差亦日限乘而泛總除，為別差。 率前少者，以總差減末率，為初率，乃別差加之； 前多者，即以總差加末率，皆為氣初日陟降數。 以別差前多者日減，前少者日加初數，得每日數。 所曆推定氣日隨算其數，陟加、降減其遲速，為各遲速數。 其後氣無同率及有數同者，皆因前末，以末數為初率，加總差為末率，及差漸加初率，為每日數，通計其秒，調而禦之。
, half , Multiply by multiply divide , obtain. Multiply subtract , divide , as. Multiply divide , as. Less , Multiply by subtract , as , add. When prior values are greater, add the total difference to the terminal rate to obtain each qi-initial-day ascent-descent value. By separate difference: subtract daily when prior is greater, add to the initial value when prior is less—yielding each day's value. , add and subtract slow , as slow. , , Multiply by as , add as , add , as , second ,.
267
求月朔弦望應平會日所入遲速：各置其經餘為辰，以入氣辰減之，乃日限乘日，日內辰為入限，以乘其氣前多之末率，前少之初率，日限而一，為總率。 其前多者，入限減泛總之殘，乘總差，泛總而一，為入差，並於總差，入限乘，倍日限除，加以總率； 前少者，入限自乘再乘別差，日限自乘，倍而除，亦加總率，皆為總數。 乃以陟加、降減其氣遲速數為定，即速加、遲減其經餘，各其月平會日所入遲速定日及餘。
：remainder is, entering qi,,,,, 1,. ,,, 1,,,,,. When prior is less: within the limit, square and multiply by the separate difference; square the day-limit, double and divide; add the total rate—all yielding total values. Multiply by add and subtract slow as , add and slow subtract remainder , conjunction slow remainder.
268
求每日所入先後：各置其氣躔衰與衰總，皆以餘通乘之，所乃躔衰如陟降率； 衰總如遲速數，亦如求遲速法，即得每所入先後及定數。
To find :,,. Slow motion, slow motion,.
269
求定氣：其每日所入先後數即為氣餘，其所曆日皆以先加之，以後減之，隨算其日，通准其餘，滿一恆氣，即為二至後一氣之數。 以加二氣，如法用別其日而命之。 又算其次，每相加命，各得其定氣日及餘也。 亦以其先後已通者，先減後加其恆氣，即次氣定日及餘。 亦因別其日，命以甲子，各得所求。
To find :,,,,, when full 1, 2 through 1. Multiply by add , count. Also compute the next entry; add and count in succession to obtain each fixed seasonal-node day and remainder. Using prior and later converted values: subtract first, then add to the constant qi to obtain the next fixed seasonal-node day and remainder. , jia-zi,.
270
求土王：距四立各四氣外所入先後加減，滿二十二日、餘八千一百五十四、秒十、麽二。 除所滿日外，即土始王日。
To find the Earth phase: from each four-establishment point, four qi outward, apply prior-later adjustment; when full: 22 days, remainder 8,154, 10 seconds, 2 micro-units. Divide when full , Saturn.
271
求侯日：定氣即初候日也。 三除恆氣，各為平候日。 餘亦以所入先後數為氣餘，所曆之日皆以先加、後減，隨計其日，通准其餘，每滿其平，以加氣日而命之，即得次候日。 亦算其次，每相加命，又得末候及次氣日。
To find :then. Divide the constant qi by 3 to obtain each mean pentad day. Remainder Multiply by as remainder , Multiply by add and subtract , , remainder , when full , Multiply by add count , obtain. Also compute the next; add and count in succession to obtain the terminal pentad and next seasonal-node day.
272
倍夜半之漏，得夜刻也。 以減百刻，不盡為晝刻。 每減晝刻五，以加夜刻，即其晝為日見、夜為不見刻數。 刻分以百為母。
Double the midnight clepsydra graduations to obtain night graduations. Subtract from 100 graduations; the remainder is day graduations. Subtract 5, Multiply by add , as and as. Clepsydra fractions use 100 as the denominator.
273
求日出入辰刻：十二除百刻，得辰刻數，為法。 半不見刻以半辰加之，為日出實，又加日出見刻，為日入實。 如法而一，命子算外，即所在辰，不滿法，為刻及分。
To find :210,,. Half Multiply by half add , as , add , as. 1, beyond tally,,,.
274
求辰前餘數：氣、朔日法乘夜半刻，百而一，即其餘也。
To find pre-chronogram remainder: multiply midnight graduations by the qi/new-moon day divisor and divide by 100.
275
求每日刻差：每氣准為十五日，全刻二百二十五為法。 其二至各前後於二分，而數因相加減，間皆六氣； 各盡於四立，為三氣。 至與前日為一，乃每日增太； 又各二氣，每日增少； 其末之氣，每日增少之小，而末六日，不加而裁焉。 二望至前後一氣之末日，終於十少； 二氣初日，稍增為十二半，終於二十太，三氣初日，二十一，終於三十少； 四立初日，三十一，終於三十五太； 五氣亦少增，初日三十六太，終四十一少； 末氣初日，四十一少，終於四十二。 每氣前後累算其數，又百八十乘為實，各泛總乘法而除，得其刻差。 隨而加減夜刻而半之，各得入氣夜定刻。 其分後十五日外，累算盡日，乃副置之，百八十乘，虧總除，為其所因數。 以減上位，不盡為所加也。 不全日者，隨辰率之。
To find :15day, 225. Each solstice precedes and follows its adjacent equinox, with numbers added and subtracted accordingly, six seasonal nodes between each interval; 4, 3. Through 1,. For each pair of seasonal nodes, the daily increment decreases. ,, 6day,. The two full moons to the last day of the preceding and following qi; ending at ten plus a fraction; 2, 12, 20, 3, 21, 30. 4, 31, 35. 5, 36, 41. , 41, 42. , 80,,. Adding and subtracting night graduations and halving yields the fixed night graduations upon entering each seasonal node. 15,, set aside, 80, obscuration,. Subtract from the upper position; the remainder is the amount added. When less than a full day, apply the chronogram rate.
276
求晨去中星：加周度一，各昏去中星減之，不盡為晨去度。
To find :1,,.
277
求每日度差：准日因增加裁，累算所得，百四十三之，四百而一，亦百八十乘，泛總除，為度差數。 滿轉法為度，隨日加減，各得所求。 分後氣間，亦求准外與前求刻，至前加減，皆因日數逆算求之。 亦可因至向背其刻，冬減夏加，而度冬加夏減。 若至前，以入氣減氣間，不盡者，因後氣而反之，以不盡日累算乘除所定，從後氣而逆以加減，皆得其數。 此但略校其總，若精存於《稽極》雲。
To find :,, 43, 4001, 80,,. When full,,. ,, through, retrograde. Through,,. If through prior, byentering qi,,,, carry to retrograde,. , 《Investigation of the Pole》.
278
轉終日，二十七； 餘，千二百五十五。
, 27. Remainder: 255.
279
終法，二千二百六十三。
Terminal divisor: 2263.
280
終實，六萬二千三百五十六。
Terminal dividend: 62356.
281
終全餘，千八。
Terminal full remainder: 8.
282
轉法，五十二。
Rotation divisor: 52.
283
篾法，八百九十七。
Bamboo-tally divisor: 897.
284
閏限，六百七十六。
Intercalation limit: 676.
285
推入轉術：終實去積日，不盡，以終法乘而又去，不如終實者，滿終法得一日，不滿為餘，即其年天正經朔夜半入轉日及餘。
Method for computing ：accumulated days,,,,,,.
286
求次日：加一日，每日滿轉終則去之，其二十八日者加全餘為夜半入初日餘。
：,, remainder is.
287
求弦望：皆因朔加其經日，各得夜半所入日餘。
To find :,.
288
求次月：加大月二日，小月一日，皆及全餘，亦其夜半所入。
To find :addlong month2day, short month1day,,.
289
求經辰所入朔弦望：經餘變從轉，不成為秒，加其夜半所入，皆其辰入日及餘。 因朔辰所入，每加日七、餘八百六十五、秒千一百六十大，秒滿日法成餘，亦得上弦。 望、下弦、次朔經辰所入徑求者，加望日十四、餘千七百三十一、秒千七十九半，下弦日二十二、餘三百三十四、秒九百九十八小，次朔日一、餘二千二百八、秒九百一十七。 亦朔望各增日一，減其全餘，望五百三十一、秒百六十二半，朔五十四、秒三百二十五。
To find :carry to rotation,,,. New moon , add 7 and remainder 865 and second , second when full day divisorremainder , obtain quarter. Full moon and quarter and new moon , add full moon 14 and remainder 731 and second 79half , quarter 22 and remainder 334 and second , new moon 1 and remainder 2,208 and second 917. New moon full moon 1, subtract remainder , full moon 531 and second 62half , new moon 54 and second 325.
290
求月平應會日所入：以月朔弦望會日所入遲速定數，亦變從轉餘，乃速加、遲減其經辰所入餘，即各平會所入日餘。
Conjunction ：Multiply by new moon quarter full moon conjunction slow , carry to remainder , add and slow subtract remainder , conjunction remainder.
291
推朔弦望定日術：
Method for computing fixed new-moon, quarter, and full-moon days:
292
各以月平會所入之日加減限，限並後限而半之，為通率； 又二限相減，為限衰。 前多者，以入餘減終法，殘乘限衰，終法而一，並於限衰而半之； 前少者，半入餘乘限衰，亦終法而一，減限衰。 皆加通率，入餘乘之，日法而一，所得為平會加減限數。 其限數又別從轉餘為變餘，朓減、朒加本入餘。 限前多者，朓以減與未減，朒以加與未加，皆減終法，並而半之，以乘限衰； 前少者，亦朓朒各並二入餘，半之，以乘限衰； 皆終法而一，加於通率，變餘乘之，日法而一。 所得以朓減、朒加限數，加減朓朒積而定朓朒。 乃朓減、朒加其平會日所入餘，滿若不足進退之，即朔弦望定日及餘。 不滿晨前數者，借減日算，命甲子算外，各其日也。 不減與減，朔日立算與後月同。 若俱無立算者，月大，其定朔算後加所借減算。 閏衰限滿閏限，定朔無中氣者為閏，滿之前後，在分前若近春分後、秋分前，而或月有二中者，皆量置其朔，不必依定。 其後無同限者，亦因前多以通率數為半衰而減之，二前少，即為通率。 其加減變餘進退日者，分為一日，隨餘初末如法求之，所得並以加減限數。 凡分餘秒篾，事非因舊，文不著母者，皆十為法。 若法當求數，用相加減，而更不過通遠，率少數微者，則不須算。 其入七日餘二千一十一，十四日餘千七百五十九，二十一日餘千五百七，二十八日始終餘以下為初數，各減終法以上為末數。 其初末數皆加減相返，其要各為九分，初則七日八分，十四日七分，二十一日六分，二十八日五分； 末則七日一分，十四日二分，二十一日三分，二十八日四分。 雖初稍弱而末微強，餘差止一，理勢兼舉，皆今有轉差，各隨其數。 若恆算所求，七日與二十一日得初衰數，而末初加隱而不顯，且數與平行正等。 亦初末有數而恆算所無，其十四日、二十八日既初末數存，而虛衰亦顯，其數當去，恆法不見。
Multiply by conjunction add subtract , half , as. Subtract the two limits to obtain the limit-decline value. ,,, 1,. ,, 1,. ,, day divisorand1,. Carry to remainder as remainder , subtract and add remainder. , Multiply by subtract subtract , Multiply by add add , subtract , half , Multiply by multiply. , 2,,. 1,,, day divisorand1. Apply the result to reduce gibbous and add crescent to the limit number; adding and subtracting the gibbous-crescent accumulation fixes the phase. Subtract and add conjunction remainder , when full retrograde , new moon quarter full moon remainder. ,, beyond tally,. Whether subtracting or not, the new-moon day tally matches the following month. When neither has a standing tally, for a long month add the borrowed reduction tally after the fixed new-moon tally. When full , new moon as , when full , and , , new moon ,. ,, 2,. , 1day,,. ,,, 10. , add subtract , , less ,. 72011, 14759, 21507, 28or below,. , 9fraction, 7day8fraction, 14day7fraction, 21day6fraction, 28day5fraction. 7day1fraction, 14day2fraction, 21day3fraction, 28day4fraction. , 1,,,. ,,, uniform motion. Initial and final values exist that constant computation omits; on the 14th and 28th days initial and final numbers remain, yet empty decline also appears—numbers that should be removed but the standard method does not account for.
293
求朔弦望之辰所加：
To find the chronogram added at new moon, first quarter, and full moon:
294
定餘半朔辰五十一大以下，為加子過； 以上，加此數，乃朔辰而一，亦命以子，十二算外，又加子初。 以後其求入辰強弱，如氣。
51or below,. ,, 1,, 12beyond tally,. For later chronogram strength-weakness, follow the qi method.
295
求入辰法度：
Method for finding entry into chronogram degree:
296
度法，四萬六千六百四十四。
Degree divisor: 46,644.
297
周數，千七百三萬七千七十六。
Circuit number: 7037076.
298
周分，萬二千一十六。
Circuit fraction: 2016.
299
轉，十三。
Rotation: 13.
300
篾，三百五十五。
Bamboo tally: 355.
301
周差，六百九半。
Circuit difference: 609 and a half.
302
在日謂之餘通，在度謂之篾法，亦氣為日法、為度法，隨事名異，其數本同。 女末接虛，謂之周分。 變周從轉，謂之轉。 晨昏所距日在黃道中，准度赤道計之。
Remainder , bamboo tally , as day divisor and as degree divisor, ,. Where the Woman lodge ends connecting to Empty is called the circuit fraction. The variation circuit follows rotation—called rotation. Morning and evening solar distance is measured on the ecliptic against the equator.
303
斗二十六牛八女十二虛十危十七室十六壁九
Dipper 26°, Ox 8°, Girl 12°, Emptiness 10°, Rooftop 17°, Encampment 16°, Wall 9°.
304
北方玄武七宿，九十八度。
7, 98degree.
305
奎十六婁十二胃十四昴十一畢十六觜二參九
Legs 16°, Bond 12°, Stomach 14°, Hairy Head 11°, Net 16°, Turtle Beak 2°, Three Stars 9°.
306
西方白虎七宿，八十度。
7, 80degree.
307
井三十三鬼四柳十五星七張十八翼十八軫十七
Well 33°, Ghost 4°, Willow 15°, Star 7°, Extended Net 18°, Wings 18°, Axletree 17°.
308
南方硃雀七宿，百一十二度。
7, 12degree.
309
角十二亢九氐十五房五心五尾十八箕十一
Horn 12°, Neck 9°, Root 15°, Room 5°, Heart 5°, Tail 18°, Winnowing Basket 11°.
310
東方蒼龍七宿，七十五度。
7, 75degree.
311
前皆赤道度，其數常定，紘帶天中，儀極攸准。
Prior entries are equatorial degrees with constant values, girdling the celestial pole as the armillary standard.
312
推黃道術：
Method for computing the ecliptic:
313
准冬至所在為赤道度，後於赤道四度為限。 初數九十七，每限增一，以終百七。 其三度少弱，平。 乃初限百九，亦每限增一，終百一十九，春分所在。 因百一十九每限損一，又終百九。 亦三度少弱，平。 乃初限百七，每限損一，終九十七，夏至所在。 又加冬至後法，得秋分、冬至所在數。 各以數乘其限度，百八而一，累而總之，即皆黃道度也。 度有分者，前後輩之，宿有前卻，度亦依體，數逐差遷，道不常定，准令為度，見步天行，歲久差多，隨術而變。
Winter Solstice, 4. 97, 1, 7. Its3,. 9, 1,, Spring Equinox. Thus for every 119, each limit reduces by 1; it also terminates at 109. Also3,. 7, 1,, Summer Solstice. Add Winter Solstice , obtain and Winter Solstice. , 81,,. , , , , , , as , , ,.
314
斗二十四牛七女十一半虛十危十七室十七壁十
Dipper 24°, Ox 7°, Girl 11.5°, Emptiness 10°, Rooftop 17°, Encampment 17°, Wall 10°.
315
北方九十六度半。
Northern lodges: 96.5°.
316
奎十七婁十三胃十五昴十一畢十五半觜二參九
Legs 17°, Bond 13°, Stomach 15°, Hairy Head 11°, Net 15.5°, Turtle Beak 2°, Three Stars 9°.
317
西方八十二度半。
Western lodges: 82.5°.
318
井三十鬼四柳十四半星七張十七翼十九軫十八
Well 30°, Ghost 4°, Willow 14.5°, Star 7°, Extended Net 17°, Wings 19°, Axletree 18°.
319
南方一百九度半。
Southern lodges: 109.5°.
320
角十三亢十氐十六房五心五尾十七箕十半
Horn 13°, Neck 10°, Root 16°, Room 5°, Heart 5°, Tail 17°, Winnowing Basket 10.5°.
321
東方七十六度半。
Eastern lodges: 76.5°.
322
前皆黃道度，步日所行。 月與五星出入，循此。
Prior entries are ecliptic degrees for stepping the sun's daily motion. The moon and five planets enter and exit by this method.
323
推月道所行度術：
Method for computing lunar path degrees:
324
准交定前後所在度半之，亦於赤道四度為限，初十一，每限損一，以終於一。 其三度強，平。 乃初限數一，每限增一，亦終十一，為交所在。 即因十一，每限損一，以終於一。 亦三度強，平。 又初限數一，每限增一，終於十一，複至交半，返前表裡。 仍因十一增損，如道得後交及交半數。 各積其數，百八十而一，即道所行每與黃道差數。 其月在表，半後交前，損減增加； 交後半前，損加增減于黃道。 其月在裡，各返之，即得月道所行度。 其限未盡四度，以所直行數乖入度，四而一。 若月在黃道度，增損于黃道之表裡，不正當於其極，可每日准去黃道度，增損于黃道，而計去赤道之遠近，准上黃道之率以求之，遁伏相消，朓朒互補，則可知也。 積交差多，隨交為正。 其五星先候，在月表裡出入之漸，又格以黃儀，准求其限。 若不可推明者，依黃道命度。
,, 11, 1, 1. Its3,. 1, 1, 11,. 11, 1, 1. Also3,. 1, 1, 11,,. Apply the eleven-fold increase-decrease to obtain later crossing and half-crossing values. , 801,. When the moon is outside, half after crossing before: apply decrease-subtract and increase-add. Half after crossing before: apply decrease-add and increase-subtract on the ecliptic. When the moon is inside, reverse each value to obtain the moon-path travel degree. 4degree,, 41. , outside and inside,,,,,, disappearance,,. When accumulated crossing difference is large, use the crossing as the standard. 5, outside and inside,,. When computation is unclear, assign degrees by the ecliptic.
325
推日度術：
Method for computing solar degree:
326
置入元距所求年歲數乘之，為積實，周數去之，不盡者，滿度法得積度，不滿為分。 以冬至餘減分； 命積度以黃道起於虛一宿次除之，不滿宿算外，即所求年天正冬至夜半日所在度及分。
Set the year sought, yielding accumulated, remove,, when full degree divisor, notwhen full. Winter Solstice. 1lodge sequence, notwhen full beyond the tally, thenthe year soughtWinter Solstice.
327
求年天正定朔度：
To find the fixed new-moon degree for the year:
328
以定朔日至冬至每日所入先後餘為分，日為度，加分以減冬至度，即天正定朔夜半日在所度分。 亦去朔日乘衰總已通者，以至前定氣除之，又如上求差加以並去朔日乃減度，亦即天正定朔日所在度。 皆日為度，餘為分。 其所入先後及衰總用增損者，皆分前增、分後損其平日之度。
Winter Solsticeremainder is,, Winter Solstice,. , by through,,. , the remainder is fraction. For advance-retreat totals used in increase-decrease, increase the mean daily degrees before the fraction and decrease after.
329
求次日：
To find the next day:
330
每日所入先後分增損度，以加定朔度，得夜半。
Daily advance-retreat fractional increase-decrease, added to the fixed new-moon degree, gives midnight position.
331
求弦望：
To find first quarter and full moon:
332
去定朔每日所入分，累而增損去定朔日，乃加定朔度，亦得其夜半。
Remove daily entry fractions from the fixed new moon, accumulate increase-decrease, subtract fixed new-moon days, then add the fixed new-moon degree to obtain midnight position.
333
求次月：
To find the next month:
334
曆算大月三十日，小月二十九日，每日所入先後分增損其月，以加前朔度，即各夜半所在至虛去周分。
Long month30day, short month29day,,, through.
335
求朔弦望辰所加：
To find the chronogram added at new moon, first quarter, and full moon:
336
各以度准乘定餘，約率而一，為平分。 又定餘乘其日所入先後分，日法而一，乃增損其平分，以加其夜半，即各辰所加。 其分皆篾法約之，為轉分，不成為篾。 凡朔辰所加者，皆為合朔日月同度。
, 1,. , day divisorand1,,,. Reduce by,,. , conjunction new moon.
337
推月而與日同度術：
Method for computing when the moon shares the sun's degree:
338
各以朔平會加減限數加減朓朒，為平會朓朒。 以加減定朔，度准乘，約率除，以加減定朔辰所加日度，即平會辰日所在。 又平會餘乘度准，約率除，減其辰所在，為平會夜半日所在。 乃以四百六十四半乘平會餘，亦以周差乘，朔實除，從之，以減夜半日所在，即月平會夜半所在。 三十七半乘平會餘，增其所減，以加減半，得月平會辰平行度。 五百二乘朓棵，亦以周差乘，朔實除而從之，朓減、朒加其平行，即月定朔辰所在度，而與日同。 若即以平會朓朒所得分加減平會辰所在，亦得同度。
Multiply by new moon conjunction add subtract add subtract , as conjunction. Multiply by add subtract new moon , multiply , divide , Multiply by add subtract new moon add , conjunction. Multiply mean-conjunction remainder by the degree standard, divide by the approximation rate, subtract the chronogram position—to obtain mean-conjunction midnight solar position. 464,,, carry to,,. ,,, uniform motion. 502multiply , Multiply by multiply , new moon divide carry to , subtract and add , new moon ,. Multiply by conjunction obtain add subtract conjunction , obtain.
339
求月弦望定辰度：
To find the fixed chronogram degree at first quarter and full moon:
340
各置其弦望辰所加日度及分，加上弦度九十一，轉分十六，篾三百一十三； 望度百八十二，轉分三十二，篾六百二十六； 下弦度二百七十三，轉分四十九，篾四十二，皆至虛，去轉周求之。
Set the degree and fraction where first-quarter and full-moon chronograms fall; for first quarter add 91°, rotating fraction 16, bamboo tallies 313; 82, 32, 626. 273, 49, 42, all through,.
341
定朔夜半入轉：
Fixed new-moon midnight entry into rotation:
342
經朔夜半所入准於定朔日有增損者，亦以一日加減之，否者因經朔為定。
,, yielding fixed.
343
其因定求朔次日、弦望、次月夜半者，如於經月法為之。
New moon and quarter full moon and half , month divisoras.
344
推月轉日定分術：
Method for computing fixed daily lunar rotation fraction:
345
以夜半入轉餘乘逡差，終法而一，為見差。 以息加、消減其日逡分，為月每日所行逡定分。
, 1,. Multiply by add and subtract , as.
346
求次日：
To find the next day:
347
各以逡定分加轉分，滿轉法從度，皆其夜半。 因日轉若各加定日，皆得朔、弦望夜半月所在定度。 其就辰加以求夜半，各以半逡差減逡分，消者，定餘乘差，終法除，並差而半之； 息者，半定餘以乘差，終法而一。 皆加所減，乃以定餘乘之，日法而一，各減辰所加度，亦得其夜半度。 因夜半亦如此求逡分，以加之，亦得辰所加度。 諸轉可初以逡分及差為篾，而求其次，皆訖，乃除為轉分。 因經朔夜半求定辰度者，以定辰去經朔夜半減，而求其增損數，乃以數求逡定分，加減其夜半，亦各定辰度。
, when full carry to degree,. Following daily rotation, add fixed days to obtain new-moon, quarter, and full-moon midnight lunar fixed degrees. Add Multiply by half , Multiply by half subtract , , remainder multiply , divide , half. ,, 1. ,, day divisorand1,,. From midnight, likewise compute the retreat fraction and add—this also gives the chronogram degree. Multiply by as bamboo tally , , , divide as rotation fraction. To find the fixed chronogram degree from mean new-moon midnight, subtract and compute the increase-decrease number, then find the fixed retreat fraction and apply it to midnight—yielding the fixed chronogram degree.
348
求月晨昏度：
To find morning and evening lunar degrees:
349
如前氣與所求每日夜漏之半，以逡定分乘之，百而一，為晨分； 減逡定分，為昏分。 除為轉度，望前以昏，後以晨，加夜半定度，得所在。
,, 1,. Subtract the circuit fixed fraction to obtain the dusk fraction. Divide for rotation degree; before full moon use dusk, after use dawn; add to midnight fixed degree to obtain position.
350
求晨昏中星：
To find morning and evening culminating stars:
351
各以度數加夜半定度，即中星度。 其朔、弦、望，以百刻乘定餘，滿日法得一刻，即各定辰近入刻數。 皆減其夜半漏，不盡為晨，初刻不滿者屬昨日。
Add the degree count to the midnight fixed degree to obtain the culminating star degree. New moon and quarter and full moon , Multiply by multiply remainder , when full day divisorobtain ,. ,, when full.
352
複月，五千四百五十八。
Return month: 5458.
353
交月，二千七百二十九。
Crossing month: 2729.
354
交率，四百六十五。
Crossing rate: 465.
355
交數，五千九百二十三。
Crossing number: 5923.
356
交法，七百三十五萬六千三百六十六。
Crossing divisor: 7,356,366.
357
會法，五十七萬七千五百三十。
Conjunction divisor: 577530.
358
交複日，二十七。 餘，二百六十三。 秒，三千四百三十五。
Crossing return days: 27. Remainder: 263. Second (time unit): 3435.
359
交日，十三。 餘，七百五十二。 秒，四千六百七十九。
Crossing days: 13. Remainder: 752. Second (time unit): 4679.
360
交限，日，十二。 餘，五百五十五。 秒，四百七十三半。
,, 12. Remainder: 555. Second (time unit): 473 and a half.
361
望差，日，一。 餘，百九十七。 秒，四千二百五半。
,, 1. Remainder: 97. Second (time unit): 4205 and a half.
362
朔差，日，二。 餘，三百九十五。 秒，二千四百八十八。
,, 2. Remainder: 395. Second (time unit): 2488.
363
會限，百五十八。 餘，六百七十六。 秒，五十半。
Conjunction limit: 58. Remainder: 676. Second (time unit): 50 and a half.
364
會日，百七十三。 餘，三百八十四。 秒，二百八十三。
Conjunction days: 73. Remainder: 384. Second (time unit): 283.
365
推月行入交表裡術：
Method for computing lunar entry into crossing outside/inside:
366
置入元積月，複月去之，不盡。 交率乘而複去，不如複月者，滿交月去之，為在裡數； 不滿為在表數，即所求年天正經入交表裡數。
Set accumulated months, remove,. ,, when full remove,. Notwhen full, thenthe year soughtoutside and insidenumber.
367
求次月：
To find the next month:
368
以交率加之，滿交月去之，前表者在裡，前裡者在表。
, when full remove,,.
369
推月入交日術：
Method for computing days of lunar entry into crossing:
370
以朔實乘表裡數，為交實； 滿交法為日，不滿者交數而一，為餘，不成為秒，命日算外，即其經朔月平入交日餘。
Multiply by new moon multiply , as crossing. Crossing divisor, 1,,, beyond tally,.
371
求望：以望差加之，滿交日去之，則月在表裡與朔同； 不滿者與朔返。 其月食者，先交與當月朔，後交與月朔表裡同。
To find full moon:byfull-moon difference, when full remove, outside and inside. Remainders that do not fill carry back to the new moon. Eclipse,, outside and inside.
372
求次月：朔差加月朔所入，滿交日去之，表裡與前月返； 不滿者，與前月同。
To find :new-moon difference, when full remove, outside and inside. Remainders that do not fill match the prior month.
373
求經朔望入交常日：
To find mean new-moon and full-moon entry into crossing constant days:
374
以月入氣朔望平會日遲速定數，速加、遲減其平入交日餘，為經交常日及餘。
Multiply by entering qinew moon full moon conjunction slow , add and slow subtract crossing remainder , as crossing remainder.
375
求定朔望入交定日：
To find fixed new-moon and full-moon entry into crossing fixed days:
376
以交率乘定朓朒，交數而一，所得以朓減、朒加常日餘，即定朔望所入定日及餘。 其去交如望差以下、交限以上者月食，月在裡者日食。
Multiply by crossing multiply , crossing 1, obtain Multiply by subtract and add remainder , new moon full moon remainder. When distance from the node is below full-moon difference but above the crossing limit, a lunar eclipse occurs; when the moon is inside the node, a solar eclipse.
377
推日入會日術：
Method for computing days of solar entry into conjunction:
378
會法除交實為日，不滿者，如交率為餘，不成為秒，命日算外，即經朔日入平會日及餘。
,,,, beyond tally,.
379
求望：加望日及餘，次月加經朔，其表裡皆准入交。
To find full moon:,, itsoutside and inside.
380
求入會常日：以交數乘月入氣朔望所平會日遲速定數，交率而一，以速加、遲減其入平會日餘，即所入常日餘。 亦以定朓朒，而朓減、朒加其常日餘，即日定朔望所入會日及餘。 皆滿會日去之，其朔望去會，如望以下、會限以上者，亦月食； 月日道表在日道裡則日食。
Conjunction ：Multiply by crossing multiply entering qinew moon full moon conjunction slow , crossing 1, Multiply by add and slow subtract conjunction remainder , remainder. Multiply by , subtract and add remainder , new moon full moon conjunction remainder. When full conjunction remove , new moon full moon remove conjunction , full moon Multiply by and conjunction Multiply by , eclipse. When the moon's path is outside and the sun's path inside the node, a solar eclipse occurs.
381
求月定朔望入交定日夜半：
To find fixed new-moon and full-moon entry into crossing fixed day at midnight:
382
交率乘定餘，交數而一，以減定朔望所入定日餘，即其夜半所定入。
, 1,,.
383
求次日：
To find the next day:
384
以每日遲速數，分前增、分後損定朔所入定日餘，以加其日，各得所入定日及餘。
Multiply by slow , and new moon remainder , Multiply by add , obtain remainder.
385
求次月：
To find the next month:
386
加定朔，大月二日，小月一日，皆餘九百七十八，秒二千四百八十八。 各以一月遲速數，分前增、分後損其所加，為定。 其入七日，餘九百九十七，秒二千三百三十九半以下者，進； 其入此以上，盡全餘二百四十四，秒三千五百八十三半者，退。 其入十四日，如交餘及秒以下者，退； 其入此以上，盡全餘四百八十九，秒千二百四十四者，進而複也。 其要為五分，初則七日四分，十四日三分； 末則七日後一分，十四日後二分，雖初強末弱，衰率有檢。
, long month2day, short month1day, 978, 2488. Multiply by slow , and add , as. 7day, remainder997, 2339or below,. , 244, 3583,. 14day, or below,. , 489, 244,. 5fraction, 7day4fraction, 14day3fraction. 71fraction, 142fraction,,.
387
求月入交去日道：皆同其數，以交餘為秒積，以後衰並去交衰，半之，為通數。 進則秒積減衰法，以乘衰，交法除，而並衰以半之； 退者，半秒積以乘衰，交法而一； 皆加通數，秒積乘，交法除，所得以進退衰積，十而一為度，不滿者求其強弱，則月去日道數。 月朔望入交，如限以上，減交日，殘為去後交數； 如望差以下即為去先交數。 有全日同為餘，各朔辰而一，得去交辰。 其月在日道裡，日應食而有不食者； 月日道表在日不應食而亦有食者。
：, remainder is,,,. ,, crossing divisor,. ,, crossing divisorand1. ,, crossing divisor,,,,. At new or full moon entry into crossing: if at the limit or above, subtract crossing days; the remainder is distance from the later crossing. If below full-moon difference, this gives the distance from the prior crossing. , 1, obtainnode distancechronogram. , eclipseeclipse. When the moon's path is outside the node, the sun should not be eclipsed—yet eclipses sometimes occur.
388
推應食不食術：
Method for computing expected eclipses that do not occur:
389
朔先後在夏至十日內，去交十二辰少； 二十日內，十二辰半； 一月內，十二辰大； 閏四月、六月，十三辰以上，加南方三辰。 若朔在夏至二十日內，去交十三辰，以加辰申半以南四辰； 閏四月、六月，亦加四辰； 穀雨後、處暑前，加三辰； 清明後、白露前，加巳半以西、未半以東二辰； 春分後秋分前，加午一辰。 皆去交十三辰半以上者，並或不食。
New moon within ten days of summer solstice, node distance less than 12 chronograms; 20, 12. 1, 12. And , Multiply by , add. Summer Solstice20, node distance13chronogram, 4chronogram. In intercalary fourth and sixth months, also add 4 chronograms. And End of Heat , add. And , add half Multiply by and half Multiply by. Spring EquinoxlaterAutumn Equinoxprior, 1chronogram. Node distances of 13.5 chronograms or more may also fail to produce an eclipse.
390
推不應食而食術：
Method for computing unexpected eclipses:
391
朔在夏至前後一月內，去交二辰； 四十六日內，一辰半，以加二辰； 又一月內，亦一辰半，加三辰及加四辰，與四十六日內加三辰； 穀雨後、處暑前，加巳少後、未太前； 清明後、白露前，加二辰； 春分後、秋分前，加一辰。 皆去交半辰以下者，並得食。
New moon within one month of summer solstice, node distance 2 chronograms; 46, 1, 2chronogram. Also1, also1, add34chronogram, 463chronogram. And End of Heat , add less and greater. After Pure Brightness and before White Dew, add 2 chronograms. After spring equinox and before autumn equinox, add 1 chronogram. Allnode distanceor below, eclipse.
392
推月食多少術：
Method for computing lunar eclipse magnitude:
393
望在分後，以去夏至氣數三之； 其分前，又以去分氣數倍而加分後者； 皆又以十加去交辰倍而並之，減其去交餘，為不食定餘。 乃以減望差，殘者九十六而一，不滿者求其強弱，亦如氣辰法，以十五為限，命之，即各月食多少。
, Summer Solstice3. Before its fraction, multiply the post-fraction qi count and add to the value after the fraction. 10node distance, node distanceremainder, eclipse. Full-moon difference, 961, notwhen full, chronogram divisor, 15,, eclipse.
394
推日食多少術：
Method for computing solar eclipse magnitude:
395
月在內者，朔在夏至前後二氣，加南二辰，增去交餘一辰太； 加三辰，增一辰少，加四辰，增太。 三氣內，加二辰，增一辰； 加三辰，增太； 加四辰，增少。 四氣內，加二辰，增太； 加三辰及五氣內，加二辰，增少。 自外所加辰，立夏後、立秋前，依本其氣內加四辰，五氣內加三辰，六氣內加二辰。 六氣內加二辰者，亦依平。 自外所加之北諸辰，各依其去立夏、立秋、清明、白露數，隨其依平辰，辰北每辰以其數三分減去交餘； 雨水後、霜降前，又半其去分日數，以加二分去二立之日，乃減去交餘； 其在冬至前後，更以去霜降、雨水日數三除之，以加霜降雨水當氣所得之數； 而減去交餘，皆為定不食餘。 以減望差，乃如月食法。 月在外者，其去交辰數，若日氣所系之限，止一而無等次者，加所去辰一，即為食數。 若限有等次，加別系同者，隨所去交辰數而返其衰，以少為多，以多為少，亦加其一，以為食數。 皆以十五為限，乃以命之，即各日之所食多少。
, Summer Solstice2, 2chronogram, node distanceremainder1. Add3chronogram, 1, add4chronogram,. 3, add2chronogram, 1chronogram. Add3chronogram,. Add4chronogram,. 4, add2chronogram,. Add35, add2chronogram,. Add , Start of Summer and , add , add , add. Within six seasonal nodes, adding two chronograms also follows the mean value. Add , remove Start of Summer and and and , , Multiply by 3subtract remove crossing remainder. Mercury and Frost Descent , half remove , Multiply by add 2remove , subtract remove crossing remainder. Winter Solstice , Multiply by remove Frost Descent and Mercury 3divide , Multiply by add Frost Descent Mercury obtain. Node distanceremainder, allyielding fixed noteclipseremainder. Full-moon difference, eclipsedivisor. , itsnode distance,, 1, 1, eclipsenumber. ,, node distance,,, 1, eclipsenumber. 15,, eclipse.
396
凡日食，月行黃道，體所映蔽，大較正交如累璧，漸減則有差，在內食分多，在外無損。 雖外全而月下，內損而更高，交淺則閑遙； 交深則相搏而不淹。 因遙而蔽多，所觀之地又偏，所食之時亦別。 月居外道，此不見虧，月外之人反以為食。 交分正等，同在南方，冬損則多，夏虧乃少。 假均冬夏，早晚又殊。 處南辰體則高，居東西傍而下視有邪正。 理不可一，由准率若實而違。 古史所詳，事有紛互，今故推其梗概，求者知其指歸。 苟地非于陽城，皆隨所而漸異。 然月食以月行虛道，暗氣所沖，日有暗氣，天有虛道，正黃道常與日對，如鏡居下，魄耀見陰，名曰暗虛，奄月則食，故稱「當月月食，當星星亡。」 雖夜半之辰，子午相對，正隔於地，虛道即虧。 既月兆日光，當午更耀，時亦隔地，無廢稟明。 諒以天光神妙，應感玄通，正當夜半，何害虧稟。 月由虛道，表裡俱食。 日之與月，體同勢等，校其食分，月盡為多，容或形差，微增虧數，疏而不漏，綱要克舉。
Eclipse,,,,, eclipse,. Though outside appears full while the moon is below, inside loses height; shallow crossings appear distant. When the crossing is deep, the bodies meet but do not fully cover each other. ,, that whicheclipse. Outer path, obscuration, eclipse. ,,, obscuration. Even assuming uniform winter and summer values, morning and evening observations differ. In southern chronograms the body appears higher; oblique viewing from east and west introduces skew. The logic cannot be unified—standard rates, though seemingly correct, still contradict observation. Ancient records conflict; here we summarize the outline for reference. Away from Yangcheng, all values gradually differ by location. Eclipse,,,,,,,, eclipse, 「eclipse,. 」,,, obscuration. Since the moon reflects sunlight, it shines brighter at noon; time also spans the earth without abolishing its received light. ,,, obscuration. , outside and insideeclipse. ,, eclipsefraction,,, obscurationnumber,,.
397
推日食所在辰術：
Method for computing the chronogram of solar eclipse:
398
置定餘，倍日限，克減之，月在裡，三乘朔辰為法，除之，所得以艮巽坤乾為次。 命艮算外，不滿法者半法減之，無可減者為前，所減之殘為後，前則因餘，後者減法，各為其率。 乃以十加去交辰，三除之，以乘率，十四而一，為差。 其朔所在氣二分前後一氣內，即為定差。 近冬至，以去寒露、驚蟄，近夏至，以去清明、白露氣數，倍而三除去交辰，增之。 近冬至，艮巽以加，坤乾以減； 近夏至，艮巽以減，坤乾以加其差為定差。 乃艮以坤加，巽以乾減定餘。 月在外，直三除去交辰，以乘率，十四而一，亦為定差。 艮坤以減，巽乾以加定餘，皆為食餘。 如氣求入辰法，即日食所在辰及大小。 其求辰刻，以辰克乘辰餘，朔辰而一，得刻及分。 若食近朝夕者，以朔所入氣日之出入刻，校食所在，知食見否之少多所在辰，為正見。
,,,, 3,,. Beyond the tally, notwhen full,,,,,. 10node distancechronogram, 3,, 141,. 21, thenyielding fixed difference. Winter Solstice , Multiply by remove and , Summer Solstice , Multiply by remove and , 3divide remove crossing ,. Winter Solstice,,. NearSummer Solstice,, yielding fixed. Then gen adds kun; xun subtracts qian from the fixed remainder. ,,, 1, alsoyielding fixed. ,, eclipseremainder. Chronogram divisor, eclipse. ,, 1,. Ifeclipse, entering qi, eclipse, eclipse,.
399
推月食所在辰術：
Method for computing the chronogram of lunar eclipse:
400
三日阻減望定餘半。 置望之所入氣日，不見刻，朔日法乘之，百而一，所得若食餘與之等、以下，又以此所得減朔日法，其殘食餘與之等、以上，為食正見數。 其食餘亦朔辰而一，如求加辰所在。 又如前求刻校之，月在沖辰食，日月食既有起訖晚早，亦或變常進退，皆於正見前後十二刻半候之。
Three days obstruct—reduce the full-moon fixed remainder by half. Full moon entering qi, , new moon day divisormultiply , 1, obtain eclipse remainder and Multiply by , Multiply by obtain subtract new moon day divisor, eclipse remainder and Multiply by , as eclipse. Itseclipse1,. , eclipse, eclipse,, 12.
401
推日月食起訖辰術：
Method for computing beginning and ending chronograms of solar and lunar eclipses:
402
准其食分十五分為率，全以下各為衰。 十四分以上，以一為衰，以盡於五分。 每因前衰每降一分，積衰增二，以加於前，以至三分。 每積增四。 二分每增四，二分增六，一分增十九，皆累算為各衰。 三百為率，各衰減之，各以其殘乘朔日法，皆率而一，所得為食衰數。 其率全，即以朔日法為衰數，以衰數加減食餘，其減者為起，加者為訖，數亦如氣。
Eclipsefraction15, or below. 14, 1, 5fraction. 1fraction, 2,, by through 3fraction. Each accumulation adds 4. 4, 6, 19,. ,, day divisor, 1,. , day divisor, eclipseremainder,,,.
403
求入辰法及求刻：以加減食所刻等，得起訖晚早之辰，與校正見多少之數。 史書虧複起訖不同，今以其全一辰為率。
To find chronogram divisor:eclipse,,. Obscuration, 1.
404
推日月食所起術：
Method for computing the initial point of solar and lunar eclipses:
405
月在內者，其正南，則起右上，虧左上。 若正東，月自日上邪北而下。 其在東南維前，東向望之，初不正，橫月高日下； 乃月稍西北，日漸東南，過於維後，南向望之，月更北，日差西南； 以至於午之後，亦南望之，月欹西北，日複東南。 西南維後，西向而望，月為東北，日則西南。 正西，自日北下邪虧，而亦後不正，橫月高日下。 若食十二分以上，起右虧左。 其正東，起上近虧下而北，午前則漸自上邪下。 維西，起西北，虧東南。 維北，起西南，虧東北； 午後則稍從下傍下。 維東，起西南，虧東北。 維南，起西北，虧東南。 在東則以上為東，在西則以下為西。
,,, obscuration. Due east: the moon descends obliquely north from above the sun. Before the southeast corner, viewing east: initially oblique—the moon high, sun below. Then the moon shifts slightly northwest, the sun gradually southeast; past the corner viewing south, the moon more north, sun skewed southwest. By through,,,. After the southwest corner, viewing west: moon northeast, sun southwest. , obscuration,,. Ifeclipse12, obscurationleft. , obscuration,. ,, obscuration. ,, obscuration. After the wu hour it gradually passes below from the side. ,, obscuration. ,, obscuration. , or below.
406
月在外者，其正南，起右下，虧左上。 在正東，月自日南邪下而映。 維北，則月微東南，日返西。 維西南，日稍移東北，以至於午，月南日北，過午之後，月稍東南，日更西北。 維北，月有西南，日複東北。 正西，月自日下邪南而上。 皆准此體以定起虧，隨其所處，每用不同。 其月之所食，皆依日虧起，每隨類反之，皆與日食限同表裡，而與日返其逆順，上下過其分。
,,, obscuration. Due east: the moon reflects obliquely south from below the sun. North of the corner: moon slightly southeast, sun returns west. ,, by through,,,,. North of the corner: moon southwest, sun again northeast. Due west: the moon rises obliquely south from below the sun. Obscuration,,. Eclipse, obscurationbegin,, eclipseoutside and inside, retrogradedirect,.
407
五星：
Five planets:
408
歲為木
The year star is Wood (Jupiter).
409
熒惑為火
Mars.
410
鎮為土
The queller star is Earth (Saturn).
411
太白為金
Venus.
412
辰為水
The chronogram star is Water (Mercury).
413
木數，千八百六十萬五千四百六十八。
Wood number (Jupiter): 8605468.
414
伏半平，八十三萬六千八百四十八。
Disappearance, 836848.
415
複日，三百九十八； 餘，四萬一千一百五十六。
Return day: 398. Remainder: 41,156.
416
歲一，殘日，三十三； 餘，二萬九千七百四十九半。
1,, 33. Remainder: NaN.
417
見去日，十四度。
, 14degree.
418
平見，在春分前，以四乘去立春日； 小滿前，又三乘去春分日，增春分所乘者； 白露後，亦四乘去寒露日； 小暑，加七日； 小雪前，以八乘去寒露日； 冬至後，以八乘去立春日，為減，小雪至冬至減七日。
Mean appearance, atSpring Equinox, Start of Spring. Lesser Fullness, Spring Equinox, Spring Equinox. White Dewafter, Cold Dew. Lesser Heat, add7day. Lesser Snow, Cold Dew. Winter Solsticeafter, Start of Spring,, Lesser SnowWinter Solstice.
419
見，初日行萬一千八百一十八分，日益遲七十分，百一十日行十八度、分四萬七百三十八而留。 二十八日乃逆，日退六千四百三十六分，八十七日退十二度、分二百四。 又留二十八日。 初日行四千一百八十八分，日益疾七十分，百一十日亦行十八度、分四萬七百三十八而伏。
, daily motion 1,818, daily decreasing slow 70, 10daily motion 18 and 40,738then stationary. 28then retrograde , retrograde 6,436, 87retrograde 12 and 204. Also stationary 28 days. Initial daily motion 4,188 fractions, increasing by 70 per day; in 110 days travels 18° 40,738 fractions, then disappears.
420
火數，三千六百三十七萬七千五百九十五。
Fire number (Mars): 36377595.
421
伏半平，三百三十七萬九千三百二十七半。
Disappearance, 3379327.
422
複日，七百七十九； 餘，四萬一千九百一十九。
Return day: 779. Remainder: 41,919.
423
歲再，殘日，四十九； 餘，萬九千一百六。
,, 49. Remainder, 6.
424
見去日，十六度。
, 16degree.
425
平見，在雨水前，以十九乘去大寒日：清明前，又十八乘去雨水日，增雨水所乘者； 夏至後，以十六乘去處暑日； 小滿後，又十五日； 寒露前，以十八乘去白露日； 小雪前，又十七乘去寒露日，增寒露所乘者； 大雪後，二十九乘去大寒日，為減，小雪至大雪減二十五日。
Mean appearance, atRain Water, Great Cold：Pure Brightness, Rain Water, Rain Water. Summer Solsticeafter, End of Heat. Lesser Fullnesslater, also15day. Cold Dew, White Dew. Lesser Snow, Cold Dew, Cold Dew. Great Snowafter, Great Cold,, Lesser SnowGreat Snow.
426
見，初在冬至，則二百三十六日行百五十八度，以後日度隨其日數增損各一； 盡三十日，一日半損一； 又八十六日，二日損一； 複三十八日，同； 又十五日，三日損一； 複十二日，同； 又三十九日，三日增一； 又二十四日，二日增一； 又五十八日，一日增一； 複三十三日，同； 又三十日，二日損一，還終至冬至，二百三十六日行百五十八度。 其立春盡春分，夏至盡立夏，八日減一日； 春分至立夏，減六日； 立秋至秋分，減五度，各其初行日及度數。 白露至寒露，初日行半度，四十日行二十度。 以其殘日及度，計充前數，皆差行，日益遲二十分，各盡其日度乃遲，初日行分二萬二千六百六十九，日益遲一百一十分，六十一日行二十五度、分萬五千四百九。 初減度五者，于此初日加分三千八百二十三、篾十七； 以遲日為母，盡其遲日行三十度，分同，而留十三日。
, at Winter Solstice, then 236 daily motion 58°, °its each 1; 30day, 11. Also86day, 21. 38day,. Also15day, 31. 12day,. Also39day, 31. Also24day, 21. Also58day, 11. 33day,. Also 30, 1, through Winter Solstice, 236 daily motion 58°. ItsStart of SpringSpring Equinox, Summer SolsticeStart of Summer,. Spring Equinox through Start of Summer, subtract6day. Start of AutumnAutumn Equinox,, initial motion. White DewCold Dew, daily motion, daily motion. Multiply by , , , daily decreasing slow 20, slow , daily motion 22,669, daily decreasing slow 110, 61daily motion 25 and 5,409. Subtract , add 3,823 and bamboo tally 17. Slow motionas , its Slow motion daily motion 30°, fractions the same, then stationary13.
427
前減日分於二留，乃逆，日退分萬二千五百二十六，六十三日退十六度、分四萬二千八百三十四。 又留十三日而行，初日萬六千六十九，日益疾百一十分，六十一日行二十五度、分萬五千四百九。 立秋盡秋分，增行度五，加初日分同前，更疾。 在冬至則二百一十三日行百三十五度； 盡三十六日，一日損一； 又二十日，二日損一； 複二十四日，同； 又五十四日，三日日增一； 又十二日，二日增一； 又四十二日，一日增一； 又十四日，一日增一半； 又十二日，增一； 複四十五日，同； 又一百六日，二日損一，亦終冬至二百一十三日，行百三十五度。
Subtract day fraction2stationary , then retrograde , retrograde 2,526, 63retrograde 16 and 42,834. Stationary 13, 6,069, daily increasing fast 10, 61daily motion 25 and 5,409. Start of AutumnAutumn Equinox, 5, day fraction,. At winter solstice, in 213 days travels 135°; 36day, 11. Also20day, 21. 24day,. Also54day, 31. Also12day, 21. Also42day, 11. Also14day, 11. Also12day, 1. 45day,. Also106day, 21, Winter Solstice213day, motion35degree.
428
前增行度五者，於此亦減五度，為疾日及數。 其立夏盡夏至初，日行半度，六十日行三十度。 夏至盡立秋，亦初日行半度，四十日行二十度。 其殘亦計充如前，皆差行，日益疾二十分，各盡其日度而伏。
5, 5degree, asfast motion. ItsStart of SummerSummer Solstice, daily motion, daily motion. Summer SolsticeStart of Autumn, daily motion, daily motion. ,, daily increasing fast, then disappears.
429
土數，千七百六十三萬五千五百九十四。
Earth number (Saturn): 7635594.
430
伏半平，八十六萬四千九百九十五。
Disappearance, 864995.
431
複日，三百七十八； 餘，四千一百六十二。
Return day: 378. Remainder: 4,162.
432
歲一，殘日，十二； 餘，三萬九千三百九十九半。
1,, 12. Remainder: NaN.
433
見去日，十六度半。
, 16.
434
平見，在大暑前，以七乘去小滿日； 寒露後，九乘去小雪日，為加，大暑至寒露加八日。 小寒前，以九乘去小雪日； 雨水後，以四乘去小滿日； 立春後，又三乘去雨水日，增雨水所乘者，為減，小寒至立春減八日。
Mean appearance, atGreat Heat, Lesser Fullness. Cold Dewafter, Lesser Snow,, Great HeatCold Dew. Lesser Cold, Lesser Snow. Rain Waterafter, Lesser Fullness. Start of Springafter, Rain Water, Rain Water,, Lesser ColdStart of Spring.
435
見，日行分四千三百六十四，八十日行七度、分二萬二千六百一十二而留三十九日乃逆，日退分二千八百二十，百三日退六度、分萬五百九十六。 又留三十九日，亦行分日四千三百六十四，八十日行七度、分二萬二千六百一十二而伏。
, daily motion 4,364, 80daily motion 7 and 22,612then stationary 39then retrograde , retrograde 2,820, 3retrograde 6 and 596. Stationary 39, 4,364, 80daily motion 7 and disappearance.
436
金數，二千七百二十三萬六千二百八。
Metal number (Venus): 27236208.
437
晨伏半平，百九十五萬七千一百四。
Disappearance, 957104.
438
複日，五百八十三； 餘，四萬二千七百五十六。
Return day: 583. Remainder: 42,756.
439
歲一，殘日，二百一十八； 餘，三萬一千三百四十九半。
1,, 218. Remainder: NaN.
440
夕見伏，二百五十六日。
Evening appearance and disappearance: 256 days。.
441
晨見伏，三百二十七日； 餘與複同。
Morning appearance and disappearance: 327 days. The remainder is the same as before.
442
見去日，十一度。
, 11degree.
443
夕平見，在立秋前，以六乘去芒種日； 秋分後，以五乘去小雪日； 小雪後，又四乘去大雪日，增小雪所乘者，為加，立秋至秋分加七日。 立春前，以五乘去大雪日； 雨水前，又四乘去立春日，增立春所乘者； 清明後，以六乘去芒種日，為減，雨水至清明減七日。
Mean appearance, atStart of Autumn, Grain in Ear. Autumn Equinoxafter, Lesser Snow. Lesser Snowafter, Great Snow, Lesser Snow,, Start of AutumnAutumn Equinox. Start of Spring, Great Snow. Rain Water, Start of Spring, Start of Spring. Pure Brightnessafter, Grain in Ear,, Rain WaterPure Brightness.
444
晨平見，在小寒前，以六乘去冬至日； 立春前，又五乘去小寒日，增小寒所乘者； 芒種前，以六乘去夏至日； 立夏前，又五乘去芒種日，增芒種所乘者，為加，立春至立夏加五日。 小暑前，以六乘去夏至日； 立秋前，又五乘去小暑日； 增小暑所乘者； 大雪後，以六乘去冬至日； 立冬後，又五乘去大雪日，增大雪所乘者，為減，立秋至立冬減五日。
Mean appearance, atLesser Cold, Winter Solstice. Start of Spring, Lesser Cold, Lesser Cold. Grain in Ear, Summer Solstice. Start of Summer, Grain in Ear, Grain in Ear,, Start of SpringStart of Summer. Lesser Heat, Summer Solstice. Start of Autumn, Lesser Heat. Lesser Heat. Great Snowafter, Winter Solstice. Start of Winterafter, Great Snow, Great Snow,, Start of AutumnStart of Winter.
445
夕見，百七十一日行二百六度。 其穀雨至小滿、白露至寒露，皆十日加一度； 小滿至白露，加三度。 乃十二日行十二度。 冬至後，十二日減日度各一，雨水盡夏至，日度七； 夏至後六日增一。 大暑至立秋，還日度十二； 至寒露，日度二十二，後六日減一。 自大雪盡冬至，又日度十二而遲。 日益疾五百二十分，初日行分二萬三千七百九十一、篾三十五，行日為母，四十三日行三十二度。
Evening appearance, 71 daily motion 206°. From Grain Rain through Lesser Fullness and White Dew through Cold Dew, add 1° every 10 days; Lesser Fullness through White Dew, add3degree. Then in 12 days travel 12°. Winter Solsticeafter, 1, Rain WaterSummer Solstice, 7. Six days after summer solstice, increase by 1. Great HeatStart of Autumn, 12. Cold Dew, 22, 1. Great SnowWinter Solstice, 12slow motion. Daily increasing fast 520, daily motion 23,791 and bamboo tally 35, as , 43daily motion 32.
446
前加度者，此依減之。 留九日乃逆，日退太半度，九日退六度，而夕伏晨見。 日退太半度，九日退六度。 複留，九日而行，日益遲五百二十分，初日行分四萬五千六百三十一、篾三十五，四十三日行三十二度。 芒種至小暑，大雪至立冬，十五日減一度； 小暑至立冬，減二度。 又十二日行十二度。 冬至後，十五日增日度各一。 驚蟄至春分，日度十七，後十五日減一，盡夏至，還日度十二。 後六日減一，至白露，日度皆盡。 霜降後，五日增一，盡冬至，又日度十二。 乃疾，百七十一日行二百六度。 前減者，此亦加之，而晨伏。
When prior entries add degrees, subtract accordingly. Stationary9 then retrograde , °, 6°, and evening disappearancemorning appearance. , 96degree. Stationary , 9, daily decreasing slow 520, daily motion 45,631 and bamboo tally 35, 43daily motion 32. Grain in EarLesser Heat, Great SnowStart of Winter,. Lesser Heat through Start of Winter, subtract2degree. Also in 12 days travel 12°. Winter Solsticeafter, 1. Awakening of InsectsSpring Equinox, 17, 1, Summer Solstice, 12. 1, White Dew,. Frost Descentafter, 1, Winter Solstice, 12. Fast motion, 71 daily motion 206°. ,, disappearance.
447
水數，五百四十萬五千六。
Water number (Mercury): 5405006.
448
晨伏半平，七十九萬九十九。
Disappearance, 790099.
449
複日，百一十五； 餘，四萬九百四十六。
Return day: 15. Remainder: 40,946.
450
夕見伏，五十一日。
Evening appearance and disappearance: 51 days。.
451
晨見伏，六十四日； 餘與複同。
Morning appearance and disappearance: 64 days. The remainder is the same as before.
452
見去日，十七度。
, 17degree.
453
夕應見，在立秋後小雪前者不見； 其白露前立夏後，時有見者。
, atStart of AutumnlaterLesser Snow. Before White Dew and after Start of Summer, it sometimes appears.
454
晨應見，在立春後小滿前者不見； 其驚蟄前立冬後，時有見者。
, atStart of SpringlaterLesser Fullness. Before Awakening of Insects and after Start of Winter, it sometimes appears.
455
夕見，日行一度太，十二日行二十度。 小暑至白露，行度半，十二日行十八度，乃八日行八度。 大暑後，二日去度一，訖十六日，而日度俱盡。 而遲，日行半度，四日行二度。 益遲，日行少半度，三日行一度。 前行度半者，去此益遲。 乃留四日而夕伏晨見，留四日，為日行少半度，三日行一度。 大寒至驚蟄，無此行，更疾，日行半度； 四日行二度； 又八日行八度。 亦大寒後，二日去度一； 訖十六日，亦日度俱盡。 益疾，日行一度太，十二日行二十度。 初無遲者，此行度半，十二日行十八度而晨伏。
Evening appearance, daily motion 1°, 12 daily motion 20°. Lesser HeatWhite Dew,, 12 daily motion, then8 daily motion. Great Heatafter, 1,,. And Slow motion, daily motion °, 4 daily motion 2°. Slow motion, daily motion °, 3 daily motion 1°. , slow motion. Stationary4 and evening disappearancemorning appearance, stationary4, as daily motion °, 3 daily motion 1°. Great ColdAwakening of Insects,,, daily motion. In 4 days travel 2°; Also in 8 days travel 8°. AlsoGreat Coldafter, 1. 16day,. Fast motion, daily motion 1°, 12 daily motion 20°. ,, 12 daily motion.
456
推星平見術：
Method for computing mean appearance.
457
各以伏半減積半實，乃以其數去之； 殘返減數，滿氣日法為日，不滿為餘，即所求年天正冬至後平見日餘。 金、水滿晨見伏日者，去之，晨平見。 求平見月日：以冬至去定朔日、餘，加其後日及餘，滿複日又去，起天正月，依定大小朔除之，不盡算外日，即星見所在。 求後平見，因前見去其歲一、再，皆以殘日加之，亦可。 其複日，金水準以晨夕見伏日，加晨得夕，加夕得晨。
Disappearance, remove. Subtract, when full day divisoras , not when full as , then the year soughtcelestial first monthWinter Solsticelater mean appearance. When Venus and Mercury fill the morning appearance-disappearance period, remove it—morning mean appearance. Mean appearance：Multiply by Winter Solstice remove new moon and remainder , add remainder , when full remove , , new moon divide , beyond tally,. To find later mean appearance: from prior appearance remove one or two year cycles and add remaining days. For return days, Venus and Mercury use morning/evening appearance-disappearance days; adding morning yields evening, adding evening yields morning.
458
求常見日：以轉法除所得加減者，為日； 其不滿，以餘通乘之，為餘； 並日，皆加減平見日、餘，即為常見日及餘。
To find :,. When full,,. , add subtract mean appearance and remainder , as remainder.
459
求定見日：以其先後已通者，先減後加常見日，即得定見日餘。
To find fixed appearanceday:,, fixed appearance.
460
求星見所在度：
To find the degree where the star appears:
461
置星定見、其日夜半所在宿度及分，以其日先後餘，分前加、分後減氣日法，而乘定見餘，氣日法而一所得加夜半度分，乃以星初見去日度數，晨減夕加之，即星初見所在宿度及分。
Fixed appearance and half , Multiply by remainder , add and subtract day divisor, multiply fixed appearanceremainder , day divisorobtain add half , Multiply by first appearanceremove , subtract add , first appearance.
462
求次日：
To find the next day:
463
各加一日所行度及分。 其有益疾、遲者副置一日行分，各以其分疾增、遲損，乃加之。 有篾者，滿法從分，其母有不等，齊而進退之。 留即因前，逆則依減入虛去分，逆出先加。 皆以篾法除，為轉分； 其不盡者，仍謂之篾，各得每日所在知去日度。 增以日所入先後分，定之。 諸行星度求水其外內，准月行增損黃道而步之； 不明者，依黃道而求所去日度。 先後分亦分明前加後減。 其金、火諸日度，計數增損定之者。 其日少度多，以日減度之殘者，與日多度少之度，皆度法乘之，日數而一，所得為分。 不滿篾，以日數為母。 日少者以分並減之一度，日多者直為度分，即皆一日平行分。 其差行者，皆減所行日數一，乃半其益疾、益遲分而乘之，益疾以減、益遲以加一日平行分，皆初日所行分。 有計日加減，而日數不滿未得成度者，以氣日法若度法乘，見已所行日即日數除之，所得以增損其氣日疾法，為日及度。 其不成者，亦即為篾。 其木、火、土，晨有見而夕有伏； 金、水即夕見還夕伏，晨見即晨伏。 然火之初行及後疾，距冬至日計日增損日度者，皆當先置從冬至日餘數，累加於位上，以知其去冬至遠近，乃以初見與後疾初日去冬至日數而增損定之，而後依其所直日度數行之也。
Add each day's motion in degrees and fractions. For auxiliary fast or slow phases, set aside one day's motion fraction; increase for fast or decrease for slow by that fraction, then add. , when full divisorcarry to fraction,,. Stationarythen , retrogradethen subtractremove parts, retrograde. All divide by the bamboo-tally divisor to obtain rotation fraction. Remainders are still called bamboo tallies—each gives the daily position and distance from the sun. Increase by the daily entry prior-later fraction to obtain the fixed value. For all planetary degrees, find Mercury's inner and outer limits and step along the yellow path using lunar increase-decrease; When not visible, find solar distance by the ecliptic. Advance-retreat fractions likewise add before and subtract after the fraction. For Venus and Mars day-degrees, compute increase-decrease counts to obtain fixed values. ,,, alldegree divisor, 1,. When full bamboo tally , Multiply by as. ,, uniform motion. , subtract 1, half fast and slow multiply , fast Multiply by subtract and slow Multiply by add 1,. ,, day divisorifdegree divisor,,,. Incomplete values are also called bamboo tallies. Jupiter and Mars and Saturn , disappearance. Venus and Mercury at evening appearance return to evening disappearance; morning appearance means morning disappearance. Thereof initial motion Fast motion, Winter Solstice°, all carry to Winter Solstice, , its remove Winter Solstice, first appearanceFast motionremove Winter Solstice and thereof , and its that which ° thereof .