Volume 82 Treatises 35: Measures and Calendar 15

乾道四年，禮部員外郎李燾言：「《統元曆》行之既久，與天不合，固宜；《大衍曆》最號精微，用之亦不過三十餘年，後之欲行遠也難矣。抑曆未差，無以知其失；未驗，無以知其是。仁宗用《崇天曆》，天聖至皇祐四年十一月日食，二曆不效，詔以唐八曆及宋四曆參定，皆以《景福》為密，遂欲改作。而劉羲叟謂：「《崇天曆》頒行逾三十年，所差無幾，詎可偶緣天變，輕議改移？」又謂：「古聖人曆象之意，止於敬授人時，雖則預考交會，不必吻合辰刻，或有遲速，未必獨是曆差。」乃從羲叟言，復用《崇天曆》。羲叟曆學為宋第一，歐陽修、司馬光輩皆遵用之。《崇天曆》既復用，又十三年，治平二年，始改用《明天曆》，曆官周琮皆遷官。後三年，驗熙寧三年七月月食不效，乃詔復用《崇天曆》，奪琮等所遷官。熙寧八年，始更用《奉元曆》，沈括實主其議。明年正月月食，遽不效，詔問修曆推恩者姓名，括具奏辨，得不廢。識者謂括強辨，不許其深於曆也。然後知羲叟之言然。願申飭曆官，加意精思，勿執今是。益募能者，熟復討論，更造密度，補治新曆。」緣燾嘗承詔監視測驗，值新曆太陰、熒惑之差，恐書成所差或多，見譏能者，乃詔諸道訪通曆者。久之，福州布衣阮興祖上言新曆差謬，荊大聲不以白部，即補興祖為局生。

In the fourth year of the Qiandao reign, Li Tao, a vice director in the Ministry of Rites, memorialized: "The Tongyuan Calendar has been in force for so long that it no longer matches the heavens—a result that is only natural; The Dayan Calendar was acclaimed as the most precise, yet even it remained serviceable for barely thirty years; anyone who hopes a calendar will stay accurate indefinitely faces a hard task. Besides, until a calendar has actually drifted, one cannot tell that it is wrong; and without testing, one cannot know that it is right. Under Emperor Renzong the Chongtian Calendar was in use; from the Tiansheng era through the eleventh month of Huangyou 4 a solar eclipse showed two calendars to be wrong. The court ordered the eight Tang calendars and four Song calendars compared, and all judged the Jingfu calendar the tightest fit—so officials wanted a new revision. Liu Yisou objected: "The Chongtian Calendar has been issued for over thirty years and has drifted only slightly. How can we, on the strength of a single heavenly irregularity, rashly propose to replace it?" He added: "The ancient sages devised calendars chiefly to set the seasons for human affairs. Even when they calculated eclipses in advance, perfect agreement to the hour was not required; discrepancies in timing need not be blamed solely on the calendar." The court accepted Liu Yisou's view and restored the Chongtian Calendar. Yisou was regarded as the foremost calendrical scholar of the Song, and men such as Ouyang Xiu and Sima Guang all adhered to his methods. Once the Chongtian Calendar was back in use, thirteen more years elapsed; in Zhiping 2 the Mingtian Calendar was adopted, and calendar officials such as Zhou Cong received promotions. Three years on, verification showed the Xining 3 seventh-month lunar eclipse wrong; the court ordered the Chongtian Calendar restored and revoked the promotions granted to Zhou Cong and his colleagues. In Xining 8 the Fengyuan Calendar was introduced, with Shen Kuo as the principal advocate. The next year's first-month lunar eclipse was wrong at once. The throne demanded the names of everyone rewarded for the calendar reform; Shen Kuo answered with a detailed defense, and the Fengyuan Calendar was not scrapped. Knowledgeable critics said Shen Kuo had argued his way out of trouble and denied that he truly mastered calendrical astronomy. Only then did people see that Liu Yisou had been right all along. I ask that calendar officers be instructed to think more rigorously and not insist that the present calendar must be right. Recruit more skilled men, debate the methods thoroughly, rebuild the density constants, and produce a revised calendar." Tao had earlier been ordered to oversee calendrical tests; when the new calendar proved wrong for the moon and Mars, he feared the finished work would contain many errors and expose him to experts' ridicule—so the court ordered every circuit to seek out masters of calendrical science. Eventually Ruan Xingzu, a Fuzhou commoner, memorialized that the new calendar was flawed. Jing Dasheng never informed his superiors but instead enrolled Xingzu directly as a bureau student.

初，新曆之成也，大聲、孝榮共為之。至是，大聲乃乙太陰九道變赤道別演一法，與孝榮立異於後。秘書少監、崇政殿說書兼權刑部侍郎汪大猷等言：「承詔于御史台監集局官，參算明年太陰宿度，箋注御覽詣實。今大聲等推算明年正月至月終九道太陰變赤道，限十二月十五日以前具稿成，至正月內，臣等召曆官上臺，用渾儀監驗疏密。」從之。

When the new calendar was first completed, Dasheng and Xiaorong had worked on it together. By then Dasheng had reworked the lunar nine-path conversion from the equator into a separate procedure and afterward broke with Xiaorong's approach. Wang Dayou, vice director of the Secretariat, lecturer at the Chongzheng Hall, and acting vice minister of justice, reported: "By edict we supervised bureau astronomers at the Censorate in calculating next year's lunar lodge positions; we now submit this memorial for the throne to verify. Dasheng and his colleagues must finish drafts of the nine-path lunar equator crossings from next year's first month through its end by the fifteenth day of the twelfth month. In the first month we shall call calendar officers to the observatory and test the results with the armillary sphere." The court agreed.

五年，國子司業兼權禮部侍郎程大昌、侍御史單時，秘書丞唐孚、秘書郎李木言：「都省下靈台郎充曆算官蓋堯臣、皇甫繼明、宋允恭等言：'厥今更造《乾道新曆》，朝廷累委官定驗，得見日月交食密近天道，五星行度允協躔次，惟九道太陰間有未密。搜訪能曆之人補治新曆，半年未有應詔者，獨荊大聲別演一法，與劉孝榮《乾道曆》定驗正月內九道太陰行度。今來二法皆未能密于天道，《乾道》太陰一法與諸曆比較，皆未盡善。今撮其精微，撰成一法，其先推步到正月內九道太陰正對在赤道宿度，願委官與孝榮、大聲驗之。如或精密，即以所修九道經法，請得與定驗官更集孝榮、大聲等同赴台，推步明年九道太陰正對在赤道宿度，點定月分定驗，從其善者用之。 '大昌等從大聲、孝榮所供正月內太陰九道宿度，已赴太史局測驗上中旬畢，及取大聲、孝榮、堯臣等三家所供正月下旬太陰宿度，參照覽視，測驗疏密，堯臣、繼明、允恭請具今年太陰九道宿度。欲依逐人所請，限一月各具今年太陰九道變黃道正對赤道其宿某度，依經具稿，送御史台測驗官不時視驗，然後見其疏密。」

In the fifth year Cheng Dachang, director of studies and acting vice minister of rites, the censor Shan Shi, Secretariat aide Tang Fu, and gentleman Li Mu reported: "Observatory calculators Gai Yaochan, Huangfu Jiming, and Song Yungong stated that in testing the new Qian Dao calendar, eclipses and planetary motions largely matched heaven—only the lunar nine-path tables still showed gaps. A six-month search for experts to fix the calendar drew no respondents; only Jing Dasheng devised an alternative method and, together with Liu Xiaorong's Qian Dao tables, submitted first-month lunar nine-path positions for verification. Neither method yet matches the heavens closely; compared with earlier calendars, the Qian Dao lunar procedure is still not the best available. We have distilled the best elements into a single method and calculated first-month lunar nine-path equator crossings; we ask that officials test it against Xiaorong's and Dasheng's results. If it proves accurate, we ask to join the verification officers, Xiaorong, Dasheng, and the rest at the observatory to calculate next year's lunar nine-path equator crossings month by month and adopt whichever method performs best. Dachang's group used Dasheng's and Xiaorong's first-month lunar tables to finish observatory tests for the first two ten-day periods, then compared all three scholars' late-first-month figures. Yaochan, Jiming, and Yungong now ask to submit the full year's lunar nine-path lodge positions. Each petitioner should within one month submit drafts giving every lunar nine-path yellow-path equator crossing for the year; the Censorate testers will check them periodically and determine which method is soundest."

裴伯壽上書言：

Pei Boshou submitted a memorial:

孝榮自陳預定丁亥歲四月朔日食、八月望月食，俱不驗。又定去年二月望夜二更五點月食九分以上，出地復滿。臣嘗言于宰相，是月之食當食既出地，《紀元曆》亦食既出地，生光在戌初二刻，復滿在戌正三刻。是夕，月出地時有微雲，至昏時見月已食既，至戌初三刻果生光，即食既出地可知；復滿在戌正三刻，時二更二點：臣所言卒驗。孝榮言見行曆交食先天六刻，今所定月食復滿，乃後天四刻，新曆謬誤為甚。

Xiaorong admitted that his predicted dinghai-year solar eclipse on the fourth-month new moon and lunar eclipse on the eighth-month full moon both failed. He had also predicted a ninth-month-or-greater lunar eclipse at the second watch on last year's second-month full moon, with the moon rising eclipsed and later returning to full brightness. I had told the chief minister that the eclipse would be total at moonrise; the Jiyuan Calendar agreed. Light would return in the early xu watch and full brightness in the middle xu watch. That evening thin clouds covered the horizon at moonrise; by dusk the moon was already in total eclipse; light returned in the third quarter of the early xu watch, confirming a total eclipse at rising; full brightness came in the middle xu watch, corresponding to the second watch second point—my prediction was borne out. Xiaorong claimed the current calendar runs eclipses six quarters early, yet his predicted time of full recovery was four quarters late—the new calendar's errors are severe.

其一曰步氣朔，孝榮先言氣差一日，觀景表方知其失，此不知驗氣者也。臣之驗氣，差一二刻亦能知之。《紀元》節氣，自崇寧間測驗，逮今六十餘載，不無少差，苟非測驗，安知其失？凡日月合朔，以交食為驗，今交食既差，朔亦弗合矣。

First, pacing the qi and new moons: Xiaorong first thought the solar term was wrong by a full day and learned his mistake only from the gnomon—he does not know how to test the qi. When I test the qi, I can detect errors of one or two quarters. The Jiyuan calendar's solar terms have been tested since Chongning; over sixty years they have drifted slightly—without observation, how could one know? New-moon conjunctions are verified by eclipses; once eclipses fail, the new moons cannot be right either.

其二曰步發斂，止言卦候而已。

Second, pacing issuing and gathering: he treats only the hexagram seasons.

其三曰步日躔，新曆乃用《紀元》二十八宿赤道度，暨至分宮，遽減《紀元》過宮三十餘刻，殊無理據。而又赤道變黃道宿度，婁、胃二宿頓減《紀元》半度。在術則婁、胃二宿合二十八度，婁當十二度太，今新曆婁作十二度半，乃棄四分度之一。室、軫二宿虛收復多，少數變宿，分宮既訛，是以乾道己丑歲太陽過宮差誤。

Third, pacing the sun's motion: the new calendar copies Jiyuan equatorial lodge degrees but arbitrarily cuts more than thirty quarters from Jiyuan's lodge-passage figures when dividing palaces—without justification. In converting equatorial to ecliptic lodge degrees, he suddenly reduced Lou and Wei by half a degree from the Jiyuan values. By calendrical theory Lou and Wei together span twenty-eight degrees, with Lou at twelve and a fraction; the new calendar sets Lou at twelve and a half, discarding a quarter-degree. He padded Shi and Zhen while trimming others, altered lodge widths, and corrupted palace division—hence the sun's palace passage was wrong in Qiandao jichou.

其四曰步晷漏，新曆不合前史。唐開元十二年測景于天下，安南測夏至午中晷在表南三寸三分，新曆算在表北七寸；其鐵勒測冬至午中晷長一丈九尺二寸六分，新曆算晷長一丈四尺九寸九分，乃差四尺二寸七分，其謬蓋若此。

Fourth, gnomon and clepsydra calculations: the new calendar contradicts historical records. Tang Kaiyuan 12 shadow surveys recorded Annan's summer-solstice noon shadow three and three-tenths inches south of the gnomon; the new calendar places it seven inches north; the Tiele measured a winter-solstice noon shadow of one zhang nine chi two cun six fen; the new calendar gives one zhang four chi nine cun nine fen—a discrepancy of four chi two cun seven fen. Its errors are typically on this scale.

其五曰步月離，諸曆遲疾、朏朒極數一同，新曆朏之極數少朒之極數四百九十三分，疾之極數少遲之極數二十分，不合曆法。

Fifth, lunar motion: every proper calendar keeps identical slow-fast and crescent-gibbous limit constants; the new calendar's crescent limits fall four hundred ninety-three fen below its gibbous limits and its fast limits twenty fen below its slow limits—violating calendrical rules.

其六曰步交會，新曆妄設陽准、陰准等差，蓋欲苟合已往交食，其間復有不合者，則遷就天道，所以預定丁亥、戊子二歲日月之食，便見差違。

Sixth, conjunctions: the new calendar invents arbitrary yang and yin standard adjustments to fit past eclipses, then tweaks further to match heaven—yet its dinghai and wuzi eclipse predictions already fail.

其七曰步五星，以渾儀測驗新曆星度，與天不合。蓋孝榮與同造曆人皆不能探端知緒，乃先造曆，後方測驗，前後倒置，遂多差失。夫立表驗氣，窺測七政，然後作曆，豈容掇拾緒餘，超接舊曆，以為新術，可乎？

Seventh, the five planets: armillary observations show the new calendar's stellar longitudes do not match the sky. Xiaorong and his colleagues never grasped first principles; they drafted the calendar before testing it—reversing the proper order and producing many errors. One erects a gnomon, tests the qi, observes the seven luminaries, and only then composes a calendar. How can patching scraps onto an old calendar and calling it new be acceptable?

新曆出於五代民間《萬分曆》，其數朔余太強，明曆之士往往鄙之。今孝榮乃三因萬分小曆，作三萬分為日法，以隱萬分之名。三萬分曆即萬分曆也。緣朔余太強，孝榮遂減其分，乃增立秒，不入曆格。前古至於宋諸曆，朔餘並皆無秒，且孝榮不知王處訥於萬分增二，為《應天曆》日法，朔餘五千三百七，自然無秒，而去王樸用秒之曆。

The new calendar stems from the Five Dynasties popular Wanfen Calendar, whose new-moon remainder is excessively large—calendrical experts commonly disdain it. Xiaorong now triples the Wanfen minor calendar, adopting thirty thousand parts as the day divisor merely to hide the Wanfen label. The thirty-thousand-part calendar is simply the Wanfen calendar under another name. Because the new-moon remainder was too large, Xiaorong trimmed the parts and invented seconds—a departure from proper calendrical practice. From antiquity through Song calendars, new-moon remainders never used seconds. Xiaorong did not know that Wang Chune had added two to the Wanfen divisor for the Yingtian Calendar, yielding a remainder of five thousand three hundred seven that required no seconds—and he discarded Wang Pu's second-based calendar.

臣與造《統元曆》之後，潛心探討復三十餘年，考之諸曆，得失曉然。誠假臣演撰之職，當與太史官立表驗氣，窺測七政，運算立法，當遠過前曆。

Since helping compose the Tongyuan Calendar, I have studied calendrical science for thirty more years and understand the strengths and weaknesses of every calendar. If I am given the task of drafting a calendar, I shall work with the Grand Astrologer to test the qi by gnomon, observe the seven luminaries, and compute new constants that will far exceed earlier calendars.

詔送監視測驗官詳之，達於尚書省。

The memorial was referred to the supervising testing officers for review and forwarded to the Ministry of Works.

時談天者各以技術相高，互相詆毀。諫議大夫單時、秘書少監汪大猷、國子司業權禮部侍郎程大昌、秘書丞唐孚、秘書郎李木言：「《乾道新曆》，荊大聲、劉孝榮同主一法，自初測驗以至權行施用，二人無異議。後緣新曆不密，詔訪求通曆者，孝榮乃訟阮興祖緣大聲補局生，自是紛紛不已。大聲官以判局提點曆書為名，乃言不當責以立法起算。不知起曆授時，何所憑據。且正月內五夜，比較孝榮所定五日並差，大聲所定五日內三日的中，兩日稍疏。繼伯壽進狀獻術，時等將求其曆書上臺測驗，務求至當，而大聲等正居其官，乃飾辭避事，測驗弗精。且大聲、孝榮同立新法，今猶反覆，苟非各具所見，他日曆成，大聲妄有動搖，即前功盡廢。請令孝榮、大聲、堯臣、伯壽各具乾道五年五月已後至年終，太陰五星排日正對赤道躔度，上之御史台，令測驗官參考。」詔從之。

Astronomers of the day each boasted of his own methods and denounced his rivals. Shan Shi, Wang Dayou, Cheng Dachang, Tang Fu, and Li Mu reported: "For the Qian Dao New Calendar, Jing Dasheng and Liu Xiaorong originally shared one method from initial testing through provisional use without dispute. When the calendar proved inaccurate, the court sought experts; Xiaorong then complained that Dasheng had installed Ruan Xingzu as a bureau student without authorization—and disputes never ended. Dasheng held the post of bureau judge overseeing calendar texts yet claimed he should not be blamed for drafting the calendar's foundational calculations. Without that competence, on what basis can one fix the seasons for the realm? In the first month's five-night test, Xiaorong's five predictions all missed, while Dasheng hit three exactly and was slightly off on two. When Boshou submitted his method, Shi and colleagues intended to test it at the observatory; yet Dasheng and his colleagues, holding those offices, evaded responsibility with polished excuses and conducted slipshod tests. Dasheng and Xiaorong co-authored the new method yet still contradict each other. Unless each records his own calculations now, Dasheng may change his mind later and waste all prior work. We ask Xiaorong, Dasheng, Yaochan, and Boshou each to submit lunar and planetary equator-crossing tables from Qiandao 5 fifth month through year's end to the Censorate for comparative testing." The court agreed.

六年，日官言：「比詔權用《乾道曆》推算，今歲頒曆于天下，明年用何曆推算？」詔亦權用《乾道曆》一年。秋，成都曆學進士賈復自言，詔求推明熒惑、太陰二事，轉運使資遣至臨安，願造新曆畢還蜀，仍進《曆法九議》。孝宗嘉其志，館於京學，賜廩給。太史局李繼宗等言：「十二月望，月食大分七、小分九十三。賈復、劉大中等各虧初、食甚分夜不同。」詔禮部侍郎鄭聞監李繼宗等測驗。是夜，食八分。秘書省言，靈台郎宋允恭、國學生林永叔、草澤祝斌、黃夢得、吳時舉、陳彥健等各推算日食時刻、分數異同。乃詔諫議大夫姚憲監繼宗等測驗五月朔日食。憲奏時刻、分數皆差舛，繼宗、澤、大聲削降有差。

In the sixth year the calendar officers asked: "We have been ordered to use the Qian Dao Calendar provisionally and issued this year's almanac nationwide—which calendar should we use next year?" The court again authorized the Qian Dao Calendar for one more year. That autumn Jia Fu, a Chengdu calendrical scholar, reported that the court sought experts on Mars and the moon; the transport commissioner sent him to Lin'an. He wished to finish a new calendar and return to Shu, presenting his Nine Discourses on Calendrical Method. Emperor Xiaozong admired his ambition, housed him at the capital academy, and granted a stipend. Li Jizong of the Astrological Bureau reported: "The twelfth-month full-moon eclipse will reach seven large parts and ninety-three small parts. Jia Fu, Liu Dazhong, and others each gave different times for first contact, maximum eclipse, and magnitude." Vice Minister of Rites Zheng Wen was ordered to supervise the observatory test. That night the eclipse reached eight-tenths magnitude. The Secretariat noted that Song Yungong, Lin Yongshu, and commoners Zhu Bin, Huang Mengde, Wu Shiju, and Chen Yanjian had each produced different solar-eclipse predictions. The court then ordered Yao Xian to supervise observatory testing of the fifth-month new-moon solar eclipse. Xian reported that all predicted times and magnitudes were wrong; Jizong, Ze, and Dasheng received demotions of varying severity.

太史局春官正、判太史局吳澤等言：「乾道十年頒賜曆日，其中十二月已定作小盡，乾道十一年正月一日注：癸未朔，畢乾道十一年正月一日。《崇天》、《統元》二曆算得甲申朔，《紀元》、《乾道》二曆算得癸未朔，今《乾道曆》正朔小餘，約得不及進限四十二分，是為疑朔。更考日月之行，以定月朔大小，以此推之，則當是甲申朔。今曆官弗加精究，直以癸未注正朔，竊恐差誤，請再推步。於是俾繼宗監視，皆以是年正月朔當用甲申。兼今歲五月朔，太陽交食，本局官生瞻視到天道日食四分半：虧初西北，午時五刻半；食甚正北，未初二刻；復滿東北，申初一刻。後令永叔等五人各言五月朔日食分數並虧初、食甚、復滿時刻皆不同。並見行《乾道曆》比之，五月朔天道日食多算二分少強，虧初少算四刻半，食甚少算三刻，復滿少算二刻已上。又考《乾道曆》比之《崇天》、《紀元》、《統元》三曆，日食虧初時刻為近；較之《乾道》，日食虧初時刻為不及。繼宗等參考來年十二月系大盡，及十一年正月朔當用甲申，而太史局丞、同判太史局荊大聲言《乾道曆》加時系不及進限四十二分，定今年五月朔日食虧初在午時一刻。今測驗五月朔日食虧初在午時五刻半，《乾道曆》加時弱四百五十分，苟以天道時刻預定乾道十二年正月朔，已過甲申日四百五十分。大聲今再指定乾道十一年正月合作甲申朔，十年十二月合作大盡，請依太史局詳定行之。」五月，詔曆官詳定。

Wu Ze of the Astrological Bureau reported: "The Qiandao 10 almanac fixed the twelfth month as a short month and marked the Qiandao 11 first-month new moon as guimao (day 20). The Chongtian and Tongyuan calendars yield jiashen (day 21) for the new moon; the Jiyuan and Qian Dao calendars yield guimao. The Qian Dao new-moon remainder falls forty-two fen short of the advance threshold, making the first-month new moon doubtful. Recomputing lunar motion to determine the month's length, the correct new moon should be jiashen. Calendar officers have not investigated carefully and simply marked guimao as the new moon; I fear this is wrong and ask for a recalculation. Jizong was ordered to supervise observations, and all agreed the year's first-month new moon should be jiashen. This year's fifth-month new-moon solar eclipse was observed at four and a half parts magnitude: first contact in the northwest at the fifth quarter of the wu hour; maximum eclipse due north in the early wei watch; full recovery in the northeast at the first quarter of the shen watch. Yongshu and four others were then asked to predict the eclipse magnitude and the times of first contact, maximum, and recovery—all differed. Compared with the current Qian Dao Calendar, the observed eclipse was about two and a fraction parts larger than predicted; first contact came four and a half quarters later than calculated; maximum three quarters later; recovery more than two quarters later. Compared with the Chongtian, Jiyuan, and Tongyuan calendars, the Qian Dao Calendar's first-contact time was closer to observation; yet it still fell short of the observed time. Jizong's group held that next year's twelfth month should be long and the eleventh year's new moon should be jiashen; but Jing Dasheng argued the Qian Dao added-time series falls forty-two fen short of the advance limit and placed this year's fifth-month eclipse first contact at the first quarter of the wu hour. Observation placed first contact at the fifth quarter of the wu hour. The Qian Dao added-time series is four hundred fifty fen too slow; projecting by observed time, the Qiandao 12 first-month new moon has already passed jiashen day by four hundred fifty fen. Dasheng now reaffirms jiashen for Qiandao 11 first month and a long twelfth month for Qiandao 10, asking that the bureau's revised tables be adopted." In the fifth month the court ordered calendar officers to settle the matter.

淳熙元年，禮部言：「今歲頒賜曆書，權用《乾道新曆》推算，明年復欲權用《乾道曆》。」詔從之。十一月，詔太史局春官正吳澤推算太陽交食不同，令秘書省敕責之，並罰造曆者。三年，判太史局李繼宗等奏：「令集在局通算曆人重造新曆，今撰成新曆七卷，《推算備草》二卷，校之《紀元》、《統元》、《乾道》諸曆，新曆為密，願賜曆名。」於是詔名《淳熙曆》，四年頒行，令禮部、秘書省參詳以聞。

In the first year of the Chunxi reign, the Ministry of Rites reported that the calendar almanacs issued that year had been calculated provisionally using the Qiandao New Calendar, and that they wished to do the same with the Qiandao Calendar the following year. The emperor approved the request by edict. In the eleventh month, an edict noted that Wu Ze, director of the Spring Office at the Bureau of Astronomy, had miscalculated solar eclipses. The Secretariat was ordered to rebuke him, and the calendar makers were punished as well. In the third year, Li Jizong, acting director of the Bureau of Astronomy, and his colleagues submitted a memorial stating that they had been instructed to gather all competent calendar specialists in the bureau to compile a new calendar. They had finished a seven-volume new calendar and a two-volume Calculation Preparatory Draft. When compared with the Jiyuan, Tongyuan, and Qiandao calendars, the new one proved more accurate, and they requested that an official name be bestowed upon it. The calendar was then officially named the Chunxi Calendar by edict. It was issued in the fourth year, and the Ministry of Rites and the Secretariat were instructed to review it and report their findings.

淳熙四年正月，太史局言：「三年九月望，太陰交食。以《紀元》、《統元》、《乾道》三曆推之，初虧在攢點九刻，食二分及三分已上；以新曆推之，在明刻內食大分空，止在小分百分中二十七。是夜，瞻候月體盛明，雖有雲而不翳，至旦不見虧食，於是可見《紀元》、《統元》、《乾道》三曆不逮新曆之密。今當預期推算淳熙五年曆，蓋舊曆疏遠，新曆未行，請賜新曆名，付下推步。」

In the first month of the fourth year of Chunxi, the Bureau of Astronomy reported that a lunar eclipse had occurred on the full moon of the ninth month of the third year. According to the Jiyuan, Tongyuan, and Qiandao calendars, the eclipse was predicted to begin at the ninth accumulated quarter, with a magnitude of two parts or more and up to three parts or greater; According to the new calendar, the eclipse would occur during daylight with no whole major parts obscured, totaling only twenty-seven hundredths of a minor part. That night observers watched the moon shine brightly; though clouds were present, they did not cover it, and no eclipse was visible even until dawn. This showed that the Jiyuan, Tongyuan, and Qiandao calendars were less accurate than the new calendar. It was now time to calculate the calendar for the fifth year of Chunxi in advance. Because the old calendar was inaccurate and the new calendar had not yet been formally adopted, they requested that the new calendar be given an official name and issued for use in astronomical calculations.

禮部驗得孟邦傑、李繼宗等所定五星行度分數各有異同。繼宗云：六月癸酉，木星在氐宿三度一十九分。邦傑言：夜昏度瞻測得木星在氐宿三度半，半系五十分，雖見月體，而西南方有雲翳之。繼宗云：是月戊寅，木星在氐宿三度四十一分；邦傑言：四望有雲，雖雲間時露月體，所可測者木星在氐宿三度太，太系七十五分。繼宗云：庚辰土星在畢宿三度二十四分，金星在參宿五度六十五分，火星在井宿七度二十七分；邦傑言：五更五點後，測見土星入畢宿二度半，半系五十分，金星入參宿六度半，火星入井宿八度多三分。繼宗云：七月辛丑，太陰在角宿初度七十一分，木星在氐宿五度七十六分；邦傑言：測見昏度太陰入軫宿十六度太，太系七十五分，木星入氐宿六度少，少系二十五分。孝宗曰：「自古曆無不差者，況近世此學不傳，求之草澤，亦難其人。」詔以《淳熙曆》權行頒用一年。

The Ministry of Rites found that Meng Bangjie, Li Jizong, and the others had recorded differing positions and fractional degrees for the five planets. Li Jizong reported that on the guiyou day of the sixth month, Jupiter stood at three degrees and nineteen parts in the Di mansion. Meng Bangjie said that evening observations placed Jupiter at three and a half degrees in the Di mansion, where "half" means fifty parts. The moon was visible, but clouds in the southwest partially obscured the view. Li Jizong reported that on the wuyin day of that month, Jupiter stood at three degrees and forty-one parts in the Di mansion; Meng Bangjie said that on the fourteenth day clouds were present; though the moon occasionally broke through, Jupiter could be measured at three and three-quarters degrees in the Di mansion, where "greater" means seventy-five parts. Li Jizong reported that on gengchen day, Saturn stood at three degrees and twenty-four parts in the Bi mansion, Venus at five degrees and sixty-five parts in the Shen mansion, and Mars at seven degrees and twenty-seven parts in the Jing mansion; Meng Bangjie said that after the fifth watch and fifth point, observations showed Saturn at two and a half degrees in the Bi mansion, Venus at six and a half degrees in the Shen mansion, and Mars at just over eight degrees and three parts in the Jing mansion. Li Jizong reported that on the xinchou day of the seventh month, the moon stood at the first degree and seventy-one parts in the Jiao mansion, and Jupiter at five degrees and seventy-six parts in the Di mansion; Meng Bangjie said that evening observations placed the moon at sixteen and three-quarters degrees in the Zhen mansion and Jupiter at six and a quarter degrees in the Di mansion. Emperor Xiaozong remarked, "No calendar in history has ever been perfectly accurate, and in recent times this science has fallen into neglect. Even searching among common scholars in the countryside, suitable experts are hard to find." By edict, the Chunxi Calendar was provisionally adopted and issued for one year.

五年，金遣使來朝賀會慶節，妄稱其國曆九月庚寅晦為己丑晦。接伴使、檢詳丘辨之，使者辭窮，於是朝廷益重曆事。李繼宗、吳澤言：「今年九月大盡，系三十日，於二十八日早晨度瞻見太陰離東濁高六十餘度，則是太陰東行未到太陽之數。然太陰一晝夜東行十三度余，乙太陰行度較之，又減去二十九日早晨度太陰所行十三度餘，則太陰尚有四十六度以上未行到太陽之數，九月大盡，明矣。其金國九月作小盡，不當見月體；今既見月體，不為晦日。乞九月三十日、十月一日差官驗之。」詔遣禮部郎官呂祖謙。祖謙言：「本朝十月小盡，一日辛卯朔，夜昏度太陰躔在尾宿七度七十分。乙太陰一晝夜平行十三度三十一分，至八日上弦日，太陰計行九十一度餘。按曆法，朔至上弦，太陰平行九十一度三十一分，當在室宿一度太。金國十月大盡，一日庚寅朔，夜昏度太陰約在心宿初度三十一分。太陰一晝夜亦平行十三度三十一分，自朔至本朝八日為金國九日，太陰已行一百四度六十二分，比之本朝十月八日上弦，太陰多行一晝夜之數。今測見太陰在室宿二度，計行九十二度餘，始知本朝十月八日上弦，密于天道。」詔祖謙復測驗。是夜，邦傑用渾天儀法物測驗，太陰在室宿四度，其八日上弦夜所測太陰在室宿二度。按曆法，太陰平行十三度余，行遲行十二度。今所測太陰，比之八日夜又東行十二度，信合天道。

In the fifth year, Jin dispatched envoys to congratulate the court on the Huaiqing Festival, falsely asserting that according to their calendar the last day of the ninth month was gengyin rather than jichou. The receiving envoy and reviewing officer Qiu challenged them in debate until the Jin envoys had no reply. The court thereafter took calendrical affairs even more seriously. Li Jizong and Wu Ze argued that the ninth month was a long month of thirty days. On the morning of the twenty-eighth, observers saw the moon more than sixty degrees above the eastern horizon, showing that the moon's eastward motion had not yet brought it to conjunction with the sun. Since the moon moves eastward more than thirteen degrees each day and night, and subtracting the distance it had already traveled by the morning of the twenty-ninth, more than forty-six degrees of motion still remained before it could reach the sun. This clearly proved the ninth month had thirty days. If Jin's calendar treated the ninth month as a short month, the moon should not have been visible at all; Since the moon was visible, that day could not have been the last day of the month. They requested that officials be sent to verify the matter on the thirtieth day of the ninth month and the first day of the tenth month. By edict, Lü Zuqian, a secretary in the Ministry of Rites, was sent to conduct the verification. Lü Zuqian reported that the Song tenth month was a short month. On the first day, the xinmao new moon, evening observations placed the moon at seven degrees and seventy parts in the Wei mansion. The moon moves thirteen degrees and thirty-one parts per day in uniform motion; by the eighth day, the first quarter, it would have traveled more than ninety-one degrees in total. By calendrical rules, the moon travels ninety-one degrees and thirty-one parts uniformly from new moon to first quarter, which should place it at one and three-quarters degrees in the Shi mansion. Jin's tenth month was a long month. On the first day, the gengyin new moon, evening observations placed the moon at roughly the first degree and thirty-one parts in the Xin mansion. The moon likewise moves thirteen degrees and thirty-one parts per day. From new moon to the Song eighth day—which was Jin's ninth day—the moon had already traveled one hundred degrees and sixty-two parts, a full day and night farther than the Song moon at first quarter on the eighth. Observations now placed the moon at two degrees in the Shi mansion, a total of more than ninety-two degrees traveled, confirming that the Song first quarter on the eighth of the tenth month accorded more closely with celestial reality. By edict, Lü Zuqian was instructed to carry out further observations. That night Meng Bangjie measured with the armillary sphere and official instruments, finding the moon at four degrees in the Shi mansion—the same instrument had placed it at two degrees in the Shi mansion on the first-quarter night of the eighth. By calendrical rules, the moon moves more than thirteen degrees in uniform motion and twelve degrees in slow motion. The moon measured that night had moved twelve degrees farther east than on the eighth, confirming the calendrical prediction.

十年十月，詔：甲辰歲曆字誤，令禮部更印造，頒諸安南國。繼宗、澤及荊大聲削降有差。

In the tenth month of the tenth year, an edict noted errors in the jiayin year's calendar and ordered the Ministry of Rites to reprint and distribute corrected copies to Annam. Li Jizong, Wu Ze, and Jing Dasheng received demotions of varying severity.

十二年九月，成忠郎楊忠輔言：「《淳熙曆》簡陋，于天道不合。今歲三月望，月食三更二點，而曆在二更二點；數虧四分，而曆虧幾五分。四月二十三日，水星據曆當夕伏，而水星方與太白同行東井間，昏見之時，去濁猶十五餘度。七月望前，土星已伏，而曆猶注見。八月未弦，金已過氐矣，而曆猶在亢。此類甚多，而朔差者八年矣。夫守疏敝之曆，不能革舊，其可哉！忠輔于《易》粗窺大衍之旨，創立日法，撰演新曆，不敢以言者，誠懼太史順過饣布非。恃刻漏則水有增損、遲疾，恃渾儀則度有廣狹、斜正。所賴今歲九月之交食在晝，而《淳熙曆》法當在夜，以晝夜辨之，不待紛爭而決矣。輒以忠輔新曆推算，淳熙十二年九月定望日辰退乙未，太陰交食大分四、小分八十五，晨度帶入漸進大分一、小分七；虧初在東北，卯正一刻一十一分，系日出前；食甚在正北，辰初一刻一十分；復滿在西北，辰正初刻，並日出後。其日日出卯正二刻後，與虧初相去不滿一刻。以地形論之，臨安在岳台之南，秋分後晝刻比嶽台差長，日當先曆而出，故知月起虧時，日光已盛，必不見食。以《淳熙曆》推之，九月望夜，月食大分五、小分二十六，帶入漸進大分三、小分四十七；虧初在東北，卯初三刻，系攢點九刻後；食甚在正北，卯正三刻後；復滿在西北，辰正初刻後，並在晝。」禮部乃考其異同，孝宗曰：「日月之行有疏數，故曆久不能無差，大抵月之行速，多是不及，無有過者。可遣台官、禮部官同驗之。」詔遣禮部侍郎顏師魯。其夜戌正二刻，陰雲蔽月，不辨虧食。師魯請詔精于曆學者與太史定曆，孝宗曰：「曆久必差，聞來年月食者二，可俟驗否？」

In the ninth month of the twelfth year, Yang Zhongfu, a Gentleman of Complete Loyalty, declared that the Chunxi Calendar was crude and failed to accord with celestial reality. This year the lunar eclipse on the March full moon occurred at the second point of the third watch, but the calendar predicted the second point of the second watch; The eclipse magnitude was four parts, but the calendar predicted nearly five. On the twenty-third of the fourth month, the calendar said Mercury should have set that evening, yet it was still traveling with Venus through the Jing mansions and remained more than fifteen degrees above the horizon at dusk. Before the July full moon, Saturn had already disappeared from view, but the calendar still listed it as visible. Before the August first quarter, Venus had already passed the Di mansion, but the calendar still placed it in Kang. Such discrepancies were numerous, and new-moon errors had persisted for eight years. How can we cling to a crude, outdated calendar and refuse to reform it! Yang Zhongfu had gained a rough understanding of the Great Expansion principle in the Book of Changes, devised a day-ratio method, and drafted a new calendar, but hesitated to present it for fear that the Bureau of Astronomy would indulge its mistakes and conceal them. Clepsydra measurements were unreliable because water flow varied in speed and volume, and armillary readings were unreliable because degree markings differed in width and alignment. Fortunately, the ninth month's nodal eclipse would occur in daylight, while the Chunxi Calendar predicted a nighttime event—a distinction that would settle the dispute without further argument. Using his new calendar, Yang Zhongfu calculated that the full moon of the ninth month of Chunxi twelve would fall back to the yiwwei day, with a lunar eclipse of four major and eighty-five minor parts and penumbral ingress of one major and seven minor parts at morning position; Initial contact would occur in the northeast at one quarter and eleven parts after maozheng, before sunrise; Maximum eclipse would occur due north at one quarter and ten parts after chenchu; Full restoration would occur in the northwest at the first quarter of chenzheng—all after sunrise. On that day the sun would rise after the second quarter of maozheng, less than one quarter before initial contact. Given Lin'an's location south of the Yue Platform, daylight lasts slightly longer there after the autumn equinox and the sun rises earlier than the calendar predicts. The eclipse would therefore begin after sunrise, when daylight was already too bright to observe it. According to the Chunxi Calendar, the lunar eclipse on the ninth-month full-moon night would have five major and twenty-six minor parts, with penumbral ingress of three major and forty-seven minor parts; Initial contact would occur in the northeast at the third quarter of maochu, after the ninth accumulated quarter; Maximum eclipse would occur due north after the third quarter of maozheng; Full restoration would occur in the northwest after the first quarter of chenzheng—all during daylight. The Ministry of Rites then compared the two predictions. Emperor Xiaozong observed that the sun and moon move at varying speeds, so no calendar could remain error-free forever. The moon generally moved quickly and tended to lag rather than run ahead. He ordered censorial and Ministry of Rites officials to verify the predictions together. By edict, Yan Shilu, Vice Minister of Rites, was sent to conduct the verification. That night at the second quarter of xuzheng, clouds covered the moon and no eclipse could be observed. Yan Shilu asked that experts in calendrical science be summoned to work with the Bureau of Astronomy on a new calendar. Emperor Xiaozong replied that calendars inevitably drifted over time and suggested waiting to verify the two lunar eclipses expected the following year.

十三年，右諫議大夫蔣繼周言，試用民間有知星曆者，遴選提領官，以重其事，如祖宗之制。孝宗曰：「朝士鮮知星曆者，不必專領。」乃詔有通天文曆算者，所在州、軍以聞。八月，布衣皇甫繼明等陳：「今歲九月望，以《淳熙曆》推之，當在十七日，實曆敝也。太史乃注於十六日之下，徇私遷就，以掩其過。請造新曆。」而忠輔乞與曆官劉孝榮及繼明等各具己見，合用曆法，指定今年八月十六日太陰虧食加時早晚、有無帶出、所見分數及節次、生光復滿方面、辰刻、更點同驗之，仰合乾象，折衷疏密。再請今年八月二十九日驗月見東方一事，苟見月餘光，則其日不當以為晦也。又今年九月十六日驗月未盈一事，苟見月體東向之光猶薄，則其日不當為望也。知晦望之差，則朔之差明矣。必使氣之與朔無毫髮之差，始可演造新曆。付禮部議，各具先見，指定太陰虧食分數、方面、辰刻，定驗折衷。詔師魯、繼周監之。既而孝榮差一點，繼明等差二點，忠輔差三點，乃罷遣之。

In the thirteenth year, Jiang Jizhou, Right Remonstrance Grandee, proposed recruiting knowledgeable commoners, appointing supervising officers, and restoring the weight given to calendrical reform under earlier emperors. Emperor Xiaozong replied that few court officials understood astronomy and calendrics, and that a dedicated supervisor was unnecessary. An edict was then issued requiring prefectures and garrisons to report anyone skilled in astronomy and calendrical calculation. In the eighth month, Huangfu Jiming and other common scholars submitted that according to the Chunxi Calendar the ninth-month full moon should fall on the seventeenth, proving the calendar was obsolete. The Bureau of Astronomy had recorded it under the sixteenth instead, bending the dates to conceal their errors. They requested that a new calendar be compiled. Yang Zhongfu then asked that he, the calendar officer Liu Xiaorong, and Huangfu Jiming and the others each submit their predictions using their respective methods, to be jointly verified on the lunar eclipse of the sixteenth of the eighth month—specifically the timing, penumbral effects, eclipse magnitude, phase sequence, directions of emergence and restoration, and hour and watch markings—so that the most accurate method could be determined by comparison with observation. He also asked for verification on the twenty-ninth of the eighth month: if the moon's remaining crescent were visible in the east, that day could not be the last day of the month. He further proposed verifying on the sixteenth of the ninth month whether the moon was truly full: if its eastern face still appeared thin, that day could not be the full moon. If errors in the last day and full moon could be established, new-moon errors would become evident as well. Only when solar terms and new moons were aligned without the slightest discrepancy could a reliable new calendar be compiled. The matter was referred to the Ministry of Rites, which required each party to submit advance predictions of the eclipse magnitude, direction, and timing for verification and comparison. By edict, Yan Shilu and Jiang Jizhou were appointed to supervise the verification. When the verification was complete, Liu Xiaorong's prediction was off by one watch-point, Huangfu Jiming and his colleagues by two, and Yang Zhongfu by three. All were then dismissed.

十四年，國學進士會稽石萬言：

In the fourteenth year, Shi Wanyan of Kuaiji, a presented scholar of the National University, submitted the following memorial:

《淳熙曆》立元非是，氣朔多差，不與天合。按淳熙十四年曆，清明、夏至、處暑、立秋四氣，及正月望、二月十二月下弦、六月八月上弦、十月朔，並差一日。如卦候、盈、虛、沒、滅、五行用事，亦各隨氣朔而差。南渡以來，渾儀草創，不合制度，無圭表以測日景長短，無機漏以定交食加時，設欲考正其差，而太史局官尚如去年測驗太陰虧食，自一更一點還光一分之後，或一點還光二分，或一點還光三分以上，或一點還光三分以下，使更點乍疾乍徐，隨景走弄，以肆欺蔽。若依晉泰始、隋開皇、唐開元課曆故事，取《淳熙曆》與萬所造之曆各推而上之於千百世之上，以求交食，與夫歲、月、日、星辰之著見於經史者為合與否，然後推而下之，以定氣朔，則與前古不合者為差，合者為不差，甚易見也。

The Chunxi Calendar's epoch was wrongly established, its solar terms and new moons were frequently in error, and it failed to accord with the heavens. In the calendar for Chunxi fourteen, Pure Brightness, Summer Solstice, End of Heat, Start of Autumn, the first-month full moon, the second- and twelfth-month last quarters, the sixth- and eighth-month first quarters, and the tenth-month new moon were all one day off. Hexagram periods, full and empty days, setting and extinguishing markers, and the five-phase assignments were similarly affected by the solar-term and new-moon errors. Since the court moved south, the armillary sphere had been crudely built to improper standards, with no gnomon to measure solar shadows and no mechanical clepsydra to fix eclipse times. When corrections were attempted, Bureau officials still manipulated last year's lunar eclipse observations—varying watch-point readings for light restoration by one, two, or three parts—to deceive their superiors. Following the calendar-evaluation precedents of Jin Taishi, Sui Kaihuang, and Tang Kaiyuan, one could project both the Chunxi Calendar and Shi's new calendar backward for thousands of years and compare their eclipse and planetary predictions against classical and historical records, then project forward to fix solar terms and new moons. Discrepancies with antiquity would reveal errors; agreement would confirm accuracy.

然其差謬非獨此耳，冬至日行極南，黃道出赤道二十四度，晝極短，故四十刻，夜極長，故六十刻；夏至日行極北，黃道入赤道二十四度，晝極長，故六十刻，夜極短，故四十刻；春、秋二分，黃、赤二道平而晝夜等，故各五十刻。此地中古今不易之法。至王普重定刻漏，又有南北分野、冬夏晝夜長短三刻之差。今《淳熙曆》皆不然，冬至晝四十刻極短、夜六十刻極長，乃在大雪前二日，所差一氣以上；自冬至之後，晝當漸長，夜當漸短，今過小寒，晝猶四十刻，夜猶六十刻，所差七日有餘；夏至晝六十刻極長、夜四十刻極短，乃在芒種前一日，所差亦一氣以上；自夏至之後，晝當漸短，夜當漸長，今過小暑，晝猶六十刻，夜猶四十刻，所差亦七日有餘；及晝、夜各五十刻，又不在春分、秋分之下。

But the calendar's errors went further. At the winter solstice the sun reaches its southernmost point, the ecliptic lying twenty-four degrees south of the equator, with the shortest daylight of forty quarters and the longest night of sixty quarters; At the summer solstice the sun reaches its northernmost point, the ecliptic twenty-four degrees north of the equator, with the longest daylight of sixty quarters and the shortest night of forty quarters; At the spring and autumn equinoxes, when the ecliptic and equator align, day and night are equal at fifty quarters each. These were the fixed principles of calendrical astronomy, unchanged since antiquity. When Wang Pu recalibrated the clepsydra, he also accounted for a three-quarter difference between northern and southern regions and between summer and winter day lengths. The Chunxi Calendar violated all of these principles: its shortest forty-quarter day and longest sixty-quarter night at the winter solstice occurred two days before Great Snow—more than one solar term off; After the winter solstice daylight should lengthen and nights shorten, yet past Lesser Cold the calendar still showed forty quarters of daylight and sixty of night—an error of more than seven days; Similarly, the longest sixty-quarter day and shortest forty-quarter night at the summer solstice fell one day before Grain in Beard—again off by more than one solar term; After the summer solstice daylight should shorten and nights lengthen, yet past Lesser Heat the calendar still showed sixty quarters of daylight and forty of night—again an error of more than seven days; Even the equal fifty-quarter day and night did not coincide with the spring and autumn equinoxes.

至於日之出入，人視之以為晝夜，有長短，有漸，不可得而急與遲也，急與遲則為變。今日之出入增減一刻，近或五日，遠或三四十日，而一急一遲，與日行常度無一合者。請考正《淳熙曆》法之差，俾之上不違于天時、下不乖於人事。

People measure day and night by the sun's rising and setting, which change gradually in length and cannot suddenly accelerate or decelerate—any such abrupt shift would be an anomaly. Yet the calendar changed sunrise and sunset times by one quarter every five to forty days, alternating abrupt advances and delays that matched none of the sun's regular motion. He requested a thorough correction of the Chunxi Calendar so that it would accord with the heavens above and serve human affairs below.

送秘書省、禮部詳之。

The matter was forwarded to the Secretariat and the Ministry of Rites for review.

皇甫繼明、史元寔、皇甫迨、龐元亨等言：「石萬所撰《五星再聚曆》，乃用一萬三千五百為日法，特竊取唐末《崇元》舊曆而婉其名爾。《淳熙曆》立法乖疏，丙午歲定望則在十七日，太史知其不可，遂注望於十六日下，以掩其過。臣等嘗陳請于太史局官對辨，置局更曆，迄今未行。今考《淳熙歷經》則又差於將來。戊申歲十一月下弦則在二十四日，太史局官必俟頒曆之際，又將妄退於二十三日矣。法不足恃，必假遷就，而朔望二弦，曆法綱紀，苟失其一，則五星盈縮、日月交會、與夫昏旦之中星、晝夜之晷刻，皆不可得而正也。渾儀、景表，壺漏之器，臣等私家無之，是以曆之成書，猶有所待。國朝以來，必假創局而曆始成，請依改造大曆故事，置局更曆，以祛太史局之敝。」事上聞，宰相王淮奏免送後省看詳，孝宗曰：「使秘書省各司同察之，亦免有異同之論。」六月，給事中兼修玉牒官王信亦言更曆事，以為曆法深奧，若非詳加測驗，無以見其疏密。乞令繼明與萬各造來年一歲之曆，取其無差者。詔從之。十二月，進所造曆。淮等奏：「萬等曆日與淳熙十五年曆差二朔，《淳熙曆》十一月下弦在二十四日，恐曆法有差。」孝宗曰：「朔豈可差？朔差則所失多矣。」乃命吏部侍郎章森、秘書丞宋伯嘉參定以聞。

Huangfu Jiming, Shi Yuanshi, Huangfu Dai, Pang Yuanheng, and others reported: "Shi Wan's Five Stars Re-gathering Calendar adopts a day divisor of 13,500—it is nothing but the late-Tang Chongyuan Calendar repackaged under a new name. The Chunxi Calendar was methodologically unsound: for the bingwu year its calculation placed the full moon on the seventeenth, but the Astronomical Bureau, knowing this was wrong, recorded it on the sixteenth to hide the mistake. We had already petitioned for a formal debate with the Bureau and the establishment of a commission to revise the calendar, but nothing had been done. A review of the Chunxi Calendar Classic shows it will miss the mark again in years to come. For the wushen year the last quarter moon falls in November on the twenty-fourth, yet Bureau officials will wait until the calendar is issued and once again arbitrarily set it back a day to the twenty-third. When the methods cannot be trusted and officials resort to ad hoc fixes, the whole system fails: new moon, full moon, and both quarter phases are the pillars of calendrical reckoning, and losing any one of them makes it impossible to fix the five planets' motions, eclipses, meridian stars at dusk and dawn, or the day's shadow divisions. We lack armillary spheres, gnomons, and water clocks in our private possession, so the finished calendar still depends on access to such equipment. Since the dynasty's founding, no calendar had been completed without a dedicated commission; we ask that, following precedent for revising the state calendar, such a bureau be established to remedy the Astronomical Bureau's defects." When the report reached the throne, Grand Counselor Wang Huai asked that it not be sent to the rear secretariat for review. Emperor Xiaozong replied: "Let the Secretariat's various offices examine it together—that will forestall conflicting views." In the sixth month, Supervising Secretary Wang Xin, who also served as compiler of the imperial genealogy, urged calendar reform, arguing that calendrical science is too abstruse to judge without rigorous testing. He proposed that Jiming and Wan each compile a calendar for the coming year, and the one without errors be adopted. The emperor assented. In the twelfth month they submitted their newly compiled calendars. Wang Huai and his colleagues reported: "Wan's calendar differs from the fifteenth Chunxi year edition by two new moons. Since the Chunxi Calendar puts November's last quarter on the twenty-fourth, the methods may be flawed." Xiaozong responded: "How can the new moon be wrong? An error in the new moon throws everything else off." He ordered Vice Minister Zhang Sen and Secretariat Assistant Director Song Bojia to adjudicate the matter and report back.

十五年，禮部言：「萬等所造曆與《淳熙曆》法不同，當以其年六月二日、十月晦日月不應見而見為驗，兼論《淳熙曆》下弦不合在十一月二十四日，是日請遣官監視。」詔禮部侍郎尤袤與森監之。六月二日，森奏：「是夜月明，至一更二點入濁。」十月晦，袤奏：「晨前月見東方。」孝宗問：「諸家孰為疏密？」周必大等奏：「三人各定二十九日早，月體尚存一分，獨忠輔、萬謂既有月體，不應小盡。」孝宗曰：「十一月合朔在申時，是以二十九日尚存月體耳。」

In the fifteenth year the Ministry of Rites argued that Wan's calendar differed methodologically from the Chunxi system and proposed verification through two anomalies: on the second day of the sixth month and the last day of the tenth month the moon had been visible when it should not have been. They also disputed the Chunxi date for November's last quarter and requested officials to observe it on that day. The emperor ordered Vice Minister You Mao and Zhang Sen to supervise the observations. On the sixth month's second day Zhang Sen reported: "The moon was bright that night; at the second mark of the first watch it entered the murk." On the last day of the tenth month You Mao reported: "The moon was visible in the east before dawn." Xiaozong asked: "Which competing calendar is more accurate?" Zhou Bida and others replied: "Three astronomers agreed the moon still showed a sliver at dawn on the twenty-ninth; only Yang Zhongfu and Shi Wan argued that any visible lunar disk ruled out a short month." Xiaozong explained: "Because November's new moon fell in the shen hour, a remnant of the moon could still be seen on the twenty-ninth."

十六年，承節郎趙渙言：「曆象大法及《淳熙曆》，今歲冬至並十二月望，月食皆後天一辰，請遣官測驗。」詔禮部侍郎李巘、秘書省鄧馹等視之。巘等請用太史局渾儀測驗，如乾道故事，差秘書省提舉一員專監之。詔差秘書丞黃艾、校書郎王叔簡。

In the sixteenth year Zhao Huan, a Gentleman for Ceremonial Duty, reported that both the Great Method and the Chunxi Calendar placed this year's winter solstice, December full moon, and lunar eclipse a full double-hour too late, and requested observational verification. The emperor assigned Vice Minister Li Yan, Secretariat official Deng Ri, and others to conduct the observations. Li Yan and his colleagues asked to use the Bureau's armillary sphere, following the Qiandao precedent, with a dedicated Secretariat commissioner assigned to oversee the work. The emperor appointed Secretariat Assistant Director Huang Ai and Proofreader Wang Shujian to the task.

紹熙元年八月，詔太史局更造新曆頒之。二年正月，進《立成》二卷、《紹熙二年七曜細行曆》一卷，賜名《會元》，詔巘序之。

In the eighth month of Shaoxi 1, the emperor ordered the Astronomical Bureau to compile a new calendar and promulgate it. In the second year they submitted two fascicles of Ready Tables and one of the Shaoxi 2 Seven Luminaries Ephemeris; the calendar was named Huiyuan, and Li Yan was ordered to write its preface.

紹熙四年，布衣王孝禮言：「今年十一月冬至，日景表當在十九日壬午，《會元曆》注乃在二十日癸未，系差一日。《崇天曆》癸未日冬至加時在酉初七十六分，《紀元曆》在醜初一刻六十七分，《統元曆》在醜初二刻二分，《會元曆》在醜初一刻二百四十分。迨今八十有七年，常在醜初一刻，不減而反增。《崇天曆》寔天聖二年造，《紀元曆》崇寧五年造，計八十二年。是時測景驗氣，如冬至後天乃減六十七刻半，方與天道協。其後陳得一造《統元曆》，劉孝榮造《乾道》、《淳熙》、《會元》三曆，未嘗測景。苟弗立表測景，莫識其差。乞遣官令太史局以銅表同孝禮測驗。」朝遷雖從之，未暇改作。」

In Shaoxi 4 the commoner Wang Xiaoli reported: "This November's winter solstice should fall on the nineteenth (renchen), but the Huiyuan Calendar places it on the twentieth (guimao)—a full day off. The Chongtian Calendar timed the solstice on guimao at you-initial plus 76 minutes; the Jiyuan at chou-initial, first quarter, 67 minutes; the Tongyuan at chou-initial, second quarter, 2 minutes; and the Huiyuan at chou-initial, first quarter, 240 minutes. Over the intervening eighty-seven years the solstice time should have shifted, yet it has stayed at chou-initial, first quarter—not shrinking as expected but growing instead. The Chongtian Calendar dated from Tiansheng 2 and the Jiyuan from Chongning 5—a span of eighty-two years. When those calendars were tested against shadow measurements and seasonal qi, the solstice had to be adjusted backward by 67½ quarters before it matched the heavens. Later Chen Deyi compiled the Tongyuan Calendar and Liu Xiaorong the Qiandao, Chunxi, and Huiyuan calendars—none of them ever measured gnomon shadows. Without erecting a gnomon and measuring shadows, no one can detect such errors. He asked that officials be sent to have the Astronomical Bureau test alongside Xiaoli using a bronze gnomon." The court agreed in principle but had not yet found time to act on it."

慶元四年，《會元曆》占候多差，日官、草澤互有異同，詔禮部侍郎胡紘充提領官，正字馮履充參定官，監楊忠輔造新曆。右諫議大夫兼侍講姚愈言：「太史局文籍散逸，測驗之器又復不備，幾何而不疏略哉！漢元鳳間，言曆者十有一家，議久不決，考之經籍，驗之帝王錄，然後是非洞見。元和間，以《太初》違天益遠，晦朔失實，使治曆者修之，以無文證驗，雜議蜂饗起，越三年始定。此無他，不得儒者以總其綱，故至於此也。《周官》馮相氏、保章氏志日月星辰之運動，而塚宰實總之。漢初，曆官猶宰屬也。熙甯間，司馬光、沈括皆嘗提舉司天監，故當是時歷數明審，法度嚴密。乞命儒臣常兼提舉，以專其責。」

In Qingyuan 4 the Huiyuan Calendar's predictions proved widely inaccurate, and court astronomers and private scholars disagreed sharply; the emperor appointed Vice Minister Hu Hong to lead the effort, Regular Scribe Feng Lu to adjudicate, and Supervisor Yang Zhongfu to compile a new calendar. Right Remonstrance Grandee Yao Yu, who also served as imperial lecturer, warned: "The Bureau's archives are scattered, its observational instruments incomplete—how can its work be anything but slipshod? Under the Han, eleven schools debated calendrical reform for years without resolution; only after consulting the classics and checking against imperial records did the truth become clear. In the Yuanhe era the Taichu Calendar had drifted so far from the heavens that new and full moons were wrong; reformers were commissioned, but without documentary proof to verify their work, debate swarmed like bees at a feast, and three years passed before a decision was reached. The reason is simple: without Confucian scholars to oversee the enterprise, matters degenerate to this. The Offices of Zhou assigned the Director of Astronomy and the Director of Omens to record celestial motions, but the Grand Minister held overall authority. In early Han, calendrical officials still answered to the chief minister. During the Xining era both Sima Guang and Shen Kuo had served as commissioners of the Directorate of Astronomy, and calendrical work at that time was clear, precise, and rigorously governed. He urged that Confucian officials be permanently assigned concurrent oversight, so that responsibility would be clearly fixed."

五年，監察御史張岩論馮履唱為詖辭，罷去。詔通曆算者所在具名來上。及忠輔曆成，宰臣京鏜上進，賜名《統天》，頒之，凡《歷經》三卷，《八曆冬至考》一卷，《三曆交食考》三卷，《晷景考》一卷，《考古今交食細草》八卷，《盈縮分損益率立成》二卷，《日出入晨昏分立成》一卷，《嶽台日出入晝夜刻》一卷，《赤道內外去極度》一卷，《臨安午中晷景常數》一卷，《禁漏街鼓更點辰刻》一卷，《禁漏五更攢點昏曉中星》一卷，《將來十年氣朔》二卷，《己未庚申二年細行》二卷，總三十二卷。慶元五年七月辛卯朔，《統天曆》推日食，雲陰不見。六年六月乙酉朔，推日食不驗。

In the fifth year Investigating Censor Zhang Yan accused Feng Lu of spreading slander, and Feng was removed from office. The emperor ordered calendrical experts throughout the realm to register their names. When Zhongfu finished his calendar, Grand Counselor Jing Tang presented it to the throne; it was named Tongtian and promulgated. The complete set comprised thirty-two fascicles: three of Calendar Classic; one each on winter solstices across eight calendars, gnomon shadows, sunrise and sunset tables, Yue Terrace day lengths, polar distances, Lin'an noon shadows, clepsydra drum schedules, and dusk-dawn meridian stars; three on eclipses across three calendars; eight of detailed eclipse calculations; two each on waxing-waning rate tables and qi-new-moon projections for the coming decade; and two of ephemerides for the jiwei and gengshen years. On the new moon of the seventh month, Qingyuan 5 (xinmao), the Tongtian Calendar predicted a solar eclipse, but clouds blocked the view. On the new moon of the sixth month, Qingyuan 6 (yiyou), the predicted solar eclipse failed to occur.

嘉泰二年五月甲辰朔，日有食之，詔太史與草澤聚驗於朝，太陽午初一刻起虧，未初刻復滿。《統天曆》先天一辰有半，乃罷楊忠輔，詔草澤通曉曆者應聘修治。

On the new moon of the fifth month, Jiatai 2 (jiachen), a solar eclipse occurred; the emperor ordered court astronomers and private scholars to observe together at court. The sun began to dim at wu-initial, first quarter, and recovered at wei-initial, first quarter. The Tongtian Calendar had been a full hour and a half too early; Yang Zhongfu was dismissed, and the emperor invited skilled private astronomers to apply for the task of revision.

開禧三年，大理評事鮑澣之言：「曆者，天地之大紀，聖人所以觀象明時，倚數立法，以前民用而詔方來者。自黃帝以來，至於秦、漢，六曆具存，其法簡易，同出一術。既久而與天道不相符合，於是《太初》、《三統》之法相繼改作，而推步之術愈見闊疏，是以劉洪，祖沖之之減破鬥分，追求月道，而推測之法始加詳焉。至於李淳風、一行而後，總氣朔而合法，效乾坤而擬數，演算之法始加備焉。故後世之論曆，轉為精密，非過於古人也，蓋積習考驗而得之者審也。試以近法言之：自唐《麟德》、《開元》而至於五代所作者，國初《應天》而至於《紹熙》、《會元》，所更者十二書，無非推求上元開闢為演紀之首，氣朔同元，而七政會于初度。從此推步，以為曆本，未嘗敢輒為截法，而立加減數於其間也。獨石晉天福間，馬重績更造《調元曆》，不復推古上元甲子七曜之會，施於當時，五年輒差，遂不可用，識者咎之。今朝廷自慶元三年以來，測驗氣景，見舊曆後天十一刻，改造新曆，賜名《統天》，進曆未幾，而推測日食已不驗，此猶可也。但其曆書演紀之始，起于唐堯二百餘年，非開闢之端也。氣朔五星，皆立虛加、虛減之數；氣朔積分，乃有泛積、定積之繁。以外算而加朔餘，以距算而減轉率，無復強弱之法，盡廢方程之舊。其餘差漏，不可備言。以是而為術，乃民間之小曆，而非朝廷頒正朔、授民時之書也。漢人以謂曆元不正，故盜賊相續，言雖迂誕，然而曆紀不治，實國家之重事。願詔有司選演撰之官，募通曆之士，置局討論，更造新曆，庶幾並智合議，調治日法，追迎天道，可以行遠。」

In Kaixi 3 Case Reviewer Bao Huan declared: "The calendar is heaven and earth's supreme chronology—the means by which sages read the heavens, mark the seasons, reckon by number, serve the people, and instruct posterity. From the Yellow Emperor through Qin and Han, six calendars survive, all sharing a single simple method. In time they drifted from the heavens, prompting successive reforms of the Taichu and Santong systems; as computational methods grew ever coarser, Liu Hong and Zu Chongzhi refined the du fractions and tracked the lunar path, bringing predictive astronomy to a new level of detail. After Li Chunfeng and the monk Yixing, calendrical science unified qi and new moons into coherent constants and modeled the cosmos in number—computation reached a new completeness. Later calendrical science grew more precise not because moderns surpassed the ancients, but because generations of observation and testing had refined the methods. Consider recent practice alone: from the Tang Lindé and Kaiyuan calendars through Five Dynasties revisions, and from the founding Yingtian to Shaoxi and Huiyuan—twelve calendars in all, each deriving its epoch from the cosmic origin, unifying qi and new moons under one epoch, and aligning the seven luminaries at the initial degree. From that epoch they computed forward as the calendar's foundation, never resorting to arbitrary truncation or ad hoc correction factors. Only Ma Chongji's Tiaoyuan Calendar of Later Jin's Tianfu era broke with tradition by abandoning the ancient jiazi epoch and the seven-luminaries conjunction; applied to the present, it failed within five years, and informed critics condemned it. Since Qingyuan 3 the court had tested seasonal qi and gnomon shadows, found the old calendar eleven quarters too slow, and commissioned the Tongtian Calendar—yet within months its solar-eclipse predictions were already failing. That much might be forgiven. But its epoch begins only two centuries into the reign of Emperor Yao—not at the true cosmic origin. For qi, new moons, and the five planets it relies on fictitious addition and subtraction factors; its accumulated qi and new-moon values require cumbersome general and fixed accumulations; it adds new-moon remainders through outer computation and subtracts rotation rates through distance computation, discarding the old strong-weak method and the traditional simultaneous equations entirely. Its other flaws are too numerous to list. Such methods belong in a farmer's almanac, not in the official calendar by which the court fixes the new moon and sets the seasons for the realm. Han scholars held that a faulty calendar epoch invited rebellion—a fanciful claim, perhaps, yet the governance of calendrical time is truly a matter of national importance. He urged the emperor to appoint drafting officials, recruit calendrical experts, establish a commission, and compile a new calendar—pooling their knowledge to refine the day divisor, realign the system with the heavens, and produce a calendar built to last."

澣之又言：「當楊忠輔演造《統天曆》之時，每與議論曆事，今見《統天曆》舛近，亦私成新曆。誠改新曆，容臣投進，與太史、草澤諸人所著之曆參考之。」七月，澣之又言：「《統天曆》來年閏差，願以諸人所進曆，令秘書省參考頒用。」

Bao Huan added: "When Yang Zhongfu was compiling the Tongtian Calendar I often debated calendrical matters with him; now that the Tongtian has recently failed, I have privately completed a calendar of my own. If the court does revise the calendar, I ask leave to submit mine for comparison with those of the Astronomical Bureau and other private scholars." In the seventh month Bao Huan reported again: "The Tongtian Calendar's intercalation for the coming year is wrong; I ask that the Secretariat review all submitted calendars and adopt the best one."

秘書監兼國史院編修官、實錄院檢討官曾漸言：「改曆，重事也，昔之主監事者，無非道術精微之人，如太史公、洛下閎、劉歆、張衡、杜預、劉焯、李淳風、一行、王樸等，然猶久之不能無差。其餘不過遞相祖述，依約乘除，舍短取長，移疏就密而已，非有卓然特達之見也。一時偶中，即復舛戾。宋朝敝在數改曆法。《統天曆》頒用之初，即已測日食不驗，因仍至今置閏遂差一月，其為當改無疑。然朝廷以一代钜典責之專司，必其人確然著論，破見行之非，服眾多之口，庶幾可見。按乾道、淳熙、慶元，凡三改曆，皆出劉孝榮一人之手，其後遂為楊忠輔所勝。久之，忠輔曆亦不驗，故孝榮安職至今。紹熙以來，王孝禮者數以自陳，每預測驗，或中或不中；李孝節、陳伯祥本皆忠輔之徒；趙達，卜筮之流；石如愚獻其父書，不就測驗晷景，止定月食分數，其術最疏；陳光則並與交食不論，愈無憑依。此數人者，未知孰為可付，故鮑澣之屢以為請。今若降旨開局，不過收聚此數人者，和會其說，使之無爭。來年閏差，其事至重。今年八月，便當頒曆外國，而三數月之間急遽成書，結局推賞，討論未盡，必生詆訾。今劉孝榮、王孝禮、李孝節、陳伯祥所擬改曆，及澣之所進曆，皆已成書，願以眾曆參考，擇其與天道最近且密者頒用，庶幾來年置閏不差。請如先朝故事，搜訪天下精通曆書之人，用沈括所議，以渾儀、浮漏、圭表測驗，每日記錄，積三五年，前後參較，庶幾可傳永久。」

Secretariat Director Zeng Jian, who also served on the National History and Veritable Records academies, observed: "Calendar reform is a grave undertaking. Past supervisors—Sima Qian, Luoxia Hong, Liu Xin, Zhang Heng, Du Yu, Liu Chuo, Li Chunfeng, Yixing, Wang Pu—were masters of their craft, yet even their work drifted over time. Lesser practitioners merely handed methods down to one another, juggling multiplication and division, trimming errors and patching gaps—they lacked any truly original vision. A single lucky prediction was soon followed by fresh failures. The Song's weakness lay in changing its calendar too often. The Tongtian Calendar failed its first solar-eclipse test at promulgation; it has persisted to the present with an intercalation now a full month wrong—revision is beyond question. Yet the court entrusts this epoch-defining task to a specialized office, and the responsible scholar must publish a convincing critique, expose the flaws of the current calendar, and win over the skeptics before reform can proceed. The Qiandao, Chunxi, and Qingyuan reforms—all three—had been Liu Xiaorong's work alone, until Yang Zhongfu displaced him. In time Zhongfu's calendar also failed, and Liu Xiaorong has held his post ever since. Since Shaoxi, Wang Xiaoli has petitioned repeatedly; his predictions sometimes hit and sometimes missed; Li Xiaojie and Chen Boxiang were both Yang Zhongfu's disciples; Zhao Da was a diviner by trade; Shi Ruyu submitted his father's work but refused gnomon testing, calculating only lunar-eclipse magnitudes by the crudest method; Chen Guang ignored eclipses entirely, leaving his work without any solid basis. It is unclear which of these men deserves the commission—which is why Bao Huan keeps pressing his case. If the court now opens a commission, it will merely assemble these same men, reconcile their theories, and suppress their quarrels. The coming year's intercalation error is a matter of the gravest urgency. The calendar must be sent to foreign states this August; yet rushing a finished text in two or three months, handing out rewards before debate is complete, will only invite recrimination. Liu Xiaorong, Wang Xiaoli, Li Xiaojie, Chen Boxiang, and Bao Huan have all completed draft calendars; I ask that these be compared and the one closest to the heavens adopted, so that next year's intercalation may be correct. For a lasting reform, follow former dynasties' precedent: seek out calendrical experts nationwide and, as Shen Kuo proposed, test daily with armillary spheres, floating clepsydras, and gnomons over three to five years, comparing results until a calendar worthy of permanence emerges."

漸又言：「慶元三年以後，氣景比舊曆有差，至四年改造新曆未成時，當頒五年曆，乃差官以測算晷景、氣朔加時辰刻附《會元曆》頒賜。今若頒來年氣朔，既有去年十月以後、今年正月以前所測晷景，已見天道冬至加時分數，來年置閏，比之《統天曆》亦已不同，兼諸所進曆並可參考。請速下本省，集判局官於本省參考，使澣之覆考，以最近之曆推算氣朔頒用。」於是詔漸充提領官，澣之充參定官，草澤精算造者、嘗獻曆者與造《統天曆》者皆延之，於是《開禧》新曆議論始定。詔以戊辰年權附《統天曆》頒之。既而婺州布衣阮泰發獻《渾儀十論》，且言《統天》、《開禧》曆皆差。朝廷令造木渾儀，賜文解罷遣之。

Zeng Jian added: "After Qingyuan 3, seasonal qi and gnomon shadows diverged from the old calendar; when the new calendar was still unfinished in Qingyuan 4, officials were dispatched to measure shadows and compute qi and new-moon times, appending the results to the Huiyuan Calendar for a five-year provisional issue. To issue next year's qi and new moons, the court already has gnomon measurements from last October through this January revealing the true winter-solstice time; next year's intercalation already differs from the Tongtian Calendar, and all submitted calendars can serve as references. I ask that orders go out at once to convene the provincial bureau here, have Bao Huan recheck the results, and promulgate qi and new moons computed from the most accurate available calendar." The emperor then appointed Zeng Jian supervising commissioner and Bao Huan adjudicating commissioner, summoning skilled private astronomers, past submitters, and the Tongtian Calendar's authors—thus the debate over the new Kaixi calendar was settled. An edict provisionally issued the wuchen year's calendar as a supplement to the Tongtian system. Soon after, a commoner from Wuzhou named Ruan Taifa submitted his Ten Discourses on the Armillary Sphere, declaring that both the Tongtian and Kaixi calendars were flawed. The court had a wooden armillary sphere built for him, conferred a civil-service degree, and sent him home.

嘉定三年，鄒淮言曆書差忒，當改造。試太子詹事兼同修國史、實錄院同修撰兼秘書監戴溪等言，請詢漸、澣之造曆故事。詔溪充提領官，澣之充參定官，鄒淮演撰，王孝禮、劉孝榮提督推算官生十有四人，日法用三萬五千四百。四年春，曆成，未及頒行，溪等去國，曆亦隨寢。韓侂胄當國，或謂非所急，無復敢言曆差者，於是《開禧曆》附《統天曆》行於世四十五年。

In Jiading 3, Zou Huai reported that the official calendar was wrong and ought to be revised. Dai Xi—acting grand mentor of the heir apparent, co-compiler of the national history and veritable records, and director of the Secretariat—and others asked that the court review how Zeng Jian and Bao Huan had handled earlier calendar reforms. An edict named Dai Xi supervising commissioner and Bao Huan adjudicating commissioner, put Zou Huai in charge of drafting, and assigned Wang Xiaoli and Liu Xiaorong to supervise fourteen calculator-clerks, with a day divisor of 35,400. In spring of Jiading 4 the calendar was finished, but before it could be promulgated Dai Xi and his colleagues left office, and the project was abandoned. With Han Tuozhou in power, calendar reform was deemed non-urgent, and no official dared raise the issue again; the Kaixi Calendar, appended to the Tongtian system, therefore remained in force for forty-five years.

嘉泰元年，中奉大夫、守秘書監俞豐等請改造新曆。監察御史施康年劾太史局官吳澤、荊大聲、周端友循默屍祿，言災異不及時，詔各降一官。臣僚言：「頒正朔，所以前民用也。比曆書一日之間，吉凶並出，異端並用，如土鬼、暗金兀之類，則添注於凶神之上猶可也，而其首則揭九良之名，其末則出九曜吉凶之法、勘昏行嫁之法，至於《周公出行》、《一百二十歲宮宿圖》，凡閭閻鄙俚之說，無所不有。是豈正風俗、示四夷之道哉！願削不經之論。」從之。二年五月朔，日食，太史以為午正，草澤趙大猷言午初三刻半日食三分。詔著作郎張嗣古監視測驗，大猷言然，曆官乃抵罪。

In Jiatai 1, Yu Feng—senior grandee of palace attendance and acting director of the Secretariat—petitioned for a new calendar. Investigating censor Shi Kangnian impeached Astronomical Bureau officials Wu Ze, Jing Dasheng, and Zhou Duanyou for drawing salaries while neglecting their duties, charging that they had failed to report celestial anomalies promptly; each was demoted one rank. A minister memorialized: "The purpose of promulgating the calendar is to guide the people in their daily affairs. Lately the daily almanac lists lucky and unlucky omens side by side and mixes in superstitious practices—Earth Ghost, Dark Golden Wu, and the like. Adding such notes beneath inauspicious spirits might be tolerable; yet the almanac opens with the Nine Good, closes with methods for the Nine Luminaries' fortunes, rules for choosing wedding dates by inspecting dusk, and even Duke of Zhou's Travels and the Hundred-and-Twenty-Year Palatial Lodging Chart—every folk superstition imaginable is included. How can this rectify public morals or instruct the foreign tributaries! I ask that these unsound entries be removed. The emperor agreed." On the first day of the fifth month in Jiatai 2, a solar eclipse occurred; the grand astrologer placed totality at the wu hour's midpoint, but the private scholar Zhao Dayou said the eclipse reached three-tenths coverage at the third quarter past the wu midpoint. The emperor ordered Compiler Zhang Sigu to oversee verification; Zhao Dayou proved correct, and the calendar officers were punished.

嘉定四年，秘書省著作郎兼權尚左郎丁端祖請考試司天生。十三年，監察御史羅相言：「太史局推測七月朔太陽交食，至是不食。願令與草澤新曆精加討論。」於是澤等各降一官。

In Jiading 4, Ding Duandu—compiler in the Secretariat and acting senior director of the Left Bureau—petitioned for examinations of the Astronomical Bureau's trainee calculators. In Jiading 13, investigating censor Luo Xiang reported: "The Astronomical Bureau had predicted a solar eclipse at the seventh month's new moon, but none occurred. I ask that they confer closely with private scholars who have submitted new calendars." Wu Ze and the others were each demoted one rank.

淳祐四年，兼崇政殿說書韓祥請召山林布衣造新曆。從之。五年，降算造成永祥一官，以元算日食未初三刻，今未正四刻，元算虧八分，今止六分故也。

In Chunyou 4, Han Xiang, lecturer at the Chongzheng Hall, petitioned to summon private scholars from the hills to draft a new calendar. The emperor agreed. In Chunyou 5, calculator Cheng Yongxiang was demoted one rank because his original prediction placed the eclipse at the third quarter of the wei hour, whereas it occurred at the fourth quarter of the wei midpoint, and his predicted magnitude of eight-tenths obscuration proved to be only six-tenths.

八年，朝奉大夫、太府少卿兼尚書左司郎中兼敕令所刪修官尹渙言：「曆者，所以統天地、侔造化，自昔皆擇聖智典司其事。後世急其所當緩，緩其所當急，以為利吾國者，惟錢谷之務；固吾圉者，惟甲兵是圖，至於天文、歷數，一切付之太史局，荒疏乖謬，安心為欺，朝士大夫莫有能詰之者。請召四方之通曆算者至都，使曆官學焉。」

In Chunyou 8, Yin Huan—grandee for closing audience, vice minister of the Court of Imperial Supplies, director of the Left Department, and editor of edicts and statutes—memorialized: "The calendar harmonizes Heaven and Earth and aligns human affairs with cosmic creation; throughout history only the wisest were entrusted with it. Later ages rush what should be slow and neglect what should be urgent, treating only fiscal revenue as the state's profit; and treat frontier defense as a matter of arms alone. Astronomy and calendrical calculation are left entirely to the Astronomical Bureau, which grows slipshod and deceitful while no court scholar dares hold them to account. I ask that skilled astronomers and mathematicians from across the realm be summoned to the capital to teach the bureau's officers."

十一年，殿中侍御史陳垓言：「曆者，天地之大紀，國家之重事。今淳祐十年冬所頒十一年曆，稱成永祥等依《開禧》新曆推算，辛亥歲十二月十七日立春在酉正一刻，今所頒曆乃相師堯等依《淳祐》新曆推算，到壬子歲立春日在申正三刻。質諸前曆，乃差六刻，以此頒行天下，豈不貽笑四方！且許時演撰新曆，將以革舊曆之失。又考驗所食分數，《開禧》舊曆僅差一二刻，而李德卿新曆差六刻二分有奇，與今頒行前後兩曆所載立春氣候分數亦差六刻則同。由此觀之，舊曆差少，未可遽廢；新曆差多，未可輕用。一旦廢舊曆而用新曆，不知何所憑據。請參考推算頒行。」

In Chunyou 11, palace attendant censor Chen Gai wrote: "The calendar is the grand chronicle of Heaven and Earth and a matter of the highest national importance. The calendar for the eleventh year issued in Chunyou 10 winter credits Cheng Yongxiang et al. with computing by the new Kaixi calendar, placing the Establishment of Spring in xinhai on the seventeenth day of the twelfth month at you-zheng one quarter; yet the calendar now issued credits Xiang Shiyao et al. with the new Chunyou calendar, placing the following year's Establishment of Spring at shen-zheng three quarters. Compared with the previous calendar, the discrepancy is six quarters. To promulgate this nationwide is to invite mockery from every quarter! A new calendar was drafted precisely to correct the old calendar's errors. In eclipse trials the old Kaixi calendar was off by only one or two quarters, yet Li Deqing's new calendar missed by six quarters and two-fifths—a discrepancy matching the six-quarter gap in Establishment of Spring between the two issued calendars. Viewed thus, the old calendar errs less and should not be rashly abandoned; while the new calendar errs more and should not be lightly adopted. To discard the old calendar overnight in favor of the new, on what grounds? I ask that both calendars be compared before issuance is decided."

十二年，秘書省言：「太府寺丞張湜同李德卿算造曆書，與譚玉續進曆書頗有牴牾，省官參訂兩曆得失疏密以聞。其一曰：玉訟德卿竊用《崇天曆》日法三約用之。考之《崇天曆》用一萬五百九十為日法，德卿用三千五百三十為日法，玉之言然。其二曰：玉訟積年一億二千二十六萬七千六百四十六，不合曆法。今考之德卿用積年一億以上。其三曰：玉訟壬子年六月，癸丑年二月、六月、九月，丙辰年七月置閏皆差一日。今秘書省檢閱林光世用二家曆法各為推算。其四曰：德卿曆與玉曆壬子年立春、立夏以下十五節氣時刻皆同，雨水、驚蟄以下九節氣各差一刻。其五曰：德卿推壬子年二月乙卯朔日食，帶出已退所見大分八；玉推日食，帶出已退所見大分七。辰當壁宿度，同。其六曰：德卿曆鬥分作三百六十五日二十四分二十八秒，玉曆鬥分作三百六十五日二十四分二十九秒，二曆鬥分僅差一秒。惟二十八秒之法，起于齊祖沖之，而德卿用之。使沖之之法可久，何以歷代增之？玉既指其謬，又多一秒，豈能必其天道合哉！請得商鹈隺推算，合眾長而為一，然後賜名頒行。」十二年，曆成，賜名《會天》，寶祐元年行之，史闕其法。

In Chunyou 12, the Secretariat reported: "Vice director Zhang Shi and Li Deqing had jointly computed a calendar that conflicted substantially with Tan Yu's subsequent submission; provincial officials were asked to compare the two calendars' strengths and weaknesses and report. First, Tan Yu charged that Li Deqing had appropriated the Chongtian Calendar's day divisor, reducing it by one-third. Examination showed the Chongtian Calendar's day divisor to be 10,590 and Li Deqing's 3,530—exactly one-third—confirming Tan Yu's charge. Second, Tan Yu objected that an accumulated epoch of 102,267,646 years violated calendrical convention. Examination confirmed that Li Deqing's epoch exceeded one hundred million years. Third, Tan Yu charged that intercalary months in renchen 6, guichou 2/6/9, and bingchen 7 were each wrong by one day. The Secretariat then had Lin Guangshi compute independently using both calendars. Fourth, the two calendars agreed on the times of fifteen seasonal qi from Establishment of Spring and Summer in renchen year, but differed by one quarter each on the nine qi from Rain Water and Waking of Insects onward. Fifth, Li Deqing's calculation of the yimao-day solar eclipse in renchen 2 predicted a carry-out after totality with visible magnitude of eight-tenths; Tan Yu's calculation predicted seven-tenths visible after carry-out. Both placed the eclipse at the Wall Lodge in the chen hour. Sixth, Li Deqing's tropical year was 365 days 24 parts 28 seconds and Tan Yu's 365 days 24 parts 29 seconds—a difference of only one second in the Dipper fraction. The twenty-eight-second fraction originated with Zu Chongzhi of Qi, yet Li Deqing adopted it. If Zu Chongzhi's value could have endured, why did every dynasty afterward revise it upward? Tan Yu exposed this error, yet his own revision added one second—how can either calendar claim to match the heavens! I ask that the two calendars be reconciled through careful computation, their strengths combined into one system, and only then named and promulgated. In Chunyou 12 the calendar was finished, named the Huitian Calendar, and adopted in Baoyou 1; the history does not record its methods.

鹹淳六年十一月三十日冬至，至後為閏十一月。既已頒曆，浙西安撫司準備差遣臧元震言：

In Xianchun 6 the winter solstice fell on the thirtieth day of the eleventh month, with an intercalary eleventh month placed after the solstice. After the calendar had been promulgated, Zang Yuanzhen, a reserve envoy of the Zhexi Pacification Commission, memorialized:

曆法以章法為重，章法以章歲為重。蓋歷數起於冬至，卦氣起於《中孚》，十九年謂之一章，一章必置七閏，必第七閏在冬至之前，必章歲至、朔同日。故《前漢·志》云：「朔旦冬至，是謂章月。」《後漢·志》云：「至、朔同日，謂之章月。」「積分成閏，閏七而盡，其歲十九，名之曰章。」《唐志》曰：「天數終於九，地數終於十，合二終以紀閏餘。」章法之不可廢也若此。

In calendrical reckoning the rule cycle is paramount, and within it the rule year is paramount. Calendrical reckoning begins at the winter solstice and hexagram qi at Zhongfu; nineteen years form one rule cycle, which must contain seven intercalations—with the seventh placed before the winter solstice and the rule year's solstice and new moon coinciding on the same day. The Former Han Treatise states: "A new moon at dawn on the winter solstice is called the rule month." The Later Han Treatise says: "When solstice and new moon fall on the same day, it is called the rule month." Accumulated fractional days become intercalations; seven intercalations complete the cycle of nineteen years, which is called a rule." The Tang Treatise says: "Heaven's number ends at nine and Earth's at ten; combining these two limits records the intercalary remainder." Such is the indispensability of the rule cycle.

今所頒庚午歲曆，乃以前十一月三十日為冬至，又以冬至後為閏十一月，莫知其故。蓋庚午之閏，與每歲閏月不同；庚午之冬至，與每歲之冬至又不同。蓋自淳祐壬子數至鹹淳庚午，凡十九年，是為章歲，其十一月是為章月。以十九年七閏推之，則閏月當在冬至之前，不當在冬至之後。以至、朔同日論之，則冬至當在十一月初一日，不當在三十日。今以冬至在前十一月三十日，則是章歲至、朔不同日矣。若以閏月在冬至後，則是十九年之內止有六閏，又欠一閏。且一章計六千八百四十日，於內加七閏月，除小盡，積日六千九百四十日或六千九百三十九日，約止有一日。今自淳祐十一年辛亥章歲十一月初一日章月冬至後起算，十九年至鹹淳六年庚午章歲十一月初一日當為冬至，方管六千八百四十日。今算造官以閏月在十一月三十日冬至之後，則此一章止有六閏，更加六閏除小盡外，實積止六千九百十二日，比之前後章歲之數，實欠二十八日。曆法之差，莫甚於此。況天正冬至乃曆之始，必自冬至後積三年餘分，而後可以置第一閏。今庚午年章歲丙寅日申初三刻冬至，去第二日丁卯僅有四分日之一，且未正日，安得遽有餘分？未有餘分，安得遽有閏月？則是後一章之始不可推算，其謬可知矣。今欲改之，有簡而易行之說。蓋曆法有平朔，有經朔，有定朔。一大一小，此平朔也；兩大兩小，此經朔也；三大三小，此定朔也。今正以定朔之說，則當以前十一月大為閏十月小，以閏十一月小為十一月大，則丙寅日冬至即可為十一月初一，以閏十一月初一之丁卯為十一月初二日，庶幾遞趲下一日置閏，十一月二十九日丁未始為大盡。然則冬至既在十一月初一，則至、朔同日矣；閏月既在至節前，則十九年七閏矣。此昔人所謂晦節無定，由時消息，上合履端之始，下得歸余於終，正謂此也。

The gengwu-year calendar now issued places the winter solstice on the thirtieth day of the preceding eleventh month and the intercalary eleventh month after the solstice—for reasons no one can explain. The gengwu intercalation differs from an ordinary leap month; and the gengwu winter solstice differs from an ordinary winter solstice as well. Counting from Chunyou renchen to Xianchun gengwu spans nineteen years—the rule year—and its eleventh month is the rule month. By the rule of seven intercalations in nineteen years, the leap month should precede the winter solstice, not follow it. If solstice and new moon must coincide, the winter solstice should fall on the first day of the eleventh month, not the thirtieth. Placing the winter solstice on the thirtieth of the preceding eleventh month means the rule year's solstice and new moon no longer coincide. If the leap month follows the winter solstice, the nineteen-year cycle contains only six intercalations—one short of the required seven. One rule cycle totals 6,840 days; adding seven intercalary months and accounting for short months yields 6,939 or 6,940 days—a discrepancy of roughly one day. Counting from the xinhai rule year of Chunyou 11—whose rule month began on the first day of the eleventh month after the winter solstice—nineteen years should bring us to gengwu of Xianchun 6 with the winter solstice on the first day of the eleventh month, yielding exactly 6,840 days. But the calendar officers placed the leap month after the solstice on the thirtieth of the eleventh month, giving this cycle only six intercalations; the actual total is 6,912 days—twenty-eight days short of the preceding and following rule years. No calendrical error could be graver than this. Moreover, the celestial winter solstice marks the calendar's starting point; fractional days must accumulate for more than three years after the solstice before the first intercalation may be placed. In the gengwu rule year the solstice falls on bingyin day at the third quarter of the shen hour—barely one-quarter day before dingmao on the following day, and not yet the formal new-moon day—how can fractional remainder have already accumulated? Without accumulated fractional days, how can a leap month suddenly be justified? The start of the next rule cycle becomes incalculable—the error is self-evident. To correct this, I propose a simple and practical remedy. Calendrical reckoning distinguishes mean new moons, canonical new moons, and true new moons. One long month and one short—this is the mean-new-moon pattern; two long and two short—the canonical-new-moon pattern; three long and three short—the true-new-moon pattern. Applying true-new-moon principles, the preceding large eleventh month should become a short intercalary tenth month, and the short intercalary eleventh month a long eleventh month; then the bingyin-day solstice becomes the first of the eleventh month, dingmao the second, with intercalation shifted back one day so that dingwei on the twenty-ninth becomes the month's last full day. With the solstice on the first of the eleventh month, solstice and new moon would coincide; and with the leap month before the solstice, the nineteen-year cycle would contain its full seven intercalations. This is what the ancients meant by 'the dark month and seasonal nodes are not fixed but follow the flux of time—aligning the year's beginning above and settling the remainder at the cycle's end.'

夫曆久未有不差，差則未有不改者。後漢元和初曆差，亦是十九年不得七閏，曆雖已頒，亦改正之。顧今何靳於改之哉！元震謂某儒者，豈欲與曆官較勝負？既知其失，安得默而不言邪！

No calendar in long use has ever remained free of error, and no error has ever gone uncorrected. In early Yuanhe of the Later Han a calendar error likewise left the nineteen-year cycle one intercalation short; though the calendar had already been promulgated, it was corrected. Why then hesitate to correct it now! Yuanzhen declared: "As a mere scholar, do I seek to best the calendar officers in debate? Having discovered the error, how could I remain silent!"

於是朝廷下之有司，遣官偕元震與太史局辨正，而太史之詞窮，元震轉一官，判太史局鄧宗文、譚玉等各降官有差。因更造曆，六年，曆成，詔試禮部尚書馮夢得序之；七年，頒行，即《成天曆》也。

The court referred the matter to the responsible offices, dispatching officials to adjudicate jointly with Yuanzhen and the Astronomical Bureau; the bureau's officers were confounded, Yuanzhen was promoted one rank, and bureau director Deng Zongwen, Tan Yu, and others were demoted by varying degrees. A new calendar was commissioned; in Xianchun 6 it was completed, and Acting Minister of Rites Feng Mengde was ordered to write its preface; and in Xianchun 7 it was promulgated as the Chengtian Calendar.

德祐之後，陸秀夫等擁立益王，走海上，命禮部侍郎鄧光薦與蜀人楊某等作曆，賜名《本天曆》，今亡。

After Deyou, Lu Xiufu and his allies enthroned the Prince of Yi and fled to sea, commissioning Vice Minister of Rites Deng Guangjian and a certain Yang of Shu to compose a calendar named the Bentian Calendar—it is now lost.

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