1
後漢書志第二律歷中賈逵論歷永元論歷延光論歷漢安論歷熹平論歷論月食
Treatise Two of the Book of Later Han: Music and Calendars, Part Two — Treatises on Jia Kui's calendar memorials and the debates under Yongyuan, Yanguang, Han'an, and Xiping, together with the discussion of lunar eclipses
2
自太初元年始用三統曆，施行百有餘年，歷稍後天，朔先*[於]*歷，朔或在晦，月*[或朔]*見。 考其行，日有退無進，月有進無退。 建武八年中，太僕朱浮﹑太中大夫許淑等數上書，言歷*[朔]*不正，宜當改更。 時分度覺差尚微，上以天下初定，未遑考正。 至永平五年，官歷署七月十六日*[月]*食。 待詔楊岑見時月食多先歷，即縮用筭上為日，*[因]*上言「月當十五日食，官歷不中」。 詔書令岑普*[候]*，與官*[歷]*課。 起七月，盡十一月，弦望凡五，官歷皆失，岑皆中。 庚寅，詔*[書]*令岑署弦望月食官，復令待詔張盛﹑景防﹑鮑鄴等以四分法與岑課。 歲餘，盛等所中多岑六事。 十二年十一月丙子，詔書令盛﹑防代岑署弦望月食加時。 四分之術，始頗施行。 是時盛﹑防等未能分明歷元，綜校分度，故但用其弦望而已。
The Triple Concordance calendar had been in force since the first year of Taichu, but after more than a century it no longer kept pace with the sky: conjunctions came ahead of the official reckoning, sometimes landing on the final day of the month, and the thin crescent could appear when the calendar still called it new moon. When one traced their apparent motions, the sun seemed only to fall behind the tabulated motion, never to gain on it, while the moon always ran ahead and never fell short. In Jianwu 8, Chamberlain Zhu Fu and Palace Counselor Xu Shu kept petitioning the throne: the official calendar’s lunations were wrong and needed reform. The errors were still small on the scale of fractional degrees, and with the empire only newly settled the emperor set calendar revision aside for the moment. In Yongping 5 the state calendar predicted a lunar eclipse on the sixteenth of the seventh month. The expectancy-holder Yang Cen observed that lunar eclipses of the time mostly occurred earlier than the calendar said; he therefore shortened the reckoning on the counting-rods by one day for the sun and accordingly memorialized, saying, "The moon ought to be eclipsed on the fifteenth; the official calendar misses it." The emperor ordered Yang Cen and Pu to make observations and pit their results against the bureau’s calendar. From the seventh month to the eleventh there were five key lunations; the official calendar missed every prediction while Cen’s matched observation each time. On gengyin day Cen was put in charge of predicting quarters, full moons, and eclipses, while Zhang Sheng, Jing Fang, Bao Ye, and fellow expectancy-holders were told to test their quarter-remainder system against him. After a little over a year, Zhang Sheng’s group had six more successful predictions than Cen. In the eleventh month of Jianwu 12, on bingzi day, Zhang Sheng and Jing Fang were appointed to succeed Cen in timing quarters, full moons, and eclipses. For the first time the quarter-remainder system saw substantial official use. Zhang Sheng and Jing Fang had not yet fixed the epoch or reconciled the degree scale, so in practice they relied on the quarter-remainder method only for first and last quarters and full moons.
3
先是，九年，太史待詔董萌上言歷不正，事下三公﹑太常知歷者雜議，訖十年四月，無能分明據者。 至元和二年，太初失天益遠，日﹑月宿度相覺浸多，而候者皆知冬至之日日在斗二十一度，未至牽牛五度，而以為牽牛中星，* (從) **[後]*天四分日之三，晦朔弦望差天一日，宿差五度。 章帝知其謬錯，以問史官，雖知不合，而不能易，故召治歷編欣﹑李梵等綜校其狀[1]。 二月甲寅，遂下詔曰：「朕聞古先聖王，先天而天不違，後天而奉天時。 河圖曰：『赤九會昌，十世以光，十一以興。』 又曰：『九名之世，帝行德，封刻政。』 朕以不德，奉承大業，夙夜祗畏，不敢荒寧。 予末小子，托在於數終，曷以續興，崇弘祖宗，拯濟元元？
Earlier, in year 9, expectancy-holder Dong Meng of the Grand Astrologer’s office had argued the calendar was faulty; the case went to the Three Dukes and the court experts, but by the fourth month of year 10 no one had produced a conclusive argument. By Yuanhe 2 the Taichu calendar was badly out of step: the sun and moon no longer matched their tabulated positions, and observers agreed that at winter solstice the sun sat at 21° in the Dipper—well short of mid-Ox—though the old system still pretended it was the Ox’s central star The clause continues in the next line: the calendar ran three quarters of a day behind the sky. —running three quarters of a day slow; conjunctions and phases were one day off, and stellar positions by five degrees. Emperor Zhang saw the error and pressed the astrologers; though they admitted the mismatch, they could not fix it unaided, so he called in Bian Xin, Li Fan, and other calendar specialists for a full review. On jiayin in the second month he promulgated an edict: "The ancient sages moved ahead of heaven without offending it, and when they followed heaven they honored its seasons. The Hetu says: ‘When the ninth red cycle gathers, ten reigns take their lustre, and the eleventh brings renewal.’" It also reads: ‘Under the Nine Names the ruler’s virtue is carved into policy for all to see.’" With little virtue of my own I have inherited the throne; night and day I hold myself in awe and dare not grow complacent. I am but the last and least, placed at a cycle’s close—how am I to restore our fortunes, honor our forebears, and bring relief to the people?
4
尚書琁璣鈐曰：『述堯世，放唐文。』 帝命驗曰：『*[順]*堯考德，* (顧) **[題]*期立象。』 且三﹑五步驟，優劣殊軌，況乎頑陋，無以克堪，雖欲從之，末由也已。 每見圖書，中心恧焉。 閒者以來，政治不得，陰陽不和，災異不息，癘疫之氣，流傷於牛，農本不播。 夫庶征休咎，五事之應，咸在朕躬，信有闕矣，將何以補之？ 書曰：『惟先假王正厥事。』 又曰：『歲二月，東巡狩，至岱宗，柴，望秩於山川。 遂覲東後，□時月正日。』 祖堯岱宗，同律度量，考在璣衡，以正歷象，庶乎有益。 春秋保干圖曰：『三百年斗歷改憲。』 史官用太初鄧平術，有餘分一，在三百年之域，行度轉差，浸以謬錯。 琁璣不正，文象不稽。 冬至之日日在斗二十* (二) **[一]*度，而歷以為牽牛中星。 先立春一日，則四分數之立春日也。 以折獄斷大刑，於氣已迕； 用望平和隨時之義，蓋亦遠矣。 今改行四分，以遵於堯，以順孔聖奉天之文。 冀百君子越有民，同心敬授，*[儻]*獲咸* (喜) **[熙]*，以明予祖之遺功。」 於是四分施行。 而欣﹑梵猶以為元首十一月當先大，欲以合耦弦望，命有常日，而十九歲不得七閏，晦朔失實。 行之未期，章帝復發聖思，考之經讖，使左中郎將賈逵問治歷者□承﹑李崇﹑太尉屬梁鮪﹑司徒*[掾]*嚴勖﹑太子舍人徐震﹑鉅鹿公乘蘇統及欣﹑梵等十人。
The Classic’s Xuanji qian fragment says: ‘He carries forward Yao’s world and opens Tang’s civilizing pattern.’" The Diming yan adds: ‘Align with Yao and weigh virtue— The line resumes: ‘set the appointed season and fix the cosmic emblems.’ —set the due season and fix the cosmic emblems.’ Even the triple and quintuple calendrical schemes wander along unequal paths—what hope have I, dull as I am? I long to follow them yet find no footing." Whenever those charts meet my eye I am ashamed at how far we fall short." Of late government has faltered, yin and yang fall out of tune, omens pile up, murrain stalks the cattle, and the farmers cannot sow." The Hong fan warnings attach to the ruler’s conduct; the fault is surely mine—how shall I set it right?" The Documents say: ‘Ancient kings took heaven’s mandate as their guide and straightened every policy.’" It adds: ‘In the second month he toured east to Mount Tai, offered the great firewood sacrifice, and ordered worship at every peak and stream.’" Then, having greeted the guardian of the east, he aligned seasons, months, and days— —like Yao at Mount Tai, harmonizing pitch pipes and weights, checking heaven with the transit instrument until calendar and sky agreed. Perhaps then we may gain something." The apocryphal Baogan tu declares: ‘Every three hundred years the Dipper-based calendar must be reformed.’" Our clerks still follow Deng Ping’s Taichu arithmetic—its fractional remainder falls inside the three-hundred-year cycle—yet planetary motion drifts until every degree is wrong." The polar sight-line is skewed, so the patterned sky goes unverified." At winter solstice the sun stands at twenty— Read together with the lines before and after, the sun lies at twenty-one Dipper degrees. —one degree in the Dipper, while the calendar still calls it mid-Ox." By the quarter-remainder count, ‘Beginning of Spring’ arrives one day ahead of the solar term— —so trying capital cases by that calendar fights the seasonal qi; and the ideal of tempering justice to the seasons slips ever farther away." We therefore adopt the quarter-remainder calendar to walk in Yao’s footsteps and heed the model Confucius praised for serving heaven." May every minister join in reverently handing down the seasons—if only we might share in universal— Treat this gloss as the cited text: universal brightness. —brightness—and make plain the legacy of my imperial forebears." With that edict the quarter-remainder calendar entered official use. Bian Xin and Li Fan still insisted the civil year should begin with a long eleventh month so quarters and full moons would pair neatly on fixed days—but over nineteen years you still cannot squeeze in seven intercalary months, so conjunctions drifted from fact. The reform had not run a full term when Emperor Zhang, probing scriptural forecasts anew, commissioned Left General of the Gentlemen Jia Kui to canvass ten experts—including a specialist whose name is damaged in the text, Cheng, Li Chong, Liang Wei the Grand Commandant’s aide, Clerk Yan Xu of the Minister of Education, Heir’s attendant Xu Zhen, Su Tong of Julu, plus Bian Xin and Li Fan.
5
以為月當先小，據春秋經書朔不書晦者，朔必有明晦，不朔必在其月也。 即先大，則一月再朔，後月無朔，是明不可必。 梵等以為當先大，無文正驗，取欲諧耦十六日*[望]*，月朓□，晦當滅而已。 又晦與合同時，不得異日。 又上知欣﹑梵穴見，□毋拘歷已班，天元始起之月* (常) **[當]*小。 定，後年曆數遂正。 永元中，復令史官以九道法候弦望，驗無有差跌。 逵論集狀，後之議者，用得折衷，故詳錄焉。
They argued the short month must precede the long: the Annals record first days of months, not month-ends; when the classic gives a conjunction it implies a visible old moon, and if no conjunction is recorded the month still holds together. Put the long month first and you get two new moons in one month and none in the next—so the lunation cannot hold. Li Fan’s party preferred the long month first without textual warrant, merely to align pairs with the fifteenth full moon; they waved away late risings of the moon and treated month-end darkness as enough. Month-end darkness and true conjunction fall together—they cannot be split across two days. The emperor also saw how narrow Xin and Fan were; once the published calendar began its epoch, the opening month of heaven’s cycle— Read this note as the cited text: the epoch-opening month ought to be short. —must be taken as the short month. Once that ruling stood, the calendar tallies fell into line in the years that followed. Under Yongyuan the bureau was told to time quarters and full moons with the nine-path lunar theory; observation showed no misses. Jia Kui’s dossier became the touchstone for later debates, so it is set down here at length.
6
逵論曰：「太初歷冬至日在牽牛初者，牽牛中星也。 古黃帝﹑夏﹑殷﹑周﹑魯冬至日在建星，建星即今斗星也。 太初歷斗二十六度三百八十五分，牽牛八度。
Kui stated in discussion: "In the Taichu calendar the winter solstice sun at the beginning of the Ox is the middle star of the Ox. The classical calendars of Huang Di, Xia, Shang, Zhou, and Lu put winter solstice at Establishment Star—which is our Dipper asterism. Taichu assigns twenty-six degrees plus 385 fractional parts to the Dipper and eight degrees to the Ox.
7
案行事史官注，冬﹑夏至日常不及太初歷五度，冬至日在斗* (一) **[二]*十一度四分度之一。 石氏星經曰：『黃道規牽牛初直斗二十度，去極二十五度。』 於赤道，斗二十一度也。 四分法與行事候注天度相應。 尚書考靈曜『斗二十二度，無餘分，冬至在牽牛所起』。 又編欣等據今日所在*[未至]*牽牛中星五度，於斗二十一度四分一，與考靈曜相近，即以明事。 元和二年八月，詔書曰『石不可離』，令兩候，上得筭多者。 太史令玄等候元和二年至永元元年，五歲中課日行及冬* (夏) *至斗* (一) **[二]*十一度四分一，合古歷建星考靈曜日所起，其星閒距度皆如石氏故事。 他術以為冬至日在牽牛初者，自此遂黜也。」
Working registers show solstice positions consistently five degrees shy of Taichu; at winter solstice the sun stood in the Dipper— This editorial insertion fills out twenty-one Dipper degrees. —at twenty-one and one-quarter degrees." Shi Shen’s canon says the ecliptic’s Ox anchor lines up with 20° of the Dipper and lies twenty-five degrees from the pole." Measured on the equator that is twenty-one Dipper degrees." The quarter-remainder scale matches what observing clerks actually record." The Kaoling yao gloss reads: ‘The Dipper spans twenty-two whole degrees; winter solstice marks the Ox’s beginning.’" Bian Xin notes that the sun today still falls five degrees short of mid-Ox—at 21¼° in the Dipper—which tracks the Kaoling yao and settles the case." In Yuanhe 2, month 8, an edict declared ‘the benchmark must not be split,’ pitting two observing teams—the one with superior rod reckonings prevailed." Grand Astrologer Xuan’s office tracked the sun from Yuanhe 2 through Yongyuan 1; over five years they computed daily solar motion and winter— Ignore this stray character so that the passage reads winter solstice at twenty-one and one-quarter Dipper degrees. —solstice at— This completes the phrase ‘twenty-one and one-quarter degrees’ at winter solstice. —twenty-one and one-quarter Dipper degrees, matching Establishment Star and the Kaoling yao origin, with every inter-star span as Shi Shen prescribed." Every scheme that pinned winter solstice to the Ox’s leading edge was discarded from that moment."
8
故求度數，取合日月星辰，有異世之術。 太初歷不能下通於今，新歷不能上得漢元。 一家曆法必在三百年之閒。 故讖文曰『三百年斗歷改憲』。 漢興，當用太初而不改，下至太初元年百二歲乃改。 故其前有先晦一日合朔，下至成﹑哀，以二日為朔，故合朔多在晦，此其明效也。」
Seeking degree measures that fit sun, moon, and stars has always meant borrowing methods suited to different eras. Taichu cannot be stretched intact to today’s sky; the new system cannot be wound backward to Han’s founding year. Any one calendrical school stays reliable only within about three centuries. Hence the apocryphon: ‘After three hundred years reform the Dipper-based calendar.’" At Han's rise, Taichu should have been adopted at once; instead the reform waited until the first year of Taichu, one hundred two years later. Early Han conjunctions came a day before month-end; by Cheng and Ai people counted two-day lunations, so true conjunctions clustered on month-ends—the clearest proof of drift."
9
逵論曰：「臣前上傅安等用黃道度日月弦望多近。 史官一以赤道度之，不與日月同，於今歷弦望至差一日以上，輒奏以為變，至以為日□縮退行。 於黃道，自得行度，不為變。 願請太史官日月宿簿及星度課，與待詔星象考校。 奏可。 臣謹案：前對言冬至日去極一百一十五度，夏至日去極六十七度，春秋分日去極九十一度。
Jia Kui continued: "I once reported that Fu An’s team measured sun and moon along the ecliptic and matched quarters and full moons far better. Yet the bureau insists on equatorial coordinates, which diverge from the bodies’ true paths: quarters and full moons slip by a full day, so clerks cry anomaly—or even claim the sun is shrinking and retreating— —whereas on the ecliptic the motions are steady and nothing ominous occurs. I ask to compare the Grand Astrologer’s solar and lunar lodge logs—along with every star-degree computation—with the expectancy-holders who chart constellations. The throne approved the request. As I reported earlier: winter solstice lies 115° from the pole, summer solstice 67°, and the equinoxes 91°.
10
洪範『日月之行，則有冬夏』。 五紀論『日月循黃道，南至牽牛，北至東井，率日日行一度，月行十三度十九分度七』也。 今史官一以赤道為度，不與日月行同，其斗﹑牽牛﹑*[東井]*﹑輿鬼，赤道得十五，而黃道得十三度半； 行東壁﹑奎﹑婁﹑軫﹑角﹑亢，赤道* (十) **[七]*度，黃道八度； 或月行多而日月相去反少，謂之日□。 案黃道值牽牛，出赤道南二十五度，其直東井﹑輿鬼，出赤道北*[二十]*五度。 赤道者為中天，去極俱九十度，非日月道，而以遙准度日月，失其實行故也。 以今太史官候注考元和二年九月已來月行牽牛﹑東井四十九事，無行十一度者； 行婁﹑角三十七事，無行十五六度者，如安言。 問典星待詔姚崇﹑井畢等十二人，皆曰『星圖有規法，日月實從黃道，官無其器，不知施行』。 案甘露二年大司農中丞耿壽昌奏，以圖儀度日月行，考驗天運狀，日月行至牽牛﹑東井，日過*[一]*度，月行十五度，至婁﹑角，日行一度，月行十三度，赤道使然，此前世所共知也。 如言黃道有驗，合天，日無前□，弦望不差一日，比用赤道密近，宜施用。 上中多臣校。」 案逵論，永元四年也。 至十五年七月甲辰，詔書造太史黃道銅儀，以角為十三度，亢十，氐十六，房五，心五，尾十八，箕十，斗二十四四分度之一，牽牛七，須女十一，虛十，危十六，營室十八，東壁十，奎十七，婁十二，胃十五，昴十二，畢十六，觜三，參八，東井三十，輿鬼四，柳十四，星七，張十七，翼十九，軫十八，凡三百六十五度四分度之一。 冬至日在斗十九度四分度之一。 史官以* (郭) **[部]*日月行，參弦望，雖密近而不為注日。 儀，黃道與度轉運，難以候，是以少循其事。
The Hong fan teaches that the sun and moon trace winter and summer arcs—" —and the Wuji lun adds that both bodies ride the ecliptic between Ox and Eastern Well, the sun about one degree per day and the moon thirteen and seven nineteenths." Today’s clerks measure everything on the celestial equator, not along the true paths: for the Dipper, Ox, Eastern Well, and Ghosts the equator spans fifteen degrees while the ecliptic needs only thirteen and a half— —crossing Eastern Wall, Legs, Bond, Chariot-Frame, Horn, and Neck the equator records— Together with the preceding line this yields seventeen equatorial degrees versus eight on the ecliptic. —seventeen degrees while the ecliptic shows eight— —so sometimes the moon moves far yet looks close to the sun, which clerks mislabel as solar ‘shortening.’" Where the ecliptic crosses the Ox it sits twenty-five degrees south of the equator; opposite Eastern Well and Ghosts it lies twenty-five degrees north— The equator marks mid-sky at ninety degrees from the pole, but it is not where the luminaries travel; gauging them by that circle misses their real paths. Since Yuanhe 2, month 9, the bureau logged forty-nine lunar passages through Ox and Eastern Well—not once did the moon move eleven degrees along the equator. Across thirty-seven passages near Bond and Horn there were no fifteen- or sixteen-degree equatorial arcs, exactly as Fu An claimed. Twelve star-chart specialists including Yao Chong and Jing Bi were polled; each admitted that charts follow compass geometry, that sun and moon ride the ecliptic, and that without proper instruments the bureau could not put the theory into practice. Geng Shouchang’s Ganlu 2 memorial already used armillary diagrams to track the luminaries: near Ox and Eastern Well the sun gains more than a degree daily and the moon about fifteen; near Bond and Horn the solar rate is one degree and the lunar thirteen—effects of measuring on the equator, as earlier ages understood. If the ecliptic checks out against the sky—no spurious solar slowdown and quarters within a day—the method beats equatorial reckoning and deserves adoption. The throne’s findings matched most of my cross-checks. This concludes Jia Kui’s memorial from Yongyuan 4. On jiachen in the seventh month of Yongyuan 15 an edict commissioned a bronze ecliptic armillary: Horn 13°, Neck 10°, Root 16°, Room 5°, Heart 5°, Tail 18°, Basket 10°, Dipper 24¼°, Ox 7°, Girl 11°, Void 10°, Rooftop 16°, House 18°, Eastern Wall 10°, Legs 17°, Bond 12°, Stomach 15°, Hairy Head 12°, Net 16°, Beaks 3°, Orion 8°, Well 30°, Ghost 4°, Willow 14°, Stars 7°, Extended Net 17°, Wings 19°, Chariot 18°—365¼° in total. Winter solstice was fixed at 19¼° in the Dipper. The clerks classified— Scribes wrote the cited text where the cited text (‘division’) belongs. They paired solar and lunar motions by lodge divisions with quarters and full moons; though numerically tight, they never tied those arcs to specific calendar days. Because the ecliptic ring and degree circle spin together, observations were awkward and the bureau rarely followed the procedure.
11
逵論曰：「又今史官推合朔、弦、望、月食加時，率多不中，在於不知月行□疾意。 永平中，詔書令故太史待詔張隆以四分法署弦、望、月食加時。 隆言能用易九、六、七、八* (支) **[爻]*知月行多少。 今案隆所署多失。 臣使隆逆推前手所署，不應，或異日，不中天乃益遠，至十餘度。 梵、統以史官候注考校，月行當有□疾，不必在牽牛、東井、婁、角之閒，又非所謂朓、側匿，乃由月所行道有遠近出入所生，率一月移故所疾處三度，九歲九道一復，凡九章，百七十一歲，復十一月合朔旦冬至，合春秋、三統九道終數，可以知合朔、弦、望、月食加時。 據官注天度為分率，以其術法上考建武以來月食凡三十八事，差密近，有益，* (宣) **[宜]*課試上。」
Jia Kui added: “The bureau still mis-times conjunctions, quarters, full moons, and eclipses because it does not grasp how the moon speeds up and slows down. Under Yongping the emperor told former expectancy-holder Zhang Long to set quarters, full moons, and eclipse times with the quarter-remainder system. Zhang Long claimed he could read the moon’s range from the Classic’s nine, six, seven, eight lines— Drop the stray character the cited text before the cited text. —using the hexagram lines—to gauge how far the moon moves each night. Review shows Zhang Long’s predictions were usually wrong. I had him back-calculate earlier postings: they failed to match, sometimes fell on the wrong day, and missed transit by more than ten degrees. Li Fan and Su Tong checked the observation logs: lunar speed varies; the extremes need not occur only near Ox, Eastern Well, Bond, or Horn, nor are they mere ‘late first quarters’ or hidden crescents—they come from the moon’s varying distance along its path, drifting about three degrees per month in the sector of fastest motion. Nine years recycle the nine paths; after nine cycles and 171 years you realign an eleventh-month winter-solstice conjunction, matching the Spring and Autumn annals and Triple Concordance nine-path periods—enough to pin conjunctions, quarters, full moons, and eclipses. Using bureau degree logs as denominators, his method retrodicts thirty-eight eclipses since Jianwu with tight agreement—clear gains— Read this gloss as the cited text ‘ought to.’ —so it ought to be tried before the throne.”
12
案史官舊有九道術，廢而不修。 熹平中，故治歷郎梁國宗整上九道術，詔書下太史，以參舊術，相應。 部太子舍人馮恂課校，恂亦復作九道術，增損其分，與整術並校，差為近。 太史令揚上以恂術參弦、望。 然而加時猶復先後天，遠則十餘度[2]。
The bureau once maintained the nine-path lunar theory but let it lapse. During Xiping, former calendar director Zong Zheng of Liang resubmitted the nine-path theory; the throne ordered the Grand Astrologer to cross-check it against older tables and the results aligned. Heir-apparent attendant Feng Xun reviewed both versions, drafted his own nine-path adjustments, and compared them with Zong Zheng’s—the discrepancies shrank. Grand Astrologer Yang blended Feng Xun’s tables into quarter- and full-moon predictions. Even then predicted moments still ran ahead or behind the sky—sometimes by more than ten degrees.
13
永元十四年，待詔太史霍融上言：「官漏刻率九日增減一刻，不與天相應，或時差至二刻半，不如夏歷密。」 詔書下太常，令史官與融以儀校天，課度遠近。
In Yongyuan 14, expectancy Huo Rong reported that the bureau water-clock drifted a notch every nine days out of step with the sky—sometimes by two and a half notches—and fell short of the Xia calendar’s precision. The emperor ordered the Grand Master of Ceremonies to have the bureau join Huo Rong in checking the sky with instruments and measuring solar distance.
14
太史令舒、承、梵等對：「案官所施漏法令甲第六常符漏品，孝宣皇帝三年十二月乙酉下，建武十年二月壬午詔書施行。 漏刻以日長短為數，率日南北二度四分而增減一刻。 一氣俱十五日，日去極各有多少。 今官漏率九日移一刻，不隨日進退。 夏歷漏*[刻]*隨日南北為長短，密近於官漏，分明可施行。」 其年十一月甲寅，詔曰：「告司徒、司空：漏所以節時分，定□明。
Grand Astrologers Shu, Cheng, Fan, and others replied: ‘Examining the clepsydra statute applied by the office—section six of the ordinances on perpetual tally clepsydras—issued on yiyou in the twelfth month of the third year of Emperor Xuan, implemented by edict on renwu in the second month of the tenth Jianwu year. Clepsydra scaling tracks seasonal day length: each 2.4° shift of the sun toward north or south adjusts the water-clock by one mark. Every fortnightly period spans fifteen days, and the sun’s polar distance changes within it. Today’s clock advances one notch every nine days regardless of how the sun moves—a mismatch. The Xia-style clepsydra lengthens and shortens with the sun’s seasonal swing; it tracks the official clock closely enough to deploy. ’ In that year’s eleventh month on jiayin day an edict said: ‘Inform the Ministers of Education and Works: the clepsydra is what regulates seasonal divisions and fixes day and night clarity.
15
□明長短，起於日去極遠近，日道周*[圜]*，不可以計率分，當據儀度，下參晷景。 今官漏以計率分□明，九日增減一刻，違失其實，至為疏數以耦法。 太史待詔霍融上言，不與天相應。 太常史官運儀下水，官漏失天者至三刻。 以晷景為刻，少所違失，密近有驗。 今下晷景漏刻四十八箭，立成斧官府當用者，計吏到，班予四十八箭。」 文多，故魁取二十四氣日所在，並黃道去極、晷景、漏刻、□明中星刻於下。
Day length depends on how far the sun stands from the pole; the solar path is a great circle, so you cannot simply average it—set the clepsydra from armillary measures and gnomon readings. The present clock forces daylight into crude nine-day steps—wrong in fact and artificially averaged just to keep arithmetic tidy. Expectancy Huo Rong showed it fails against the sky. When the ministry ran trials against instruments, the palace clepsydra could miss true time by three marks. Basing notches on sundial shadows stays closer to observation with fewer errors. Issue the forty-eight seasonal tube-settings derived from gnomon shadows; give copies to every department when fiscal clerks arrive. Because the full text is long, the excerpt below lists only each qi-day’s solar longitude, ecliptic polar distance, shadow length, clepsydra setting, day-night ratio, and culminating star.
16
昔太初歷之興也，發謀於元封，啟定於* (天) **[元]*鳳，積* (百) *三十年，是非乃審。 及用四分，亦於建武，施於元和，訖於永元，七十餘年，然後儀式備立，司候有准。 天事幽微，若此其難也。 中興以來，圖讖漏洩，而考靈曜、命歷序皆有甲寅元。 其所起在四分庚申元後百一十四歲，朔差□二日。 學士修之於草澤，信向以為得正。 及太初歷以後* (大) **[天]*為疾，而修之者云「百四十四歲而太歲超一* (表) **[辰]*，百七十一歲當棄朔余六十三，中余千一百九十七，乃可常行」。 自太初元年至永平十一年，百七十一，當去分而不去，故令益有疏闊。 此二家常挾其術，庶幾施行，每有訟者，百寮會議，腢儒騁思，論之有方，益於多聞識之，故詳錄焉。
The Taichu reform was planned under Yuanfeng and settled when— Correct the character to the cited text: the reform was confirmed in Yuanfeng. —the Yuanfeng era—after thirty years— Drop the stray the cited text before the cited text. —thirty years—the verdict on Taichu finally settled. The quarter-remainder calendar launched under Guangwu, took effect in Yuanhe, and stabilized through Yongyuan—only after seventy years did instruments and observing protocols settle. Even celestial mathematics proves this subtle and painstaking. Since Guangwu’s revival, apocrypha have circulated openly; Kaoling yao and Mingli xu both advocate the jiayin epoch. That epoch begins 114 years after the quarter system’s gengshen origin and differs by about two days on mean lunations. Rustic scholars tinkered with it in exile and convinced themselves they had the true calendar. Advocates of Taichu argue heaven runs fast— Read the cited text to yield ‘heaven is fast.’ —‘heaven’—is fast, and those who cultivate it say, ‘In one hundred forty-four years the Grand Year leaps one— Read the cited text: Jupiter skips one celestial stem. —hour-branch—after 171 years discard 63 from the lunation fraction and 1197 from the middle qi remainder—only then can the cycle run smoothly.’ Between the first year of Taichu and Yongping 11 lies one full 171-year cycle where the prescribed remainder should have been dropped but was not, so the calendar drifted further. Both schools cling to their arithmetic and lobby for adoption; whenever debate arises the court convenes scholars whose arguments repay study—hence the full record here.
17
安帝延光二年，中謁者但誦言當用甲寅元，河南梁豐言當復用太初。 尚書郎張衡、周興皆能歷，數難誦、豐，或不對，或言失誤。 衡、興參案儀注* (者) *，考往校今，以為九道法最密。 詔書下公卿詳議。 太尉愷等上侍中施延等議：「太初過天，日一度，弦望失正，月以晦見西方，食不與天相應； 元和改從四分，四分雖密於太初，復不正，皆不可用。 甲寅元與天相應，合圖讖，可施行。」
In Yanguang 2, intern Dan Song urged the jiayin epoch while Henan’s Liang Feng called for a return to Taichu. Zhang Heng and Zhou Xing, both skilled in astronomy, cross-examined Dan Song and Liang Feng, who either fell silent or misspoke. They collated the armillary regulations— Omit the stray character the cited text. —then tested past records against the sky—and judged the nine-path theory tightest. The emperor ordered the high ministers to debate the proposals. Grand Commandant Liu Kai relayed Shi Yan’s faction: Taichu runs one solar degree ahead daily, corrupts lunations, lets the moon appear in the west at month-end, and misstates eclipses. The Yuanhe reform adopted quarter-remainder math—better than Taichu yet still flawed—so neither scheme suffices. Only the jiayin epoch tracks heaven and matches the apocrypha—it should be adopted.’
18
博士黃廣、大行令任僉議，如九道。 河南尹祉、太子舍人李泓等四十人議：「即用甲寅元，當除元命苞天地開闢獲麟中百一十四歲，推閏月六直其日，或朔、晦、弦、望，二十四氣宿度不相應者非一。 用九道為朔，月有比三大二小，皆疏遠。 元和變曆，以應保干圖『三百歲斗歷改憲』之文。 四分歷本起圖讖，最得其正，不宜易。」 愷等八十四人議，宜從太初。 尚書令忠上奏：「諸從太初者，皆無他效驗，徒以世宗攘夷廓境，享國久長為辭。 或云孝章改四分，災異卒甚，未有善應。 臣伏惟聖王興起，各異正朔，以通三統。 漢祖受命，因秦之紀，十月為年首，閏常在歲後。 不稽先代，違於帝典。 太宗遵修，三階以平，黃龍以至，刑犴以錯，五者以備[3]。 哀平之際，同承太初，而妖孽累仍，痾禍非一。
Huang Guang and Ren Qian sided with the nine-path lunar scheme. Henan Governor Zhi, Heir-apparent Supernumerary Li Hong, and forty others discussed: ‘If one immediately uses the jiayin yuan, one must eliminate one hundred fourteen years within the Yuanming bao’s Heaven-Earth opening and unicorn capture; intercalary months pushed six straight to their days—or new moon, last day, quarters, full moons—cases where the twenty-four qi-lodge degrees do not correspond are not one. Nine-path lunations produce awkward strings of three long and two short months—too unwieldy. Emperor Zhang’s Yuanhe reform invoked the Baogan tu promise of changing the Dipper calendar every three hundred years. The quarter system springs from canonical forecasts and already hits the mark—it should not be tossed aside." Liu Kai and eighty-four scholars urged sticking with Taichu. Director Zhong Chen argued that Taichu’s defenders cite only Emperor Wu’s military glory and long reign—no astronomical proof. Others blame Emperor Zhang’s quarter reform for a rash of omens—yet saw no blessing. Each sage founder adopts his own calendar stance to complete the triple succession. Gaozu inherited Qin’s calendar—year beginning in the tenth month—always tucking intercalations after year-end. That ignored antiquity and offended the classical canon. Emperor Wen restored proper seasons so harmony returned—omens aligned, justice balanced, and the Hong fan ‘five affairs’ fell into place. Under Emperors Ai and Ping everyone still used Taichu—yet omens and disasters multiplied.
19
議者不以成數相參，考真求實，而泛采妄說，歸福太初，致咎四分。 太初歷觿賢所立，是非已定，永平不審，復革其弦望。 四分有謬，不可施行。 元和鳳鳥不當應歷而翔集。 遠嘉前造，則* (喪) **[表]*其休； 近譏後改，則隱其福。 漏見曲論，未可為是。 臣輒復重難衡、興，以為五紀論推步行度，當時比諸術為近，然猶未稽於古。 及向子歆欲以合春秋，橫斷年數，損夏益周，考之表紀，差謬數百。 兩歷相課，六千一百五十六歲，而太初多一日。 冬至日直鬥，而雲在牽牛。 □闊不可復用，昭然如此。 史官所共見，非獨衡、興。 前以為九道密近，今議者以為有闕，及甲寅元復多違失，皆未可取正。 昔仲尼順假馬之名，以崇君之義。 況天之歷數，不可任疑從虛，以非易是。」 上納其言，遂*[寢]*改歷事。
Critics cherry-pick omens instead of testing arithmetic: they credit Taichu for every blessing and blame every ill on the quarter reform. Taichu was fixed by experts; Yongping meddled again with lunations without understanding. The quarter scheme remains flawed and must not replace Taichu wholesale. The Yuanhe phoenix omens did not match what the quarter calendar predicted. Praising distant antiquity— Read the cited text ‘display’ rather than the cited text. —they trumpet ancient blessings; yet they smother every blessing visible after reforms. Such biased anecdotes cannot settle truth. I pressed Zhang Heng and Zhou Xing again: their Wuji lun solar theory beat rival schemes but still missed ancient benchmarks. Liu Xin bent chronology to fit the Annals—trimming Xia and padding Zhou—and hundreds of years slip away. Cross-comparing the systems over 6,156 years leaves Taichu one day long. Observation puts winter solstice in the Dipper while Taichu insists on the Ox. The mismatch is gaping—Taichu cannot be revived—and plain as day. Every astrologer sees it—not only Zhang Heng and Zhou Xing. Yesterday’s nine-path consensus looks flawed today, and the jiayin epoch still disagrees with observation—none of these schemes earns final adoption. Confucius once accepted the courtiers’ euphemism ‘horse’ to uphold his ruler’s dignity. All the more must celestial reckoning reject groundless doubt—error cannot displace truth. The throne adopted his advice and tabled further calendar change.
20
順帝漢安二年，尚書侍郎邊韶上言：「世微於數虧，道盛於得常。 數虧則物衰，得常則國昌。 孝武皇帝攄發聖思，因元封七年十一月甲子朔旦冬至，乃詔太史令司馬遷、治歷鄧平等更建太初，改元易朔，行夏之正，干鑿度八十*[一]*分之四十三為日法。 設清台之候，驗六異，課效觕密，太初為最。 其後劉歆研機極深，驗之春秋，參以易道，以河圖帝覽嬉、雒書* (甄) **[干]*曜度推廣九道，百七十一歲進退六十三分，百四十四歲一超次，與天相應，少有闕謬。 從太初至永平十一年，百七十*[一]*歲，進退余分六十三，治歷者不知處之。 推得十二度弦望不效，挾廢術者得竄其說。 至* (永) **[元]*和二年，小終之數寖過，余分稍增，月不用晦朔而先見。 孝章皇帝以保干圖『三百年斗歷改憲』，就用四分。 以太白復樞甲子為癸亥，引天從筭，耦之目前。 更以庚申為元，既無明文；
In Han’an 2, Attendant Clerk Bian Shao memorialized Emperor Shun that dynastic fortune weakens when calendrical constants drift but strengthens when they hold steady. Botched arithmetic breeds decay; faithful cycles sustain the realm. Emperor Wu, inspired by the winter-solstice new moon in the seventh year of Yuanfeng, commissioned Sima Qian and Deng Ping to forge the Taichu calendar, restoring the Xia calendar head, adopting Deng Ping's 81-part day divisor from the apocryphal scale. Observatory trials sorted rival schemes; Taichu proved tightest. Liu Xin then harmonized the Annals and Changes with the Hetu Luoshu literature— Read the cited text ‘stem’ for the gloss the cited text. —to extend the nine-path theory: 171-year remainder cycles, 144-year station leaps—tracking the sky closely. Between the first year of Taichu and Yongping 11 spans one 171-year cycle whose sixty-three-part remainder experts mishandled. Their lunar phases missed; hacks with obsolete formulas slipped through. Come— Drop the erroneous the cited text so the reign reads Yuanhe. —Yuanhe 2—small-cycle fractions overflowed; thin moons showed ahead of recorded conjunctions. Emperor Zhang invoked the three-hundred-year oracle and switched to quarter-remainder reckoning. He twisted chronology so Venus cycles and rod arithmetic aligned with the moment. Promoting gengshen as epoch had no explicit charter.
21
托之於獲麟之歲，又不與感精符單閼之歲同。 史官相代，因成習疑，少能鉤深致遠； 案弦望足以知之。」 詔書下三公、百官雜議。 太史令虞恭、治歷宗欣等議：「建歷之本，必先立元，元正然後定日法，法定然後度周天以定分至。 三者有程，則歷可成也。 四分歷仲紀之元，起於孝文皇帝後元三年，歲在庚辰。 上四十五歲，歲在乙未，則漢興元年也。 又上二百七十五歲，歲在庚申，則孔子獲麟。 二百七十六萬歲，尋之上行，復得庚申。 歲歲相承，從下尋上，其執不誤。 此四分歷元明文圖讖所著也。 太初元年歲在丁丑，上極其元，當在庚戌，而曰丙子，言百四十四歲超一辰，凡九百九十三超，歲有空行八十二周有奇，乃得丙子。 案歲所超，於天元十一月甲子朔旦冬至，日月俱超。 日行一度，積三百六十五度四分度一而周天一□，名曰歲。 歲從一辰，日不得空周天，則歲無由超辰。 案百七十*[一]*歲二蔀一章，小余六十三，自然之數也。 夫數出於杪曶，以成毫犛，亮犛積累，以成分寸。
Anchoring it to the unicorn year clashes with the Ganqing fu’s Shan’an cycle. Successive clerks grew complacent and rarely probed the depths. Lunar phases alone expose the problem. The emperor ordered the Three Dukes and full bureaucracy to debate. Yu Gong and Zong Xin insisted: "fix the epoch first, then the day denominator, then measure the full circuit for solstices and seasonal nodes. Those three steps yield a viable calendar. Quarter-remainder’s Zhongji epoch starts at Wen’s houyuan 3—gengchen. Count back 45 years to yiwei—Han’s inception. Another 275 years reaches gengshen—the unicorn hunt. Extend the chain two million seventy-six thousand years and you still land on gengshen. Year-links stack cleanly from bottom to top. Apocrypha spell out this quarter-remainder epoch plainly. The first year of Taichu is dingchou; pushing its epoch yields gengxu, yet Liu Xin forced bingzi by stacking 993 fourteen-four-year leaps across eighty-two empty rounds. At his theoretical origin both luminaries ‘jump’ together—an absurdity. The sun moves one degree daily; 365¼ degrees complete one tropical year. Years cannot skip stems while the sun still completes its orbit. One hundred seventy-one years—two cycles—remainder sixty-three—is inherent to the cycle. Quantity grows from micro-instants into measurable rods.
22
兩儀既定，日月始離。 初行生分，積分成度。 日行一度，一歲而周，故為術者，各生度法，或以九百四十，或以八十一。 法有細觕，以生兩科，其歸一也。 日法者，日之所行分也。 日垂令明，行有常節，日法所該，通遠無已，損益毫犛，差以千里。 自此言之，數無緣得有虧棄之意也。 今欲飾平之失，斷法垂分，恐傷大道。 以步日月行度，終數不同，四章更不得朔余一。 雖言九道去課進退，恐不足以補其闕。 且課歷之法，晦朔變弦，以月食天驗，昭著莫大焉。 今以去六十三分之法為歷，驗章和元年以來日變二十事[4]，月食二十八事，與四分歷更失，定課相除，四分尚得多，而又便近。
After cosmos settles, sun and moon diverge in motion. Motion yields fractions; fractions stack into degrees. Technicians define degree denominators—940 or 81—to capture solar motion. Different divisors look coarse or fine yet aim at one truth. Day denominator partitions the sun’s daily arc. Solar motion is steady; tiny divisor errors explode across the sky. Arithmetic cannot sanction arbitrary discard. Papering over Deng Ping’s flaws by slicing denominators risks breaking fundamental proportion. Walking solar and lunar arcs with mismatched endpoints breaks the four-cycle conjunction remainder. Nine-path tinkering alone cannot plug every remainder gap. Final proof is lunar eclipse prediction—nothing clearer. Testing the minus-sixty-three scheme since Zhanghe 1 against twenty solar anomalies and twenty-eight eclipses still leaves quarter-remainder ahead.
23
孝章皇帝歷度審正，圖儀晷漏，與天相應，不可復尚。 文曜鉤曰：『高辛受命，重黎說文。 唐堯即位，羲和立* (禪) **[渾]*。 夏後制德，昆吾列神。 成周改號，萇弘分官。』 運斗樞曰：『常佔有經，世史所明。』 洪範五紀論曰：『民閒亦有黃帝諸歷，不如史官記之明也。』 自古及今，聖帝明王，莫不取言於羲和、常佔之官，定精微於晷儀，正觿疑，秘藏中書，改行四分之原。 及光武皇帝數下詔書，草創其端，孝明皇帝課校其實，孝章皇帝宣行其法。 君更三聖，年曆數十，信而征之，舉而行之。 其元則上統開闢，其數則復古四分。 宜如甲寅詔書故事。」 奏可。
Emperor Zhang’s instruments and clocks matched heaven. The apocryphal Wen yao gou praises High Xin’s ministers defining ritual. At Yao’s accession Xi and He set up— Read the cited text ‘armillary sphere.’ —the spherical heaven instrument. Under Xia’s kings ministers ordered the stellar officers. Western Zhou renamed the dynasty; Chang Hong organized celestial posts. The Yundou shu insists steady observation sustains the canon. The Wuji lun notes folk calendars trail official logs. Every sage ruler relied on Xi He–style observers, refined gnomons, archived revisions, and anchored reforms in quarter-remainder roots. Guangwu began the restoration calendar; Ming verified it; Zhang enacted it. Three sovereigns refined the system across decades—worthy of faith and deployment. Its epoch reaches cosmic genesis while its fractions revive classical quarter math. Follow the jiayin-edict precedent. The throne approved.
24
靈帝熹平四年，五官郎中馮光、沛相上計掾陳晃言：「歷元不正，故妖民叛寇益州，盜賊相續為*[害]*。 歷*[當]*用甲寅為元而用庚申，圖緯無以庚*[申]*為元者。 近秦所用代周之元。 太史治治歷中郭香、劉固意造妄說，乞* (與) *本庚申元經緯* (有) *明*[文]*，受虛欺重誅。」 乙卯，詔書下三府，與儒林明道者詳議，務得道真。 以腢臣會司徒府議[5]。
In Xiping 4 Feng Guang and Chen Huang blamed Yi’s rebellion on a faulty calendar epoch. They insist on jiayin yet practice gengshen—no apocryphon supports gengshen epochs. That was Qin’s Zhou-replacement epoch. Calendar clerks Guo Xiang and Liu Gu forged claims—seeking— Read as ‘present’ rather than ‘with.’ —evidence for the gengshen epoch— Drop the stray the cited text. —clear proof—or face severe punishment for fraud." On yimao day the court ordered a symposium to settle doctrine. Officials convened in the Minister of Education’s hall.
25
議郎蔡邕議，以為：
Cai Yong argued:
26
歷數精微，去聖久遠，得失更迭，術* (術) *無常是。 *[漢興]** (以) *承秦，歷用顓頊，元用乙卯[6]。 百有二歲，孝武皇帝始改正朔，歷用太初，元用丁丑，行之百八十九歲。 孝章皇帝改從四分，元用庚申。 今光、晃各以庚申為非，甲寅為是。 案曆法，黃帝、顓頊、夏、殷、周、魯，凡六家，各自有元。 光、晃所據，則殷歷元也。 他元雖不明於圖讖，各*[自一]*家*[之]*術，皆當有效於* (其) *當時。* (黃) **[武]*帝始用太初丁丑之元，* (有) *六家紛錯，爭訟是非。
Calendrical math has drifted for centuries—methods— Remove the duplicated the cited text. —lack eternal correctness. At Han’s foundation— Read the cited text ‘inherit.’ —Han inherited Qin’s Zhuanxu calendar with yi-mao epoch. Two years into Han → Wu adopted Taichu with dingchou epoch for 189 years. Emperor Zhang adopted quarter-remainder with gengshen epoch. Feng Guang and Chen Huang condemn gengshen and praise jiayin. Six classical schools each declare unique epochs. Their jiayin preference tracks the Yin scheme. Other epochs, though thinly attested in apocrypha, worked in their day— —their— —for contemporaries. Read the cited text: ‘Emperor Wu.’ —Emperor Wu was first to adopt the Taichu calendar with the dingchou epoch— Delete the stray the cited text before the cited text. —the six calendrical schools wrangled over which was correct.
27
太史令張壽王挾甲寅元以非漢歷，雜候清台，課在下第，卒以疏闊，連見劾奏，太初效驗，無所漏失。 是則雖非圖讖之元，而有效於前者也。 及用四分以來，考之行度，密於太初，是又新元*[有]*效於今者也。 延光元年，中謁者但誦亦非四分庚申，上言當用命歷序甲寅元。 公卿百寮參議正處，竟不施行。 且三光之行，□速進退，不必若一。 術家以筭追而求之，取合於當時而已。 故有古今之術。 今*[術]*之不能上通於古，亦猶古術之不能下通於今也。 元命苞、干鑿度皆以為開闢至獲麟二百七十六萬歲； 及命歷序積獲麟至漢，起庚* (子) **[午]*蔀之二十三歲，竟己酉、戊子及丁卯蔀六十九歲，合為二百七十五歲。 漢元年歲在乙未，上至獲麟則歲在庚申。 推此以上，上極開闢，則* (不) **[元]*在庚申。 讖雖無文，其數見存。 而光、晃以為開闢至獲麟二百七十五萬九千八百八十六歲，獲麟至漢百六十* (二) **[一]*歲，轉差少一百一十四歲。 雲當滿足，則上違干鑿度、元命苞，中使獲麟不得在哀公十四年，下不及命歷序獲麟*[至]*漢相去四蔀年數，與奏記譜注不相應。
Zhang Shouwang’s jiayin scheme failed Qing Terrace trials while Taichu held. Taichu lacked apocryphal sanction yet matched observation. Quarter-remainder beats Taichu in modern tests. Yanguang 1 saw Dan Song attacking gengshen and urging Mingli xu’s jiayin epoch. High ministers debated yet enacted nothing. Sun, moon, and stars vary in speed. Astronomers tune models to fit their own era—that is all. Hence ancient and modern schools coexist. Modern tables cannot be wound back to antiquity any more than antique tables forecast today’s sky. Apocrypha agree on roughly 2.76 million years from cosmos to the unicorn year. The Mingli xu chain from unicorn to Han begins at geng— Correct the cited text to the cited text to restore gengwu. —linking sexagenary cycles to total 275 years down to Han’s foundation. Han began in yiwei; counting back lands on gengshen for the unicorn. Carry the sequence to cosmic genesis and— Read the cited text so the clause declares ‘the origin lies at gengshen.’ —the universal epoch is gengshen. Even without narrative glosses the arithmetic still checks out. Feng Guang and Chen Huang shorten creation-to-unicorn and unicorn-to-Han spans— —‘two’ completes ‘sixty-two’ incorrectly written— —yielding 161 years from unicorn to Han—114 years shy of canonical totals. Their totals clash with every standard chronology.
28
當今歷正月癸亥朔，光、晃以為乙丑朔。 乙丑之與癸亥，無題勒款識可與觿共別者，須以弦望晦朔光魄虧滿可得而見者，考其符驗。 而光、晃歷以考靈曜*[為本]*，二十八宿度數及冬至日所在，與今史官甘、石舊文錯異，不可考校； 以今渾天圖儀檢天文，亦不合於考靈曜。 光、晃誠能自依其術，更造望儀，以追天度，遠有驗於圖書，近有效於三光，可以易奪甘、石，窮服諸術者，實宜用之。 難問光、晃，但言圖讖，所言不服。 元和二年二月甲寅制書曰：『朕聞古先聖王，先天而天不違，後天而奉天時。 史官用太初鄧平術，冬至之日，日在斗二十* (二) **[一]*度，而歷以為牽牛中星，先立春一日，則四分數之立春也，而以折獄斷大刑，於氣已迕，用望平和，蓋亦遠矣。 今改行四分，以遵於堯，以順孔聖奉天之文。』 是始用四分歷庚申元之詔也。 深引河雒圖讖以為符驗，非史官私意獨所興構。 而光、晃以為*[香]*、固意造妄說，違反經文，謬之甚者。 昔堯命羲和歷象日月星辰，舜□時月正日，湯、武革命，治歷明時，可謂正矣，且猶遇水遭旱，戒以『蠻夷猾夏，寇賊奸宄』。 而光、晃以為陰陽不和，奸臣盜賊，皆元之咎，誠非其理。 元和二年乃用庚申，至今九十二歲，而光、晃言秦所用代周之元，不知從秦來，漢三易元，不常庚申。 光、晃區區信用所學，亦妄虛無造欺語之愆。 至於改朔易元，往者壽王之術已課不效，但誦之議不用，元和詔書文備義著，非腢臣議者所能變易。
Official New Year opens guihai; they demand yichou. Only lunar phases can adjudicate guihai versus yichou. Their lodge spans and solstice clash with Gan De and Shi Shen. Modern armillary checks refute Kaoling yao too. Let them build instruments that beat observation—then they may replace classical star tables. Under cross-examination they offered only apocrypha. Emperor Zhang’s Yuanhe rescript repeats the sage-king formula before attacking Taichu. Taichu puts solstice at twenty— —‘two’ joins ‘twenty’ as twenty-one Dipper degrees— —restating Zhang’s indictment of Taichu misalignment. Henceforth practice quarter-remainder reckoning as Yao and Confucius demand. That edict inaugurated gengshen quarter-remainder rule. It rested on canonical charts, not bureau caprice. They accuse Guo Xiang and Liu Gu of forging doctrine against scripture. Even perfectly tuned calendars did not banish flood or rebellion. They cannot blame banditry on the sexagenary epoch. Han has rotated epochs repeatedly since Qin; gengshen is not eternal. Their pedantry repeats the same fraud. Prior failed reforms and the Yuanhe charter settle the matter.
29
太尉耽、司徒隗、司空訓以邕議劾光、晃不敬，正鬼薪法。 詔書勿治罪[7]。
The Three Dukes indicted Feng Guang and Chen Huang for disrespect. The emperor pardoned them.
30
太初歷推月食多失。 四分因太初法，以河平癸巳為元，施行五年。 永元元年，天以七月後閏食，術以八月。 其* (十) *二年正月十二日，蒙公乘宗紺上書言：「今月十六日月當食，而歷以二月。」 至期如紺言。 太史令巡上紺有益官用，除待詔。 甲辰，詔書以紺法署。 施行五十六歲。 至本初元年，天以十二月食，歷以後年正月，於是始差。 到熹平三年，二十九年之中，先歷食者十六事。 常山長史劉洪上作七曜術。 甲辰詔屬太史部郎中劉固、舍人馮恂等課效，復作八元術，固等作月食術，並已相參。 固術與七曜術同。 月食所失，皆以歲在己未當食四月，恂術以三月，官歷以五月。 太史上課，到時施行中者。 丁巳，詔書報可。
Taichu eclipse math was loose. The interim quarter scheme used a guisi epoch for five years. Observation saw a seventh-month eclipse; tables said eighth. In— —ten— —second year first month twelfth day—the ranked commoner Zong Gan of Meng memorialized: ‘This month’s sixteenth day the moon ought to eclipse—but the calendar puts it in the second month.’ The eclipse arrived as Gan predicted. Grand Astrologer Xun recommended Gan and he became an expectancy-holder. A jiachen edict adopted Gan’s tables. Gan’s scheme ran fifty-six years. Benchu 1 saw another mismatch between observation and calendar. Between those dates official predictions jumped ahead sixteen times. Liu Hong of Changshan offered his planetary theory. The court tasked Liu Gu and Feng Xun to cross-check Hong’s work with new eclipse algorithms. Liu Gu’s math aligned with Hong’s. For yiwei-year misses Feng Xun predicted month 3, the bureau month 5. The bureau ran trials and favored whichever hit. The emperor approved on dingsi.
31
其四年，紺孫誠上書言：「受紺法術，當復改，今年十二月當食，而官歷以後年正月。」 到期如言，拜誠為舍人。 丙申，詔書聽行誠法。
Zong Cheng refined his grandfather’s formula and challenged the next eclipse timing. The prediction succeeded and Cheng earned an appointment. A bingshen edict adopted Cheng’s revision.
32
光和二年歲在己未，三月、五月皆陰，太史令修、部舍人張恂等推計行度，以為三月近，四月遠。 誠以四月。 奏廢誠術，施用恂術。 其三年，誠兄整前後上書言：「去年三月不食，當以四月。 史官廢誠正術，用恂不正術。」 整所上* (五) **[正]*屬太史，太史主者終不自言三月近，四月遠。 食當以見為正，無遠近。 詔書下太常：「其詳案注記，平議術之要，效驗虛實。」 太常就耽上選侍中韓說、博士蔡較、谷城門候劉洪、右郎中陳調於太常府，覆校注記，平議難問。 恂、誠各對。 恂術以五千六百四十* (日) **[月]*有九百六十一食為法，而除成分，空加縣法，推建武以來，俱得三百二十七食，其十五食錯。 案其官素注，天見食九十八，與兩術相應，其錯辟二千一百。 誠術以百三十五月二十三食為法，乘除成月，從建康以上減四十一，建康以來減三十五，以其俱不食。
Clouds obscured two candidate nights in Guanghe 2. Cheng favored April. The bureau dropped Cheng for Xun. That third year Cheng’s elder Zheng repeatedly memorialized: ‘Last March saw no eclipse—it ought have been April. The bureau preferred Xun over Cheng." Zheng argued— —five— Proper protocol rested with the Grand Astrologer’s verdict. Observation—not arithmetic distance—settles eclipse timing. An edict went down to the Grand Master of Ceremonies: ‘Examine annotations closely—fairly debate each method’s essentials—verify truth or emptiness.’ Grand Minister of Ceremonies Jiu Dan memorialized to select Attendant Han Shuo, Erudite Cai Jiao, Gucheng Gate Watcher Liu Hong, and Right Gentleman Chen Diao to meet at the office of the Grand Minister of Ceremonies, review and check the notes, and fairly discuss the difficulties and questions. Feng Xun and Zong Cheng presented defenses. Xun’s denominator counts five thousand six forty— —days— His eclipse canon miscounts fifteen of 327 eclipses since Jianwu. Against ninety-eight observed eclipses his avoidance errors balloon. Cheng’s interval arithmetic trims phantom eclipses before and after Jiankang.
33
恂術改易舊法，誠術中復減損，論其長短，無以相踰。 各引書緯自證，文無義要，取追天而已。 夫日月之術，日循黃道，月從九道。 以赤道儀，日冬至去極俱一百一十五度。 其入宿也，赤道在斗中十一，而黃道在斗十九。 兩儀相參，日月之行，曲直有差，以生進退。 故月行井、牛，十四度以上； 其在角、婁，十二度以上。 皆不應率不行。 以是言之，則術不差不改，不驗不用。 天道精微，度數難定，術法多端，歷紀非一，未驗無以知其是，未差無以知其失。 失然後改之，是然後用之，此謂允執其中。 今誠術未有差錯之謬，恂術未有獨中之異，以無驗改未失，是以檢將來為是者也。 誠術百三十五月月二十三食，其文在書籍，學者所修，施行日久，官守其業，經緯日月，厚而未愆，信於天文，述而不作。 恂久在候部，詳心善意，能揆儀度，定立術數，推前校往，亦與見食相應。 然協歷正紀，欽若昊天，宜率舊章，如甲辰、丙申詔書，以見食為比。 今宜施用誠術，棄放恂術，史官課之，後有效驗，乃行其法，以審術數，以順改易。 耽以說等議奏聞，詔書可。 恂、整、誠各復上書，恂言不當施誠術，整言不當復* (棄) *恂術。 為洪議所侵，事下永安台覆實，皆不如恂、誠等言。 劾奏謾欺。 詔書報，恂、誠各以二月奉贖罪，整適作左校二月。 遂用洪等，施行誠術。
Both schemes fiddle constants without decisive superiority. They cite apocrypha yet aim only at fitting the sky. Sun rides the ecliptic; moon follows lunar corridors. Equatorial measures fix winter solstice polar distance at 115°. Equator midpoints Dipper differently than the ecliptic. Combining frames reveals why motions diverge. Near Ox and Well the moon moves fastest. Near Horn and Bond it slows. Uniform rates fail along these arcs. Revise only after proved error. No criterion exists except empirical check. That is the doctrine of holding the mean. Neither Cheng nor Xun yet proves decisive superiority. Cheng’s cycle is long-attested and conservative. Feng Xun’s observational skill matches eclipses too. Dynastic precedent prefers eclipse testimony. Adopt Cheng until further tests dictate change. Jiu Dan forwarded the verdict; the emperor agreed. The litigants appealed again— —abandon— —finish ‘must not abandon Xun.’ Liu Hong’s office reviewed and rejected both extremes. The court found deceit. Fines and corvée punished Feng Xun, Zong Cheng, and Zong Zheng. The bureau adopted Cheng’s scheme.
34
不能，對。」 洪上言：「推* (元) *漢己巳元，則考靈曜旃蒙之歲乙卯元也，與光、晃甲寅元相經緯。 於以追天作歷，校三光之步，今為疏闊。 孔子緯一事見二端者，明歷興廢，隨天為節。 甲寅歷於孔子時效； 己巳顓頊秦所施用，漢興草創，因而不易，至元封中，迂闊不審，更用太初，應期三百改憲之節。 甲寅、己巳讖雖有文，略其年數，是以學人各傳所聞，至於課校，罔得厥正。 夫甲寅元天正正月甲子朔旦冬至，七曜之起，始於牛初。 乙卯之元人正己巳朔旦立春，三光聚天廟五度。 課兩元端，閏余差* (自) **[百]*五十*[二]*分* (二) *之三，朔三百四，中節之餘二十九。 以效信難聚，漢不解說，但言先人有書而已。 以漢成注參官施行，術不同二十九事，不中見食二事。 案漢習書，見己巳元，謂朝不聞，不知聖人獨有興廢之義，史官有附天密術。 甲寅、己巳，前已施行，效後格而* (已) *不用。 河平疏闊，史官已廢之，而漢以去事分爭，殆非其意。 雖有師法，與無同。 課又不近密。 其說蔀數，術家所共知，無所採取。」 遣漢歸鄉里[8]。
He had no answer. Liu Hong continued: “If we derive— Omit the stray the cited text after the cited text. —Wang Han’s jisi epoch—matches the Kaoling yao yimao anchor and cross-cuts Feng Guang’s jiayin scheme. That path is too loose to discipline the three lights. Apocrypha admit two ends to one case: calendars live or die with the sky. Jiayin fit the Spring and Autumn era. Qin’s jisi Zhuanxu held until Taichu; Han first kept it, then reformed. Apocrypha name both epochs but omit years, so experts guess. Jiayin posits a grand conjunction at Ox’s lead. Yimou clusters three lights at the Temple five degrees. The two epoch openings disagree on intercalary fraction— Read the cited text for the cited text. —152 points— —two—thirds completes the fraction— —remainder arithmetic diverges on lunations and qi. Han archivists cite old books without proving alignment. Wang Han’s tables clash with the office on twenty-nine points and miss two eclipses. Han tradition preserves jisi records unseen at court. Both epochs were tried; later rejected— Read as ‘already’ completing ‘later invalidated.’ —they fell out of use. Heping-era looseness was discarded—relitigating it misses the point. Transmission without verification equals none. Trials still lack precision. His cycle arithmetic is common coin—no novelty." The emperor dismissed Wang Han to his home county.
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校勘記
Editorial apparatus begins here.
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三０二五頁四行朔先*[於]*歷集解引盧文弨說，謂「先」下脫「於」字，依御覽補。 今據補。
Page 3025, line 4: "before the cited text for the cited text[the cited text]the cited text-the Jijie cites Lu Wenchao saying after the cited text the graph the cited text is missing-supplemented per Yuolan." Adopted into the received text.
37
三０二五頁四行月*[或朔]*見集解引盧文弨說，謂「月」下脫「或朔」二字，依御覽補。 今據補。
Page 3025, line 4: "moon [or new moon] appear-the Jijie cites Lu Wenchao saying after the cited text graphs the cited text are missing-supplemented per Yuolan." Adopted.
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三０二五頁五行建武八年中按：集解引惠棟說，謂北宋本無「中」字。
Page 3025, line 5: “within Jianwu 8”-note: Jijie cites Hui Dong saying Northern Song editions lack the cited text.
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三０二五頁六行歷*[朔]*不正集解引盧文弨說，謂「言」下脫「朔」字，依御覽補。 今據補。
Page 3025, line 6: "calendar [new moon] incorrect-the Jijie cites Lu Wenchao saying after the cited text graph the cited text is missing-supplemented per Yuolan." Adopted.
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三０二五頁六行官歷署七月十六日*[月]*食集解引盧文弨說，謂「日」下脫「月」字，依御覽補。 王先謙謂以下文證之，當有「月」字。 今據補。 按：印影宋本御覽「月」斗「日」。
Page 3025, line 6: "official calendar registers seventh month sixteenth day [moon] eclipse-the Jijie cites Lu Wenchao saying after the cited text graph the cited text is missing-supplemented per Yuolan." "Wang Xianqian says following text confirms there ought be graph the cited text." Adopted. Note: "Song imperial Yuolan photographic edition mixes the cited text with the cited text."
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三０二五頁七行*[因]*上言月當十五日食集解引盧文弨說，謂御覽「上言」上有「因」字。 今據補。
Page 3025, line 7: "[therefore] memorialized moon ought feast fifteenth day eclipse-the Jijie cites Lu Wenchao saying Yuolan above the cited text has graph the cited text." Adopted.
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三０二五頁八行詔書令岑普*[候]*與官*[歷]*課集解引盧文弨說，謂「普」下脫「候」字，「官」下脫「歷」字，御覽有。 今據補。
Lu Wenchao restores the observation verb after Pu’s name and the bureau calendar noun after office before comparing Yang Cen against the official reckoning. Adopted.
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三０二五頁九行詔*[書]*令岑署弦望月食官集解引盧文弨說，謂「詔」下脫「書」字，依御覽補。 今據補。
Page 3025, line 9: "edict [document] ordered Cen administer quarters full moons eclipse office-the Jijie cites Lu Wenchao saying after the cited text graph the cited text missing-supplemented per Yuolan." Adopted.
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三０二六頁二行* (從) **[後]*天四分日之三集解引李銳說，謂「後天」誤「從天」，當改。 今據改。
Line break before marginal gloss. Variant graph before emendation. "[behind] heaven three quarters day-the Jijie cites Li Rui saying the cited text was wrongly written the cited text-ought change." Adopted.
45
三０二六頁七行*[順]*堯考德* (顧) **[題]*期立象集解引惠棟說，謂「顧」一作「題」。 又引盧文弨說，謂緯書所載作「順堯考德，題期立象」。 按：曹□傳作「順堯考德，題期立象」，今據以補改。
Lemma opening gloss block. Variant for following lemma. "[topic] period establish images-the Jijie cites Hui Dong saying the cited text sometimes reads the cited text." Also cites Lu Wenchao saying weft texts read ‘follow Yao examine virtue; fix period establish images.’ Note: Cao biography reads ‘follow Yao examine virtue; fix period establish images’—now supplemented per this.
46
三０二六頁一三行日在斗二十* (二) **[一]*度據集解引盧文弨說改。
Lemma split across lines. Numeral gloss completing twenty-two versus twenty-one debate. Lu Wenchao settles Dipper degree reading.
47
三０二六頁一五行*[儻]*獲咸* (喜) **[熙]*集解引惠棟說，謂「獲」上一有「儻」字，「喜」作「熙」，宋志同。 又引盧文弨說，謂南宋本有「儻」字。 今據以補改。
Lemma on exalted brightness clause. Variant character note. Hui Dong’s variants for the cited text and the cited text. "Also cites Lu Wenchao saying Southern Song edition has graph the cited text." Adopted.
48
三０二七頁三行司徒*[掾]*嚴勖集解引錢大昕說，謂此嚴勖亦司徒之掾屬，非司徒也，史脫文。 今據補。
Page 3027, line 3: Minister of Education [clerk] Yan Xu—the Jijie cites Qian Daxin saying this Yan Xu is Minister clerk—not Minister—history drops text. Adopted.
49
三０二七頁四行朔必有明晦不朔必在其月也按：集解引盧文弨說，謂「明」字衍，「不朔」當作「朔不」。
Page 3027, line 4: new moon necessarily bright last day / no new moon necessarily in its month-note: "Jijie cites Lu Wenchao saying the cited text is redundant-the cited text should read the cited text."
50
三０二七頁五行是明不可必按：集解引盧文弨說，謂唐一行大衍曆議引「明」作「朔」。
Page 3027, line 5: therefore brightness not certain-note: "Jijie cites Lu Wenchao saying Tang Yihang Great Expansion Calendar discussion cites the cited text as the cited text."
51
三０二七頁六行十六日*[望]*按各本俱無「望」字，今依歷理及文義補。
Page 3027, line 6: sixteenth day [full moon]-note: "all editions lack the cited text-now supplemented per calendrical principle and sense."
52
三０二七頁七行天元始起之月* (常) **[當]*小據汲本、殿本改。
Lemma on epoch month. Variant graph. The Ji and Palace editions substitute the verb ought for the variant constant in this lemma.
53
三０二七頁一二行冬至日在斗* (一) **[二]*十一度四分度之一據汲本、殿本改。
Lemma continues. Numeral gloss. Editions fix twenty-one degrees reading.
54
三０二七頁一四行日所在*[未至]*牽牛中星五度集解引盧文弨說，謂「在」下當脫「未至」二字。 今據補。
Page 3027, line 14: "sun location [not yet] Ox middle star five degrees-the Jijie cites Lu Wenchao saying after the cited text graphs the cited text missing." Adopted.
55
三０二八頁一行五歲中課日行及冬* (夏) *至斗* (一) **[二]*十一度四分一集解引惠棟說，謂李本「一十」作「二十」。 按：上屢見冬至日在斗二十一度，明作「一十」者鬥，今據改。 又按文義「夏」字當衍，今刪。
Lemma on winter solstice clause. (summer) [extraneous character per commentary] Connects broken phrase to Dipper degrees. Numeral gloss. "[two] eleven degrees quarter one—the Jijie cites Hui Dong saying Li edition writes twenty for ten." Note: "above repeatedly winter solstice sun at Dipper twenty-one degrees—clearly writing ten as twenty is error—now changed." "Also press sense shows graph the cited text ought be redundant-now deleted."
56
三０二八頁四行日* (朔) **[食]*二十三事據集解引盧文弨說改。
Lemma on solar eclipse count. Variant before emendation. [eclipse] twenty-three cases—according to Jijie cites Lu Wenchao saying change.
57
三０二八頁五行二得* (三) **[二]*日按：各本並作「三日」，於歷理為舛，今改正。
Lemma on second-day eclipses. Wrong numeral. [two] days—note: "all editions read three days—contrary to calendrical principle—now corrected."
58
三０二八頁八行治歷者方以七十六歲斷之按：集解引盧文弨說，謂「方」疑當作「乃」。
Page 3028, line 8: calendar makers then use seventy-six years to cut it-note: "Jijie cites Lu Wenchao saying the cited text perhaps ought read the cited text."
59
三０二八頁九行則余分* (稍) **[消]*長集解引惠棟說，謂「稍」李本作「消」。 今按：依文義作「消」是，各本作「稍」，蓋涉下「稍」字而誤，今據改。
Lemma on remainder wax/wane. Variant graph. "[wane] grow-the Jijie cites Hui Dong saying Li edition writes the cited text for the cited text." Now press: "according to sense the cited text is correct-all editions read the cited text-probably contamination from following the cited text-now changed."
60
三０二九頁五行其斗牽牛*[東井]*輿鬼集解引錢塘說，謂「牽牛」下脫「東井」二字。 斗、牽牛冬至日所在，東井、輿鬼夏至日所在也。 今據補。
Page 3029, line 5: "its Dipper Ox [Eastern Well] Ghosts—the Jijie cites Qiantang saying Ox lacks Eastern Well pair." Comment clarifies lodge pairing. Adopted.
61
三０二九頁六行行東壁按：於文義「行」字當衍。
Page 3029, line 6: traveling Eastern Wall-note: "according to sense graph the cited text redundant."
62
三０二九頁六行赤道* (十) **[七]*度集解引李光地說，謂「十」當作「七」。 今按：壁、奎、婁、軫、亢閒在黃道斜交赤道之附近，以赤道標準度之，則赤道得度多而黃道得度少，其大較為七與八之比，李說是，今據改。
Lemma on degree counts. Wrong tens digit. "[seven] degrees—the Jijie cites Li Guangdi saying ten ought read seven." Because Wall, Legs, Bond, Chariot-Frame, and Horn lie where the ecliptic slices obliquely across the equator, equatorial measures overshoot ecliptic spans in about a seven-to-eight ratio; Li Guangdi’s emendation from ten to seven degrees is adopted.
63
三０二九頁七行出赤道南二十五度按：「五」當作「四」，說詳下。
Page 3029, line 7, “emerging south of the equator twenty-five degrees”: the note says “five” should read “four”; the argument is set out below.
64
三０二九頁七行出赤道北*[二十]*五度據集解引李光地說補。 按：當作「二十四度」，說詳下。
Li Guangdi’s gloss supplies the missing tens place so the northern latitude reads twenty-five degrees. Note: the figure should read “twenty-four degrees”; explanation follows.
65
三０二九頁八行去極俱九十度當作「九十一度」，脫「一」字。 按：四分歷以周天為三百六十五度又四分一，赤道去極為其四分之一，約為九十一度。 張衡渾儀謂「赤道橫帶渾天之腹，去極九十一度十六分之五，黃道斜帶其腹，出赤道表裡各二十四度，故夏至去極六十七度而強，冬至去極百一十五度亦強也」。 上文亦言「冬至日去極一百十五度，夏至日去極六十七度，春秋分日去極九十一度」。 並足證當時以赤道去極為九十一度，黃道於牽牛及東井各距赤道南北二十四度也。
Page 3029, line 8: “distance from the pole in both cases ninety degrees” should read ninety-one; the graph “one” has dropped out. The Fractional Quarter reckoning implies about ninety-one degrees from pole to equator. Zhang Heng’s armillary sphere states: “The equator girdles the belly of the armillary heaven at ninety-one and five-sixteenths degrees from the pole; the ecliptic crosses it obliquely and emerges twenty-four degrees inside and outside the equator on either side; hence at summer solstice the distance from the pole is a little over sixty-seven degrees, and at winter solstice a little over one hundred fifteen.” The passage above likewise says: “At winter solstice the sun is one hundred fifteen degrees from the pole; at summer solstice sixty-seven; at the spring and autumn equinoxes ninety-one.” The numbers align equatorial polar distance with ecliptic latitude at the relevant lodges.
66
三０二九頁一二行日過*[一]*度據殿本考證補。
The palace edition adds the missing “one” in the sun’s daily motion.
67
三０三０頁三行史官以* (郭) **[部]*日月行參弦望按：集解引齊召南說，謂「郭」當作「部」。 今據改。
Line begins the historians’ clause on reckoning the luminaries. The received graph Guo is suspect; see the next entry. Qi Zhaonan restores bu (“office,” here the calendrical bureau) where the text wrongly wrote Guo. Adopted.
68
三０三０頁六行能用易九六七八* (支) **[爻]*知月行多少據集解引盧文弨說改。
Line cites hexagram numerology before the emendation. The graph stands for the wrong character; see the restoration of yao. Lu Wenchao reads yao (hexagram lines), not zhi (branch).
69
三０三０頁一一行* (宣) **[宜]*課試上據集解引盧文弨說改。
Broken line before the variant gloss. The graph Xuan is a corruption awaiting correction. Lu Wenchao reads yi (“fitting”) where the text had xuan.
70
三０三一頁一行而月日行十三度十九分度之*[七]*有畸據集解引盧文弨說補。
Lu Wenchao supplies the missing numerator seven in the fractional degree.
71
三０三一頁三行事□而不悖按：集解引惠棟說，謂杜集「事」上有「則」字，「悖」作「□」。
Hui Dong compares Du’s edition for particles and the damaged graph bei.
72
三０三一頁六行日食*[亦得朔]*據集解引盧文弨說補。
Lu Wenchao adds that eclipses likewise align with conjunction.
73
三０三一頁八行*[明]*此*[食]*非用幣伐鼓常月據集解引盧文弨說補。
Lu Wenchao marks this eclipse as outside the routine ritual month.
74
三０三一頁一０行*[而]*先儒所未喻也據集解引盧文弨說補。
Lu Wenchao supplies the concessive “yet.”
75
三０三一頁一二行而三統曆唯*[得]*一食據集解引盧文弨說補。
Lu Wenchao inserts “only” or “got” to qualify the single matched eclipse.
76
三０三一頁一四行皆不* (得) *諧合據集解引盧文弨說刪。
Clause on failed agreement opens mid-line. Manuscript variant for the verb. Lu Wenchao excises the redundant phrase xiehe.
77
三０三一頁一七行累日為月*[累月為歲]*據集解引盧文弨說補。
Lu Wenchao restores the parallel phrase about years.
78
三０三一頁一七行以新故相序不得不有毫毛之差此自然*[之]*理也集解引惠棟說，謂「序」原作「涉」，「毛」作「末」，「然」下有「之」字。 按：晉志引長歷與惠校同，今以「相序」與「相涉」，「毫毛」與「毫末」，文異而義同，故但補一「之」字。
Hui Dong’s collation differs on three graphs before the closing formula. Note: the Jin treatise’s Chang calendar agrees with Hui’s collation; here xiangxu versus xiangshe and haomao versus haomo differ in wording but match in sense, so only the particle zhi is supplied.
79
三０三一頁一七行*[有]*曠年不食者據集解引盧文弨說補。
Lu Wenchao supplies “there are” before years without eclipses.
80
三０三二頁一行而筭守* (從) **[恆]*數據汲本、殿本改。
Line criticizes conservative calculators. Received graph; emendation follows. Editions read heng (“constant”) for the corrupted graph.
81
三０三二頁二行非為合以驗天* (者) *也據集解引盧文弨說刪。
Phrase challenges merely forcing the model to match observations. Particle zhe is judged intrusive. Lu Wenchao deletes the superfluous ye.
82
三０三二頁四行以考朔晦* (也) *據集解引盧文弨說刪。
Clause on checking lunation endpoints. Final particle targeted for deletion. Lu Wenchao removes the stray particle.
83
三０三二頁四行而*[見]*皆不然據集解引盧文弨說補。
Lu Wenchao inserts jian (“see,” “observe”).
84
三０三二頁五行善筭李修夏顯按：集解引盧文弨說，謂「善筭」本作「有善筭者」。 又引惠棟說，謂「夏」杜集作「卜」。
Page 3032, line 5: “skilled reckoner Li Xiu, Xia Xian”—the note: the Jijie cites Lu Wenchao, who says the original had “there was one skilled at reckoning.” It also cites Hui Dong, who says “Xia” in Du’s collection reads “Bu.”
85
三０三二頁七行以干度與* (太) **[泰]*始歷參校古今記注據盧文弨腢書拾補校改。
Line begins comparison of stem measures with a calendar. Wrong graph for the calendar name. Taishi is the standard calendar title; texts miswrote tai.
86
三０三二頁七行干度歷殊勝*[泰始歷上勝官歷四十五事]*集解引盧文弨說，謂「勝」下脫「泰始歷上勝官歷四十五事」十一字，依晉志補。 今據補。
Lu Wenchao restores a long testimonial clause dropped after “superior.” Adopted.
87
三０三二頁八行今*[其]*術具存據汲本、殿本補。
Editions add qi (“its”) before “method.”
88
三０三二頁八行今具列其* (時) *得失之數據集解引盧文弨說刪。
Introduces a detailed tabulation. Corrupt graph for the intended word. Lu Wenchao deletes the trailing phrase as gloss.
89
三０三二頁八行又據經傳微旨* (證據及失閏旨) *考日辰朔晦據集解引盧文弨說刪。
Line cites exegetical warrant. Spurious gloss wrongly inserted after “subtle meaning”; see deletion. Lu Wenchao removes the corrupt trailing clause.
90
三０三二頁九行及失閏*[違]*時據集解引盧文弨說補。
Lu Wenchao adds wei (“go against”) before seasons.
91
三０三二頁一０行蓋*[是]*春秋當時之歷也據集解引盧文弨說補。 按：「之」原斗「文」，逕改正。
Lu Wenchao supplies shi (“this”) before the nominal predicate. Note: "for zhi the manuscript wrongly had wen—corrected outright."
92
三０三二頁一五行夏歷漏*[刻]*隨日南北為長短集解引惠棟說，謂「漏」下脫「刻」字，當依隋志增。 今據補。
Page 3032, line 15: “Xia-calendar clepsydra markings [ke] lengthen and shorten with the sun’s north and south”—the Jijie cites Hui Dong, who says ke is missing below lou and should be added per the Sui treatise. Adopted.
93
三０三三頁二行日道周*[圜]*集解引惠棟說，謂「周」下宋志有「圜」字。 今據補。
Hui Dong adds yuan to complete “circuit ring.” Adopted.
94
三０三三頁五行立成斧官府當用者計吏到班予四十八箭文多故魁取二十四氣日所在集解引盧文弨說，謂「立成」至「魁取」二十二字宋志無。 今按：文有斗奪，難句讀。 疑詔書至「班予四十八箭」止，下為史官□述之文。 「魁」字衍。 言文多，故僅取二十四氣日所在等刻於下也。
Page 3033, line 5: from “ready reckonings” through “take only … twenty-four qi positions”—the Jijie cites Lu Wenchao, who says those twenty-two graphs are absent in the Song treatise. The sentence is corrupt and unclear. Probably the edict ends at “issue the forty-eight arrows”; what follows is the historian’s paraphrase. "The graph kui is redundant." The apparatus explains why only a subset is printed.
95
三０三三頁七行發謀於元封啟定於* (天) **[元]*鳳積* (百) *三十年是非乃審集解引李銳說，謂前志云「自漢歷初起，至元鳳六年，而是非堅定」。 案自太初元年至元鳳六年，正得三十年，此文「天鳳」當作「元鳳」，「百」字衍。 今據改。 按：依前書則「啟」當作「堅」。
Historical phrase about calendral reform epochs. Wrong graph in the reign title. The reign name should be Yuanfeng, not Tianfeng. Superfluous numeral. "...thirty years, then right and wrong were finally clear": the Jijie cites Li Rui, who says the Former Han treatise states that from the rise of the Han calendar to the sixth year of Yuanfeng, right and wrong were fixed. Editors restore Yuanfeng and delete the stray “hundred.” Adopted. Note: by the Former Han text qi here should read jian (“firm,” “settled”).
96
三０三三頁八行亦於建武施於元和按：集解引張文虎說，謂「亦」下疑脫一字，謂始於建武，而施行於元和也。
Page 3033, line 8: “also applied in Jianwu and deployed in Yuanhe”—the note: the Jijie cites Zhang Wenhu, who suspects one graph is missing below yi, meaning it began in Jianwu and was implemented in Yuanhe.
97
三０三三頁一０行及太初歷以後* (大) **[天]*為疾據集解引李銳說改。
Clause compares later calendars’ speed. Wrong graph in “great excess.” Li Rui reads tian (“celestial”) for da in the formula.
98
三０三三頁一一行百四十四歲而太歲超一* (表) **[辰]*據集解引錢大昕說改。
Lemma on suichaò cycle. Wrong graph for the astrological term. Qian Daxin restores chen for the year-branch jump.
99
三０三四頁二行或不對按：集解引惠棟說，謂「不」下宋志有「能」字。
Song editions add neng after “not.”
100
三０三四頁二行衡興參案儀注* (者) *集解引惠棟說，謂「者」字衍，從宋志刪。 今據刪。
Line cites official calibration. Particle deemed intrusive. "—the Jijie cites Hui Dong, who says zhe is redundant and deletes it per the Song treatise." Adopted.
101
三０三四頁五行太子舍人李泓按：殿本「泓」作「弘」。
Palace edition varies the graphs for Li Hong’s name.
102
三０三四頁六行推閏月六直其日按：尋文義，疑「六」為「不」之鬥，「日」為「月」之鬥。
Editors suspect two copying errors in numerals and “day/month.”
103
三０三四頁一０行災異卒甚汲本﹑殿本「卒」作「率」。 按：盧文弨雲北宋本作「卒」。
Page 3034, line 10: “disasters and anomalies ended severely”—Ji and Palace editions write shuai (“generally”) for zu. Note: "Lu Wenchao says Northern Song editions read zu."
104
三０三四頁一二行五是以備按：汲本﹑殿本「是」作「者」。 集解引錢大昕說，謂洪範「五者來備」一作「五是」，蓋漢儒傳本異也。 閩本﹑汲古閣本作「五者」，則後人據今本尚書易之。 李雲傳「五氏來備」，氏古是字。 荀爽傳「五韙來備」，韙亦訓是。
Variant particles between “five” and the verb. The Jijie cites Qian Daxin, who says in Hong Fan “the five kinds arrive complete” also reads “five are”—Han exegetes transmitted both wordings. "Min and Mao editions write zhe, reflecting later editors harmonizing to the current Book of Documents." In Li Yun’s memorial “five clans arrive complete”—shi in antiquity is the graph for “this.” In Xun Shuang’s memorial “five wei arrive complete”—wei likewise glosses as “this.”
105
三０三四頁一五行則* (喪) **[表]*其休集解引盧文弨說，謂錢氏改「喪」為「表」。 按：詳文義當作「表」，表與喪形近，今據改。
Line breaks before a gloss. Graph targeted for emendation. Lu Wenchao follows Qian in reading “display” rather than “mourning.” Note: examining the wording, it should read “manifest” (biao); biao resembles sang (mourning) in form; now emended accordingly.
106
三０三五頁五行遂*[寢]*改歷事集解引錢大昕說，謂詳文義，是安帝納尚書令忠言，仍用四分，不復議改。 宋志亦云「但等遂寢」。 此文「遂」下當有「罷」字，或是「寢」字。 今據錢說並參宋志，補一「寢」字。
Qian explains that the court shelved further calendar reform and kept the Quarter-Day system. The Song treatise likewise says “but Deng and the others then let it drop.” Editors suspect a missing verb after “then.” Now, following Qian’s argument and cross-checking the Song treatise, one graph qin (“shelve”) is supplied.
107
三０三五頁九行干鑿度八十*[一]*分之四十三為日法據集解引錢大昕說補。
Qian Daxin restores the missing “one” in eighty-one.
108
三０三五頁一０行雒書* (甄) **[干]*曜度據殿本改。 按：集解引惠棟說，謂「干」作「甄」當是避太子承干諱改。
Line begins a citation from the Luo document. Manuscript variant graph before emendation to gan. The palace text restores gan where Zhen had appeared. Note: the Jijie cites Hui Dong, who says gan was written as Zhen to avoid the taboo on the heir apparent’s name Cheng Qian.
109
三０三五頁一二行從太初至永平十一年百七十*[一]*歲據集解引錢大昕說補。
Qian Daxin adds the missing “one” to one hundred seventy-one.
110
三０三五頁一三行至* (永) **[元]*和二年據集解引錢大昕說改。
Lemma continues the interval phrase. Partial reign-name graph before yuan. Qian Daxin restores Yuanhe rather than a corrupt reading.
111
三０三六頁九行案百七十*[一]*歲二蔀一章據集解引錢大昕說補。
The parallel calculation includes one hundred seventy-one years.
112
三０三七頁二行羲和立* (禪) **[渾]*集解引盧文弨說，謂「禪」乃「渾」之鬥，渾謂渾儀，與韻協。 今據改。
Line cites the ancient calendar masters. Received graph; next entry restores hun. "[armillary sphere]—the Jijie cites Lu Wenchao, who says chan is a corruption of hun (armillary sphere); hun matches the rhyme scheme." Adopted.
113
三０三七頁九行盜賊相續為*[害]*王先謙謂「為」下疑有「害」字。 宋志作「歷元不正，故盜賊為害」。 今據王說參宋志，補一「害」字。
Wang proposes supplying hai (“harm,” “damage”). The Song treatise reads: “the calendar origin was not correct, so bandits caused harm.” The commentary adds the missing character for harm, following Wang’s argument and the Song calendar treatise.
114
三０三七頁一０行歷*[當]*用甲寅為元而用庚申王先謙謂宋志作「歷當以甲寅為元，不用庚申」。 今依宋志補一「噹」字。
Page 3037, line 10: “the calendar [ought] to use jiayin as origin yet uses gengshen”—Wang Xianqian says the Song treatise reads: “the calendar ought to take jiayin as origin and not use gengshen.” Now the particle dang (“ought”) is supplied after the Song treatise.
115
三０三七頁一０行圖緯無以庚*[申]*為元者據集解引盧文弨說補。
Lu Wenchao restores shen after geng.
116
三０三七頁一一行乞* (與) *本庚申元經緯* (有) *明*[文]*據集解引盧文弨說刪補。
Petition clause begins mid-line. Variant preposition in the broken phrase. Connects the plea to apocryphal authority. Verb “there is” in the lemma. Lu Wenchao adjusts ming and wen.
117
三０三七頁一一行受虛欺重誅按：集解引盧文弨說，謂此句上亦有脫文。
Lu Wenchao reports a larger lacuna above.
118
三０三八頁一行術* (術) *無常是據集解引惠棟說刪。
Line on competing calendar methods. Duplicate or gloss graph. Hui Dong excises the redundant phrase.
119
三０三八頁一行*[漢興]** (以) *承秦集解引惠棟說，謂「以」字誤，宋志「漢興承秦」。 今據宋志改。 按：盧文弨腢書拾補改作「漢承秦正」。
Historical opening clause. Particle yi judged erroneous. “inherited Qin”—the Jijie cites Hui Dong, who says yi is wrong; the Song treatise reads “when Han rose it inherited Qin.” Adopted. Note: Lu Wenchao’s Omitted Characters reads “Han inherited Qin’s first month.”
120
三０三八頁四行各*[自一]*家*[之]*術皆當有效於* (其) *當時據集解引盧文弨說補刪。 今按：御覽卷十六引作「各自一家之說，皆當有效於當時」。
Introduces plural schools of reckoning. Possessive before “time.” Lu Wenchao adjusts ziji and shishang. Taiping yulan supports “doctrine” and “their time.”
121
三０三八頁五行* (黃) **[武]*帝始用太初丁丑之元* (有) *六家紛錯據盧文弨腢書拾補校改。 按：宋志作「昔始用太初丁丑之後」。 御覽一六引作「昔太初始用丁丑之後」。
Broken line before emperor clause. Wrong graph for Martial Emperor. Restores Wu di for Huang. Continuative particle. Lu fixes the six-calendar confusion. Note: the Song treatise reads “formerly began using Taichu dingchou thereafter.” Imperial Readings 16 cites “formerly Taichu first used dingchou thereafter.”
122
三０三八頁八行是又新元*[有]*效於今者也據宋志及御覽一六補。
Editions add you (“there is”) before “effect.”
123
三０三八頁一０行今*[術]*之不能上通於古集解引惠棟說，謂「今」下宋志有「術」字。 今據補。
Hui Dong inserts shu after “now.” Adopted.
124
三０三八頁一二行起庚* (子) **[午]*蔀之二十三歲據集解引錢大昕說改。
Epoch calculation continues. Wrong Earthly Branch in sequence. Qian Daxin restores wu for zi in the bu reckoning.
125
三０三八頁一三行則* (不) **[元]*在庚申集解引錢大昕說，謂自獲麟至開闢二百七十六萬歲，以六十除之，恰盡獲麟之歲，既是庚申，則開闢之始亦必庚申矣。 當云「元在庚申」，「不」乃「元」字之鬥。 又引李銳說，謂上文雲二百七十六萬歲，尋之上行，復得庚申，「不」當作「復」。 按：錢﹑李兩家之說並是，今從錢說改「不」字為「元」字。
Hypothesis about world origin. Graph suspected as corruption. Qian Daxin argues gengshen closure for the grand cycle. It should read “the origin lies in gengshen”; bu is a corruption of yuan. It also cites Li Rui, who says the text above gives 2,760,000 years; tracing upward in the line again reaches gengshen—bu should read fu (“again”). Note: both Qian’s and Li’s views hold; now following Qian, bu is changed to yuan.
126
三０三八頁一四行獲麟至漢百六十* (二) **[一]*歲集解引李銳說，謂邕於甲寅元開闢至漢元年數內減去庚申元開闢至獲麟年數，余一百六十一為獲麟至漢元年數，因謂光﹑晃差少一百一十四歲。 今按：甲寅元開闢至獲麟積年二百七十五萬九千八百八十歲，獲麟至漢二百七十五歲，共二百七十六萬一百六十一歲，邕以庚申元開闢至獲麟積年二百七十六萬歲減之，則獲麟至漢為百六十一歲，明「百六十二歲」之「二」字當作「一」，今據改。
Span from unicorn to Han founding. Erroneous tens digit. Li Rui explains Yong’s subtraction and the 114-year gap. Cai Yong’s residual is 161 years from the unicorn omen to Han, not 162, so the second graph in the numeral is emended to the first, changing one hundred sixty-two to one hundred sixty-one.
127
三０三九頁一行下不及命歷序獲麟*[至]*漢相去四蔀年數據集解引盧文弨說補。
Lu Wenchao adds zhi (“to”) in the span phrase.
128
三０三九頁二行蔡邕* (命) **[月令]*論曰集解引惠棟說，謂「命論」未詳。 案邕明堂月令論有之，「令」誤「命」，落「月」字也。 今據改。
Citation of Yong’s treatise. Wrong graph for monthly ordinance. [Hall Monthly Ordinance] treatise states—the Jijie cites Hui Dong, who says “decree treatise” is unclear. Case: Yong’s Hall Monthly Ordinance treatise contains this passage; ling was miswritten ming and yue dropped. Adopted.
129
三０三九頁五行而光晃歷以考靈曜*[為本]*集解引惠棟說，謂「曜」下宋志有「為本」二字。 今據補。
Page 3039, line 5: “yet Guang and Huang’s calendar took the Kaoling Yao [as its basis]”—the Jijie cites Hui Dong, who says below yao the Song treatise has “as basis.” Adopted.
130
三０三九頁一０行日在斗二十* (二) **[一]*度按：三０二六頁一三行「日在斗二十二度」，已據盧文弨說改「二十二」為「二十一」，此與上同。
Solar longitude lemma. Wrong tens digit. [one] degree—note: page 3026, line 13, “sun at Dipper twenty-two degrees,” was already changed from twenty-two to twenty-one per Lu Wenchao; this matches.
131
三０三九頁一三行而光晃以為*[香]*固意造妄說據集解引盧文弨說補。
Lu Wenchao restores the missing graph in the clerk Guo Xiang’s name.
132
三０四０頁二行亦妄虛無造欺語之愆按：集解引盧文弨說，謂「亦」下文有鬥。
Page 3040, line 2: “likewise empty false forged deceitful fault”—the note: the Jijie cites Lu Wenchao, who says below yi the text has lacuna.
133
三０四０頁八行其* (十) *二年正月十二日集解引李銳說，謂「十二年」當作「二年」，與下「十二日」相涉，誤衍「十」字。 案下文云「以紺法署施行五十六歲」，自永元二年至本初元年，正得五十六年，故知「十」字衍也。 今據刪。
Date phrase for promulgation. Superfluous tens digit. “two years, first month, twelfth day”—the Jijie cites Li Rui, who says “twelve years” should read “two years,” conflated with “twelfth day” to insert ten. The narrative’s fifty-six-year interval confirms that the extra ten is a copying error. Adopted.
134
三０四一頁二行部舍人張恂按：「張恂」疑當作「馮恂」。 上文言「熹平中，故治歷郎梁國宗整上九道術，詔書下太史，以參舊術，相應。 部太子舍人馮恂課校，恂亦復作九道術，增損其分，與整術並校，差為近。 太史令揚上以恂術參朔望」。 此處雖言課校恂﹑誠二術，整為誠兄，且先後上書為誠術辨，則所謂整術﹑誠術實同為一事，而參與推計行度者為馮恂也。
Page 3041, line 2: “department attendant Zhang Xun”—note: Zhang Xun probably should be Feng Xun. Above it says: “During Xiping, former calendar gentleman Zong Zheng of Liang submitted the nine-path method; an edict sent it to the Grand Clerk to compare with the old method—they agreed. Feng Xun’s revision and comparison. Closing the cited edict narrative." Editorial note reconciles names and roles.
135
三０四一頁四行整所上* (五) **[正]*屬太史據汲本改。 按：「五屬太史」不可解，尋文義以「正屬太史」為長。
Lemma on submission routing. Wrong numeral for zheng. Ji edition reads zheng (“proper”) not wu (“five”). Note: “five belongs to the Grand Clerk” makes no sense; following the wording, “proper jurisdiction of the Grand Clerk” fits.
136
三０四一頁七行恂術以五千六百四十* (日) **[月]*有九百六十一食為法據集解引錢大昕說改。 按：「法」原斗「注」，逕改正。
Numerator–denominator setup. Wrong graph for month. Qian Daxin restores month for day in the ratio. Note: "fa was corrupted as zhu—corrected outright."
137
三０四二頁六行恂言不當施誠術整言不當復* (棄) *恂術按：整﹑恂各挾己術相攻訐，恂言不當施誠術，整言不當復恂術，「棄」字當涉上「棄放恂術」而斗衍，今刪。
Debate between the two scholars. Graph implicated in duplication; see next sentence. “… Xun’s method”—the note: Zheng and Xun each pressed their own methods in mutual accusation; Xun said Cheng’s method ought not be applied, Zheng said Xun’s method ought not be revived; qi was dragged in from “banish Xun’s method” above and wrongly duplicated—now deleted.
138
三０四二頁七行整適作左校二月殿本「適」作「輸」。 按：適同□，原不鬥，殿本以意改也。
Page 3042, line 7: “Zheng happened to be assigned Left Corrector in second month”—Palace edition writes shu (“deliver,” “escort”) for shi (“fit,” “happened to”). The apparatus keeps shi as exile-style reassignment rather than adopting shu.
139
三０四二頁七行遂用洪等按：下疑脫一「議」字。
Editors suspect yi fell out before “Hong.”
140
三０四二頁一二行推元* (謂) **[課]*分據集解引盧文弨說改。
Broken lemma on epoch derivation. Particle wei gloss before ke. Lu Wenchao reads ke (“trial reckoning”) not wei.
141
三０四二頁一三行推* (元) *漢己巳元集解引盧文弨說，謂「推」下「元」字衍，漢即王漢。 今據刪。
Continuation of epoch lemma. Redundant graph targeted for deletion. “Han jisi origin”—the Jijie cites Lu Wenchao, who says yuan below tui is redundant; Han is Wang Han. Adopted.
142
三０四三頁三行閏余差* (自) **[百]*五十*[二]*分* (二) *之三集解盧文弨說，謂「自」當作「百」，又引李銳說，謂當作「百五十二分之三」。 今據改。
Lemma on fractional intercalation. Wrong graph for bai. Numerator–denominator phrase. Digit variant. … out of three—the Jijie cites Lu Wenchao, who says zi should read bai; also citing Li Rui, who reads “three one hundred fifty-second parts.” Adopted.
143
蔡邕議云：「梵，清河人。」
Cai Yong identifies Fan’s home commandery.
144
杜預長歷曰：「書稱『開三百六旬有六日，以閏月定四時成歲，允厘百工，庶績咸熙』。 是以天子必置日官，諸侯必置日御，世修其業，以考其術。 舉全數而言，故曰六日，其實五日四分之一。 日日行一度，而月日行十三度十九分度之*[七]*有畸。 日官當會集此之遲疾，以考成晦朔，錯綜以設閏月。 閏月無中氣，而北斗邪指兩辰之閒，所以異於他月也。 積此以相通，四時八節無違，乃得成歲。 其微密至矣。 得其精微，以合天道，事□而不悖。 故傳曰：『閏以正時，時以作事，事以厚生，生民之道，於是乎在。』 然陰陽之運，隨動而差，差而不已，遂與歷錯。 故仲尼、丘明每於朔閏發文，蓋矯正得失，因以宣明歷數也。 桓十七年，日食得朔，而史闕其日，單書朔。 僖十五年，日食*[亦得朔]*，而史闕朔與日。 故傳因其得失，並起時史之謬，兼以明其餘日食，或歷失其正也。 莊二十五年，經書『六月辛未朔，日有食之，鼓用牲於社』。 周之六月，夏之四月，所謂正陽之月也。 而時歷誤，實是七月之朔，非六月。 故傳云：『非常也。 唯正月之朔，慝未作，日有食之，於是乎有用幣於社，伐鼓於朝。』 *[明]*此*[食]*非用幣伐鼓常月，因變而起，歷誤也。 文十五年經文皆同，而更復發，傳曰『非禮』。 明前傳欲以審正陽之月，後傳發例，欲以明諸侯之禮也。 此乃聖賢之微旨，*[而]*先儒所未喻也。 昭十七年夏六月，日有食之，而平子言非正陽之月，以誣一朝，近於指鹿為馬。 故傳曰『不君君』，且因以明此月為得天正也。 劉子駿造三統曆，以修春秋。 春秋日食有甲乙者三十四，而三統曆唯*[得]*一食，歷術比諸家既最□。 又六千餘歲輒益一日。 凡歲當累日為次，而無故益之，此不可行之甚者。 班固前代名儒，而謂之最密。 非徒班固也，自古以來，諸論春秋者，多述謬誤，或造家術，或用黃帝以來諸歷，以推經傳朔日，皆不* (得) *諧合。 日食於朔，此乃天驗，經傳又書其朔食，可謂得天，而劉、賈諸儒說，皆以為月二日或三日，公違聖人明文。 其蔽在於守一元，不與天消息也。 余感春秋之事，嘗著歷論，極言歷之通理。 其大指曰：天行不息，日月星辰，各運其捨，皆動物也。 物動則不一，雖行度大量，可得而限。 累日為月，*[累月為歲]*，以新故相序，不得不有毫毛之差，此自然*[之]*理也。 故春秋日有頻月而食者，*[有]*曠年不食者，理不得一，而筭守* (從) **[恆]*數，故歷無不有差失也。 始失於毫毛，而尚未可覺，積而成多，以失弦望朔晦，則不得不改憲以從之。 書所謂『欽若昊天，歷象日月星辰』，易所謂『治歷明時』，言當順天以求合，非為合以驗天* (者) *也。 推此論之，春秋二百餘年，其治歷變通多矣。 雖數術絕滅，還尋經傳微旨，大量可知。 時之違謬，則經傳有驗。 學者固當曲循經傳月日日食，以考朔晦* (也) *，以推時驗。 而*[見]*皆不然，各據其學以推春秋。 此無異度己之跡，而欲削他人之足也。 余為歷論之後，至咸寧中，善筭李修、夏顯，依論體為術，名干度歷，表上朝廷。 其術合日行四分之數，而微增月行。 用三百歲改憲之意，二元相推，七十餘歲，承以強弱，強弱之差蓋少，而適足以遠通盈縮。 時尚書及史官以干度與* (太) **[泰]*始歷參校古今記注，干度歷殊勝*[泰始歷，上勝官歷四十五事]*，今*[其]*術具存。 時又並考古今十歷，以驗春秋，知三統曆之最□也。 今具列其* (時) *得失之數，又據經傳微旨* (證據及失閏旨) *，考日辰朔晦，以相發明，為經傳長歷。 諸經傳證據，及失閏*[違]*時，文字謬誤，皆甄發之。 雖未必其得天，蓋*[是]*春秋當時之歷也。 學者覽焉。」
Du Yu’s Comprehensive Calendar states: “The Document says: ‘Define three hundred sixty-six days by the cycle to set the four seasons and complete the year, thus regulating all offices so every task flourishes. Du Yu on institutional responsibility for calendrics. The six-day round figure versus the quarter-day remainder. Daily solar and lunar motion in degrees. Reconciling lunar speed with lunation boundaries. Why an intercalary month stands apart. Interlocking intercalation with the seasonal markers. Du Yu stresses precision. Sound calendrics keeps policy aligned with the sky. Quoted Zuo gloss on intercalary necessity. Calendars drift from celestial motion over time. Du Yu on exegetical purpose in Chunqiu dating. Example of terse eclipse record. Second lacunose record. Zuo commentary uses errors pedagogically. Classic line on summer eclipse. Seasonal alignment between Zhou and Xia months. Du Yu corrects the month count. Zuo flags abnormality. Quoted rule for ritual month eclipses. Supplemented graphs clarify irregular ritual timing. Parallel eclipse entry with added commentary. Du Yu distinguishes two exegetical aims. Lu Wenchao’s supplement adds the concessive particle. Pingzi misrepresented the month of the eclipse, traducing the whole court with an absurd denial akin to calling a deer a horse. Zuo’s moral and calendrical verdict. Liu Xin’s project. Lu’s supplement adds “only.” Du Yu criticizes systematic drift. Condemns gratuitous intercalation. Irony toward Ban Gu’s praise. Sweeping critique of prior Chunqiu calendrics. Verb in broken phrase. Closing the negative clause. Du Yu sides with conjunction records. Single-epoch dogmatism. Du Yu’s own monograph. Du Yu opens his thesis by insisting that Heaven never stops and every luminous body moves within its station. Motion admits variation. Parallel phrase and particle supplied per apparatus. Broken lemma on eclipse frequency. Corrupt graph before heng. Lu restores heng for rigid adherence. Why periodic reform is unavoidable. Classical warrant for observation-first calendrics. Particle judged redundant. Closing quotation. Historical flexibility. Recovery from internal evidence. Self-correcting texts. Methodological advice. Particle for deletion. Completes the broken sentence. Lu adds jian (“see,” “observe”). Du Yu’s metaphor for mismatched systems. Jin-era sequel to Du’s work. Technical summary of stem-degree adjustments. The stem-degree method borrows the three-hundred-year reform principle, alternates two epochs through seventy-odd-year handoffs of strong and weak phases, and lets small imbalances absorb long-run lunar slack. Court evaluation begins. Wrong graph in calendar name. Lu’s supplements restore Taishi and missing comparative clause. Parallel testing of ten systems against the Annals proved Liu Xin’s Triple Concordance the least reliable of the lot. Introduces tabulation. Corrupt graph. Parallel to apparatus elsewhere. Corrupt gloss wrongly inserted in Du Yu’s manuscript trail. Title of Du’s digest. Editorial scope of the digest. Lu supplies shi before nominal predicate. Closing Du Yu quotation."
145
洪範：「庶征，曰雨，曰暘，曰燠，曰寒，曰風。 五者來備，各以其敘。」
Hong Fan: “Among the many omens: rain, clear sky, heat, cold, and wind. Completion of the quotation."
146
案五行志，章和元年訖漢安二年日變二十三事，古今注又長。
Cross-reference on sun omens.
147
蔡邕集載：「三月九日，百官會府公殿下，東面，校尉南面，侍中、郎將、大夫、千石、六百石重行北面，議郎、博士西面。 戶曹令史當坐中而讀詔書，公議。 蔡邕前坐侍中西北，近公卿，與光、晃相難問是非焉。」
Cai Yong’s collected works record: “On the ninth day of the third month all officials assembled below the minister’s hall; the east faced the commandant of the guard, the south the attendant, general of the household, grandees, thousand- and six-hundred-bushel ranks in staggered rows to the north; deliberation gentlemen and erudites to the west. Procedure for reading the imperial charge. Yong’s placement in the debate.
148
蔡邕命論曰：「顓頊歷術曰：『天元正月己巳朔旦立春，俱以日月起於天廟營室五度。』 今月令孟春之月，日在營室。」
Cai Yong’s Treatise on the Mandate states: “The Zhuanxu calendar method says: ‘Heaven’s origin, first month, jisi new moon at dawn, Beginning of Spring—sun and moon alike rise from the Celestial Temple in Encampment at five degrees. Harmonizing Ordinance with Zhuanxu line."
149
臣昭曰：不有君子，其能國乎？ 觀蔡邕之議，可以言天機矣。 賢明在朝，弘益遠哉！ 公卿結正，足懲淺妄之徒，詔書勿治，亦深「盍各」之致。
Pei Zhao’s editorial aside opens. Praise for Yong’s sophistication. Commends the debate’s quality. Ministers reached a just verdict—enough to chastise shallow bluster; the edict ordering no prosecution also honors the Analects’ “why not each follow his own course.”
150
袁山松書曰：「劉洪字符卓，泰山蒙陰人也。 魯王之宗室也。 延熹中，以校尉應太史征，拜郎中，遷常山長史，以父憂去官。 後為上計掾，拜郎中，檢東觀著作律歷記，遷謁者，谷城門候，會稽東部都尉。 征還，未至，領山陽太守，卒官。 洪善筭，當世無偶，作七曜術。 及在東觀，與蔡邕共述律歷記，考驗天官。 及造干象術，十餘年，考驗日月，與象相應，皆傳於世。」 博物記曰：「洪篤信好學，觀乎六藝腢書意，以為天文數術，探賾索隱，鉤深致遠，遂專心銳思。 為曲城侯相，政教清均，吏民畏而愛之，為州郡之所禮異。」
Yuan Shansong’s Book states: “Liu Hong, courtesy name Zhuo, was a man of Mengyin in Taishan. Liu Hong belonged to the Lu royal house. In Yanxi he entered imperial service through the astrological bureau, rose to Changshan chief clerk, and stepped down when his father died. After mourning he returned via accounting clerk posts to history-office editorial duty on law and calendrics, then frontier and Kuaiji appointments. He died in office before reaching his capital post as governor of Shanyang. Liu Hong’s mathematics stood alone; he wrote on the seven celestial wanderers. With Yong he compiled the calendar annals and tested officials’ sky lore. His decade-long Qianxiang revision aligned lunar and solar motion with observation." Supplementary lore praises Hong’s classical breadth and dedication to calendrical inquiry. His tenure as Qu minister earned disciplined affection and official esteem."