1
夫天地之所貴者生也，萬物之所尊者人也，役智窮神，無幽不察，是以動作云為，皆應天地之象。 古先聖哲，擬辰極，制渾儀。 夫陰陽二氣，陶育羣品，精象所寄，是為日月。 羣生之性，章為五才，五才之靈，五星是也。 曆所以擬天行而序七耀，紀萬國而授人時。 黃帝使大撓造六甲，容成制曆象，羲和占日，常儀占月。 少昊氏有鳳鳥之瑞，以鳥名官，而鳳鳥氏司曆。 顓頊之代，南正重司天，北正黎司地。 堯復育重黎之後，使治舊職，分命羲、和，欽若昊天。 故虞書曰：「朞三百有六旬六日，以閏月定四時成歲。」 其後授舜，曰：「天之曆數在爾躬。」 舜亦以命禹。 爰及殷、周二代，皆創業革制，而服色從之。 順其時氣，以應天道，萬物羣生，蒙其利澤。 三王既謝，史職廢官，故孔子正春秋以明司曆之過。 秦兼天下，自以為水德，以十月為正，服色上黑。
Life is what Heaven and Earth prize most; among the myriad creatures, it is humanity that commands the highest regard. By applying intellect and pushing the mind to its limits, nothing concealed lies beyond notice; thus every deed and utterance answers to the patterns of Heaven and Earth. The sage rulers of antiquity took the celestial pole as their model and devised the armillary sphere. Yin and yang—the two vital breaths—shape and nourish all living kinds; the luminous bodies in which their essence is embodied are the sun and the moon. The character of all living things displays itself in the five endowments, and the animating power of those five endowments resides in the five planets. The calendar exists to mirror the course of Heaven, set the seven luminaries in sequence, register the affairs of the myriad realms, and grant humankind its proper seasons. The Yellow Emperor charged Da Rao with devising the six jia cycle; Rong Cheng established the calendar and its celestial models; Xi He took charge of observing the sun, and Chang Yi of observing the moon. Under Shaohao the phoenix appeared as a favorable omen; he named his ministries after birds, and the Ministry of the Phoenix Bird oversaw the calendar. In Zhuanxu's time Chong, rectifier of the south, governed affairs of Heaven, while Li, rectifier of the north, governed affairs of Earth. Yao in turn raised up the line of Chong and Li, restored them to their former offices, and separately appointed Xi and He with the charge to honor the great Heaven. Hence the Yu section of the Documents states: "The full cycle is three hundred sixty-six days and a fractional sixth; intercalary months establish the four seasons and complete the year." He then passed the charge to Shun, saying, "Heaven's calendrical mandate is vested in you alone." Shun in turn entrusted the same charge to Yu. Through the Yin and Zhou dynasties alike, each new house founded its rule and revised its institutions, and court ritual colors followed suit. By aligning with the seasonal breaths they answered Heaven's Way, and the myriad creatures reaped the benefit of that harmony. Once the Three Kings were gone, the historiographical service lapsed and its posts stood empty; for this reason Confucius rectified the Spring and Autumn to lay bare the errors of those charged with the calendar. When Qin united the empire, it claimed the Water phase as its own, took the tenth month as the year's beginning, and elevated black in ritual dress.
2
漢興，襲秦正朔，北平侯張蒼首言律曆之事，以顓頊曆比於六曆，所失差近。 施用至武帝元封七年，太中大夫公孫卿、壺遂、太史令司馬遷等，言曆紀廢壞，宜改正朔，易服色，所以明受之於天也。 乃詔遂等造漢曆。 選鄧平、長樂司馬可及人間治曆者，二十餘人。 方士唐都分天部，落下閎運算轉曆。 其法積八十一寸，則一日之分也。 閎與鄧平所治同。 於是皆觀星度，日月行，更以算推，如閎、平法，一月之日二十九日八十一分日之四十三。 詔遷用鄧平所造八十一分律曆，以平為太史丞。 至元鳳三年，太史令張壽王上書，以為元年用黃帝調曆，「今陰陽不調，更曆之過」。 詔下主曆使者鮮于妄人與治曆大司農中丞麻光等二十餘人雜候晦朔弦望二十四氣。 又詔丞相、御史、大將軍、右將軍史各一人雜候上林清臺，課諸疏密，凡十一家。 起三年盡五年。 壽王課疏遠。 又漢元年不用黃帝調曆，劾壽王逆天地，[1]大不敬。 詔勿劾。 復候，盡六年，太初曆第一。 壽王曆乃太史官殷曆也。 壽王再劾不服，竟下吏。 至孝成時，劉向總六曆，列是非，作五紀論。 向子歆作三統曆以說春秋，屬辭比事，雖盡精巧，非其實也。 班固謂之密要，故漢曆志述之。 校之何承天等六家之曆，雖六元不同，分章或異，至今所差，或三日，或二日數時，考其遠近，率皆六國及秦時人所造。 其術斗分多，上不可檢於春秋，下不驗於漢、魏，雖復假稱帝王，祇足以惑時人耳。
At the founding of Han the dynasty retained Qin's calendar and new year's day; Zhang Cang, Marquis of Beiping, was the first to address matters of pitch-pipes and the calendar, comparing the Zhuanxu system with the six rival calendars—and the discrepancies proved almost equally slight. This system remained in use until the seventh year of Yuanfeng, when Gongsun Qing, Hu Sui, Sima Qian as Grand Astrologer, and others declared that the calendrical cycle had broken down and that the new year's month and ritual colors should be revised to show plainly that the mandate came from Heaven. The emperor accordingly ordered Sui and his colleagues to draft the Han calendar. Deng Ping, Director of Changle, Sima Ke, and more than twenty calendar specialists drawn from the general population were chosen for the task. The adept Tang Du apportioned the celestial regions, while Luoxia Hong performed the calculations that set the calendar in motion. By that method, eighty-one fractional parts accumulate to constitute one day's fractional share. Luoxia Hong and Deng Ping worked from the same principles. They then observed stellar positions and the courses of the sun and moon; further reckoning confirmed Hong and Ping's formula—a lunar month comprised twenty-nine days and forty-three eighty-firsts of a day. An edict directed Sima Qian to adopt the eighty-first-part pitch-pipe calendar that Ping had devised, and Ping was appointed Assistant Grand Astrologer. In the third year of Yuanfeng, Grand Astrologer Zhang Shouwang memorialized the throne, arguing that the inaugural year had employed the Yellow Emperor's tuning calendar: "Yin and yang are now out of harmony because the calendar was changed." The court ordered the Chief Calendar Officer Xianyu Wangren, Assistant Director of Agriculture Ma Guang, and more than twenty colleagues jointly to observe new moons, full moons, quarter moons, and the twenty-four seasonal nodes. A further edict assigned one clerk each from the chancellor, imperial secretary, grand marshal, and right general to observe at the Clear Terrace in Shanglin and grade every system for tightness or slack—eleven schools in all. Observations began in the third year and ran through the fifth. Shouwang's calendar ranked lowest for being loose and far off the mark. Moreover, Han's inaugural year had not in fact used the Yellow Emperor's tuning calendar; Shouwang was charged with defying Heaven and Earth—a grave offense of supreme irreverence. An edict forbade proceeding with the impeachment. Observations were resumed through the sixth year, and the Taichu calendar placed first. Shouwang's system turned out to be the Yin calendar preserved in the Grand Astrologer's archives. When Shouwang was impeached a second time and refused to yield, he was at last remanded to the magistrates. Under Emperor Cheng, Liu Xiang surveyed the six calendars, weighed their merits and faults, and wrote the Treatise on the Five Cycles. Xiang's son Xin devised the Triple Concordance calendar to interpret the Spring and Autumn; for all its ingenious phrasing and parallel structures, it did not accord with fact. Ban Gu pronounced it subtle and essential, which is why the Han Treatise on the Calendar preserves it. Set against the calendars of He Tiancheng and five other schools—though their six origins differed and their fractional schemes diverged—the accumulated error down to the present runs to three days, or two days and several hours; judged by antiquity, nearly all were the work of scholars from the Six States and Qin. Their methods employ oversized dipper fractions: they cannot be tested against the Spring and Autumn in earlier times nor confirmed against Han and Wei in later ones; even when they falsely attribute themselves to ancient emperors, they are fit only to delude the men of their own day.
3
光武建武八年，太僕朱浮上言曆紀不正，宜當改治。 時所差尚微，未遑考正。 明帝永平中，待詔楊岑、張盛、景防等典治曆，但改易加時弦望，未能綜校曆元也。 至元和二年，太初失天益遠，宿度相覺浸多，候者皆知日宿差五度，冬至之日在斗二十一度，晦朔弦望，先天一日。 章帝召治曆編訢、李梵等綜校其狀。 [2]遂下詔書稱：「春秋保乾圖曰：『三百年斗曆改憲。』 史官用太初鄧平術，有餘分一，在三百年之域，行度轉差，浸以繆錯，琁璣不正，文象不稽。 冬至之日，日在斗二十一度，[3]先立春一日，則四分之立春日也。 而以折獄斷大刑，於氣已逆； 用望平和，蓋亦遠矣。 今改行四分，以遵堯順孔，奉天之文，同心敬授，儻獲咸熙。」 於是四分法施行。 黃帝以來諸曆以為冬至在牽牛初者皆黜焉。
In the eighth year of Jianwu under Emperor Guangwu, Grand Minister Zhu Fu submitted a memorial stating that the calendrical cycle was wrong and should be corrected. The error was still small, and the court had no leisure to undertake a full correction. Under Emperor Ming in the Yongping period, Yang Cen, Zhang Sheng, Jing Fang, and other awaiting-edict scholars oversaw the calendar but merely adjusted the hours at which quarter moons were marked; they could not carry out a full verification of the calendrical origin. By the second year of Yuanhe the Taichu system's drift from Heaven had grown pronounced; stellar lodge positions showed a steadily widening gap; every observer knew the sun's lodge was five degrees in error, winter solstice stood in Dipper 21, and new moons, full moons, and quarters appeared a day too early. Emperor Zhang summoned the calendar experts Bian Xin, Li Fan, and others to conduct a comprehensive review. He then issued an edict citing the Spring and Autumn and the Baojian tu: "Every three hundred years the Dipper calendar must revise its standard. The court astronomers still employ the Taichu method of Deng Ping, which retains a surplus fraction of one; over the three-hundred-year span its computed motion drifts ever further astray; the sighting tube no longer points true, and the celestial patterns fail to match. On winter solstice the sun stood in Dipper 21—one day before Establishment of Spring, which in the Quarter-Day calendar is itself the day of Establishment of Spring. Yet to settle lawsuits and pronounce capital sentences when the seasonal breath had already reversed course— to take the full moon as the moment of evenhanded justice was equally wide of the mark. We now adopt the Quarter-Day system to follow Yao, accord with Confucius, uphold Heaven's pattern, and with one accord reverently grant the seasons—perhaps thereby attaining universal radiance." The Quarter-Day calendar was thereupon put into effect. Every calendar since the Yellow Emperor that had placed winter solstice at the start of the Ox constellation was set aside.
4
和帝永元十四年，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，不如夏曆密。」 其年十一月甲寅，詔曰：「漏所以節時分，定昏明。 昏明長短，起於日去極遠近，日道周圜，不可以計率分。 官漏九日增減一刻，違失其實，以晷景為刻，密近有驗。 今下晷景漏刻四十八箭。」 其二十四氣日所在，并黃道去極、晷景、漏刻、昏明中星，並列載于續漢律曆志。
In the fourteenth year of Yongyuan under Emperor He, Grand Astrologer Huo Rong, then an awaiting-edict, wrote: "The official water clock gains or loses one notch every nine days on average, failing to match Heaven; at times the discrepancy reaches two and a half notches—it is less exact than the Xia calendar." In the eleventh month of that year, on the day jia-yin, an edict declared: "The clepsydra exists to regulate the hours and fix dusk and dawn. The varying length of dusk and dawn stems from how far the sun stands from the pole; the sun's path is circular and cannot be apportioned by simple proportional calculation. The official practice of adding or subtracting one notch every nine days misses the reality; deriving the notches from gnomon shadows is tighter and can be verified. Let the gnomon-based clepsydra of forty-eight arrow-tubes be issued." The sun's lodge on each of the twenty-four qi, together with the yellow path's polar distance, gnomon shadow, clepsydra gradations, hours of dusk and dawn, and midnight culminating stars, are all tabulated in the Continued Han Treatise on Pitch-pipes and the Calendar.
5
安帝延光三年，[4]中謁者亶誦上書言當用甲寅元，河南梁豐云當復用太初。 尚書郎張衡、周興皆審曆，數難誦、豐，或不能對，或云失誤。 衡等參案儀注，考往校今，以為九道法最密。 詔下公卿詳議。 太尉愷等參議：「太初過天一度，月以晦見西方。 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。」 議者不同。 尚書令忠上奏：「天之曆數，不可任疑從虛，以非易是。」 亶等遂寢。
In the third year of Yanguang under Emperor An, Palace Attendant Xuan recited a memorial arguing for the jia-yin origin, while Liang Feng of Henan urged a return to the Taichu system. Secretarial Gentlemen Zhang Heng and Zhou Xing were both expert calendarists; they repeatedly cross-examined Xuan and Feng by calculation, and sometimes the latter could not answer, or their figures were shown to be wrong. Heng and his colleagues reviewed the observatory regulations, compared past with present, and judged the Nine Paths method the most precise. An edict referred the matter to the chief ministers for detailed deliberation. Grand Marshal Kai and others reported jointly: "The Taichu runs one degree ahead of Heaven, and the moon is seen in the west on the last day of the month. Under Yuanhe the court switched to the Quarter-Day system; although the Quarter-Day is tighter than the Taichu, it too is in error. Neither may be adopted. The jia-yin origin accords with Heaven and agrees with the prognostic texts—it should be put into practice." The disputants could not agree. Secretarial Director Zhong memorialized: "Heaven's calendrical numbers must not be left to doubt and groundless speculation, substituting error for truth." Xuan and his party accordingly let the proposal drop.
6
靈帝熹平四年，五官郎中馮光、沛相上計掾陳晃等言：「曆元不正，故盜賊為害。 曆當以甲寅為元，不用庚申，乞本庚申元經緯明文。」 詔下三府，與儒林明道術者詳議。 羣臣會司徒府集議。 議郎蔡邕曰：「曆數精微，術無常是。 漢興承秦，曆用顓頊，元用乙卯。 百有二歲，孝武皇帝始改太初，元用丁丑。 行之百八十九歲，孝章帝改從四分，元用庚申。 今光等以庚申為非，甲寅為是。 按曆法，黃帝、顓頊、夏、殷、周、魯，各自有元。 光、晃所援，則殷曆元也。 昔始用太初丁丑之後，六家紛錯，爭訟是非。 張壽王挾甲寅元以非漢曆，雜候清臺，課在下第。 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。 故延光中，亶誦亦非四分，言當用甲寅元，公卿參議，竟不施行。 且三光之行，遲速進退，不必若一。 故有古今之術。 今術之不能上通於古，亦猶古術不能下通於今也。 又光、晃以考靈耀為本，二十八宿度數至日所在，錯異不可參校。 元和二年用至今九十二歲，而光、晃言陰陽不和，姦臣盜賊，皆元之咎。 元和詔書，文備義著，非羣臣議者所能變易。」 三公從邕議，以光、晃不敬，正鬼薪法。 詔書勿治罪。
In the fourth year of Xiping under Emperor Ling, Palace Gentleman Feng Guang and Chen Huang, the Pei commandery reporting officer, declared: "The calendrical origin is wrong, and that is why robbers and rebels inflict harm. They argued that the calendar should adopt jia-yin as its origin rather than geng-shen, and requested the canonical and weft passages underlying the geng-shen origin." An edict referred the question to the Three Excellencies and to Confucian scholars versed in the Way and its arts for thorough debate. The officials gathered at the Minister of Education's headquarters to debate jointly. Consulting Gentleman Cai Yong declared: "Calendrical reckoning is exquisitely subtle; no method remains permanently correct. At Han's founding the dynasty took over Qin's institutions; it employed the Zhuanxu calendar with the yi-mao origin. One hundred and two years later Emperor Wu first adopted the Taichu calendar, with the ding-chou origin. That system remained in use for one hundred eighty-nine years until Emperor Zhang switched to the Quarter-Day calendar with the geng-shen origin. Guang and his allies now declare geng-shen wrong and jia-yin correct. By calendrical reckoning, the Yellow Emperor, Zhuanxu, Xia, Yin, Zhou, and Lu each possessed its own origin. The origin that Guang and Huang invoke is that of the Yin calendar. From the first adoption of the Taichu ding-chou origin, six rival schools quarreled without resolution, contesting right and wrong. Zhang Shouwang had wielded the jia-yin origin against the Han calendar; joint observations at the Clear Terrace ranked his system last. The Taichu system's empirical tests showed no gaps or failures. Thus, although it was not the origin favored by prognostic literature, it had proved effective in its own day. Moreover, since the Quarter-Day calendar came into use, examination of celestial motion has shown it tighter than the Taichu—the newer origin has again proved effective in our own time. That is why in the Yanguang period Xuan's memorial likewise rejected the Quarter-Day and urged the jia-yin origin—yet after joint deliberation by the chief ministers, the change was never carried out. Besides, the courses of the sun, moon, and planets—now slower, now swifter, advancing and retreating—need not be uniform. Hence there are methods suited to antiquity and methods suited to the present. Methods of today cannot be extended upward to antiquity, just as ancient methods cannot be extended downward to the present. Furthermore, Guang and Huang treat the Kaoling yao as their authority, yet the lodge-degree figures for the sun's daily position contain irreconcilable discrepancies. The Quarter-Day calendar adopted in Yuanhe 2 has been in use ninety-two years, yet Guang and Huang blame every disharmony of yin and yang and every traitor and thief on the calendrical origin. The Yuanhe edict is complete in wording and firm in intent; it is not a matter that debating ministers may overturn." The Three Excellencies sided with Yong; Guang and Huang were judged irreverent and sentenced to convict labor under the ghost-labor statute. An edict forbade imposing punishment.
7
何承天曰：夫曆數之術，若心所不達，雖復通人前識，無救其為敝也。 是以多歷年歲，未能有定。 四分於天，出三百年而盈一日。 積代不悟，徒云建曆之本，必先立元，假言讖緯，遂關治亂，此之為蔽，亦已甚矣。 劉歆三統法尤復疏闊，方於四分，六千餘年又益一日。 揚雄心惑其說，采為太玄，班固謂之最密，著于漢志； 司彪因曰「自太初元年始用三統曆，施行百有餘年。」 曾不憶劉歆之生，不逮太初，二三君子言曆，幾乎不知而妄言歟。
He Tiancheng remarked: In the art of calendrical reckoning, when the mind fails to comprehend the method, even the keenest experts cannot keep the system from breaking down. That is why, across many years and generations, no lasting standard has been settled. Measured against Heaven, the Quarter-Day calendar accumulates one full day's error every three hundred years. Age after age failed to notice the drift, insisting that every new calendar must begin by fixing an origin, invoking prognostic texts to tie calendrical choice to the rise and fall of states—a delusion already carried to extremes. Liu Xin's Triple Concordance system is especially loose; set beside the Quarter-Day, it accumulates yet another day's error every six thousand years or so. Yang Xiong was beguiled by the system and wove it into his Taixuan; Ban Gu pronounced it the most exact and entered it in the Han Treatises. Sima Biao then wrote, "The Triple Concordance calendar came into use in the first year of Taichu and remained in force for more than a century. Yet he forgot that Liu Xin was born only after the Taichu reform—were these worthies, one wonders, debating the calendar in near-total ignorance?
8
光和中，穀城門候劉洪始悟四分於天疏闊，更以五百八十九為紀法，百四十五為斗分，造乾象法，又制遲疾曆以步月行。 方於太初、四分，轉精微矣。 魏文帝黃初中，太史丞韓翊以為乾象減斗分太過，後當先天，造黃初曆，以四千八百八十三為紀法，一千二百五為斗分。 其後尚書令陳羣奏，以為「曆數難明，前代通儒多共紛爭。 黃初之元，以四分曆久遠疏闊，大魏受命，宜正曆明時。 韓翊首建黃初，猶恐不審，故以乾象互相參校。 歷三年，更相是非，舍本即末，爭長短而疑尺丈，竟無時而決。 按三公議，皆綜盡曲理，殊塗同歸，欲使效之璿璣，各盡其法，一年之間，得失足定，合於事宜。」 奏可。 明帝時，尚書郎楊偉制景初曆，施用至于晉、宋。 古之為曆者，鄧平能修舊制新，劉洪始減四分，又定月行遲疾，楊偉斟酌兩端，以立多少之衷，因朔積分設差，以推合朔月蝕。 此三人，漢、魏之善曆者。 然而洪之遲疾，不可以檢春秋，偉之五星，大乖於後代，斯則洪用心尚疏，偉拘於同出上元壬辰故也。
During Guanghe, Liu Hong of Gucheng Gate first saw that the Quarter-Day calendar no longer hugged the sky; he reset the era divisor to 589 and the dipper fraction to 145, created the Qianxiang system, and added a slow-fast lunar ephemeris. Set beside the Taichu and Quarter-Day systems, it marked a clear advance in precision. Early in Wei Wendi's Huangchu reign, Deputy Astrologer Han Yi argued that Qianxiang had pared the dipper fraction too sharply and would eventually run ahead of the sky; he therefore drafted the Huangchu calendar with an era divisor of 4883 and a dipper fraction of 1250. Minister Chen Qun then submitted a memorial: "Calendar theory is notoriously opaque, and fine scholars of past dynasties rarely agreed. The Huangchu reform began from the premise that the old Quarter Remainder had grown hopelessly slack; now that Wei held the mandate, the seasons needed a new calendar. Han Yi opened the discussion, yet his work still lacked rigorous vetting, so the Qianxiang system was brought in for a side-by-side test. For three years the rival camps traded contradictory verdicts, quibbling over minutiae while missing the main issue, disputing measurements without end—and never reached a settlement. The three high ministers replied that every proposal drew on solid classical principle and aimed at the same end by different routes; the thing to do was to run each system on the armillary instrument for a full year—then success and failure would show plainly, which fit the practical need." The throne approved the memorial. Under Emperor Ming, Secretariat official Yang Wei produced the Jingchu calendar, which remained in use through Jin and Song. The great calendar makers of old each advanced the art in a different way: Deng Ping renewed inherited practice; Liu Hong first trimmed the quarter-day excess and modeled the moon's variable speed; Yang Wei split the difference between rival systems, then built correction tables from accumulated conjunction data to predict new moons and eclipses. These three were the foremost calendrical experts of Han and Wei. Yet Liu Hong's lunar tables fail when tested against the Spring and Autumn, and Yang Wei's planetary positions drift badly in later times—Hong's approach was still insufficiently rigorous, and Wei was hamstrung by forcing every calculation back to the shared renchen upper origin.
9
魏明帝景初元年，改定曆數，以建丑之月為正，改其年三月為孟夏四月。 其孟仲季月，雖與正歲不同，至於郊祀、迎氣，祭祠、烝嘗，巡狩、蒐田，分至啟閉，班宣時令，皆以建寅為正。 三年正月，帝崩，復用夏正。
In Jingchu 1 under Wei Emperor Ming the court reset the calendar: the chou month became month one, and what had been the third month was relabeled early summer, the fourth month. Seasonal names no longer matched the Xia calendar, yet for suburban rites, qi-welcoming ceremonies, temple offerings, seasonal sacrifices, royal tours and hunts, the four turning points of the year, and the publication of seasonal edicts, the yin month still counted as the true pivot of the year. When the emperor died in the first month of the third year, the court reverted to the Xia-style first month.
10
楊偉表曰：「臣攬載籍，斷考曆數，時以紀農，月以紀事，其所由來，遐而尚矣。 乃自少昊，則玄鳥司分，顓頊帝嚳，則重、黎司天，唐帝、虞舜則羲、和掌日。 三代因之，則世有日官。 日官司曆，則頒之諸侯，諸侯受之，則頒于境內。 夏后之代，羲、和湎淫，廢時亂日，則書載胤征。 由此觀之，審農時而重人事者，歷代然也。 逮至周室既衰，戰國橫騖，告朔之羊，廢而不紹，登臺之禮，滅而不遵。 閏分乖次而不識，孟陬失紀而莫悟，大火猶西流，而怪蟄蟲之不藏也。 是時也，天子不協時，司曆不書日，諸侯不受職，日御不分朔，人事不恤，廢棄農時。 仲尼之撥亂於春秋，託褒貶糾正，司曆失閏，則譏而書之，登臺頒朔，則謂之有禮。 自此以降，暨于秦、漢，乃復以孟冬為歲首，閏為後九月，中節乖錯，時月紕繆，加時後天，蝕不在朔，累載相襲，[5]久而不革也。 至武帝元封七年，始乃寤其繆焉。 於是改正朔，更曆數，使大才通人，造太初曆，校中朔所差，以正閏分，課中星得度，以考疏密，以建寅之月為正朔，以黃鍾之月為曆初。 其曆斗分太多，後遂疏闊。 至元和二年，復用四分曆，施而行之。 至于今日，考察日蝕，率常在晦，是則斗分太多，故先密後疏而不可用也。 是以臣前以制典餘日，推考天路，稽之前典，驗之食朔，詳而精之，更建密歷，則不先不後，古今中天。 以昔在唐帝，協日正時，允釐百工，咸熙庶績也。 欲使當今國之典禮，凡百制度，皆韜合往古，郁然備足，乃改正朔，更曆數，以大呂之月為歲首，以建子之月為曆初。 臣以為昔在帝代，則法曰顓頊，曩自軒轅，則曆曰黃帝。 暨至漢之孝武，革正朔，更曆數，改元曰太初，因名太初曆。 今改元為景初，宜曰景初曆。 臣之所建景初曆，法數則約要，施用則近密，治之則省功，學之則易知。 雖復使研桑心算，隸首運籌，重、黎司晷，羲、和察景，以考天路，步驗日月，究極精微，盡術數之極者，皆未如臣如此之妙也。 是以累代曆數，皆疏而不密，自黃帝以來，改革不已。
Yang Wei wrote: "I have combed the sources and worked through the calendar's numbers: seasons organize agriculture, months organize administration—and that arrangement goes back to remote antiquity. Already under Shaohao the swallow asterism marked the equinoxes; under Zhuanxu and Di Ku, Chong and Li were charged with celestial regulation; under Yao and Shun, Xi and He took charge of the solar calendar. The three early dynasties kept the office, so every reign maintained specialists charged with the calendar. Those officers set the calendar and issued it to the regional lords, who in turn published it inside their own territories. Later in the Xia era, Xi and He neglected their duty and deranged the calendar, which is why the Book of Documents preserves the punitive campaign against them. From this it is clear that every dynasty has treated careful timing for agriculture and attention to civic order as a standing principle. By late Zhou the realm fractured: monthly temple announcements lapsed, and observatory ceremonies died out. Leap months drifted out of place, the civil year's first month wandered, and while Antares still hung in the western sky, courtiers marveled at insects that failed to burrow—blind to the calendar's chaos. The king no longer coordinated the seasons, official scribes stopped reliably dating events, regional rulers ignored their responsibilities, astronomers lost track of true conjunctions, and governance neglected both people and the agricultural timetable. Confucius used the Spring and Autumn to restore order through moral judgment: he mocked missed intercalations, yet praised rulers who climbed the terrace to proclaim the civil calendar as behaving ritually. Down through Qin and Han the year again began in early winter, leap months were tacked after the ninth month, solar terms and lunar months slipped badly out of alignment, ephemeris corrections ran behind the sky, eclipses missed conjunction, and the mistake persisted reign after reign [5] without overhaul. Only in the seventh year of Yuanfeng did Emperor Wu recognize the fault. He reformed the civil calendar, commissioned the Taichu system, adjusted the leap fraction against true conjunctions, checked stellar longitudes, adopted jianyin as the first month of the year, and anchored the epoch in the yellow-bell month. The Taichu system's fractional remainder for the Dipper was too big, so in later reigns it grew increasingly imprecise. From Yuanhe 2 the Quarter-Day calendar returned and was put back into use. Eclipse records down to our own day show conjunctions drifting to month-end—proof the fractional excess was too large, yielding an apparently tight fit that soon loosened into uselessness. Hence I reworked the official surplus-day reckoning, checked it against eclipses and prior canons, and drew up a denser calendar that tracks the sky evenly—neither ahead nor behind—between antiquity and today. This matches Yao's age, when aligning the days and seasons allowed every office to function and every undertaking to thrive. He wants current state ritual and every institution to echo ancient models; so he resets the civil year, revises the calendrical constants, begins the year in the great-lü month, and sets the computational epoch in the jianzi month. I note that each great reform named its system—the Zhuanxu method in the age of the emperors, the Yellow Emperor's calendar from Xuanyuan onward. Han Emperor Wu likewise reformed the new year, revised the constants, and took the era name Taichu—giving the system its title. Since the reign era is now Jingchu, the new system ought to be called the Jingchu calendar. The Jingchu scheme I propose keeps the parameters lean yet numerically tight, runs efficiently in practice, and is straightforward to learn. No amount of legendary arithmetical skill—merchant Yan's mental sums, Lishou's rods, Chong and Li at the sundial, Xi and He at the solstice shadow—could match the precision I have reached. That is why earlier dynasties' calendars always erred on the loose side, and why reform has followed reform ever since the Yellow Emperor.
11
壬辰元以來，至景初元年丁巳，歲積四千四十六，算上。 此元以天正建子黃鍾之月為曆初，元首之歲夜半甲子朔旦冬至。
From the Gengchen origin through the Jingchu era's first year (a Dingsi year in the cycle), the accumulated count is 4046, inclusive on the upper stem count. The epoch uses the true winter month (jianzi) aligned with the yellow-bell pitch month as calendar zero, so at the origin year's start a Jiazi day begins at midnight exactly at winter solstice.
12
元法，萬一千五十八。
The origin divisor (yuanfa): 11,058.
13
紀法，千八百四十三。
The era divisor (jifa): 1,843.
14
紀月，二萬二千七百九十五。
Months per full era cycle (jiyue): 22,795.
15
章歲，十九。
Rule years (zhang sui): nineteen.
16
章月，二百三十五。
Rule months (zhang yue): 235.
17
章閏，七。
Rule intercalations (zhang run): seven.
18
通數，十三萬四千六百三十。
The communication number (tong shu): 134,630.
19
日法，四千五百五十九。
The day divisor (ri fa): 4,559.
20
餘數，九千六百七十。
The remainder number (yu shu): 9,670.
21
周天，六十七萬三千一百五十。
The circuit-of-heaven constant (zhou tian): 673,150.
22
歲中，[6]十二。
Mid-year qi [6]: twelve.
23
氣法，十二。
The qi divisor (qi fa): twelve.
24
沒分，六萬七千三百一十五。
The extinction fraction (mo fen): 67,315.
25
沒法，九百六十七。
The extinction divisor (mo fa): 967.
26
月周，二萬四千六百三十八。
The lunar circuit constant (yue zhou): 24,638.
27
通法，四十七。
The communication divisor (tong fa): forty-seven.
28
會通，七十九萬一百一十。 [7]
The conjunction communication constant (hui tong): 790,110. Editorial note 7.
29
朔望合數，六萬七千三百一十五。
The new- and full-moon combined number (shuo wang he shu): 67,315.
30
入交限數，七十二萬二千七百九十五。
The node-crossing limit number (ru jiao xian shu): 722,795.
31
通周，十二萬五千六百二十一。
The communication circuit constant (tong zhou): 125,621.
32
周日日餘，二千五百二十八。
The circuit-day remainder (zhou ri ri yu): 2,528.
33
周虛，二千三十一。
The circuit void (zhou xu): 2,031.
34
斗分，四百五十五。
The dipper fraction (dou fen): 455.
35
甲子紀第一：
Era series I: Jiazi.
36
紀首合朔，月在日道裏。
At the initial conjunction of the era the moon is north of the ecliptic.
37
交會差率，四十一萬二千九百一十九。
The eclipse-node difference rate for this era: 412,919.
38
遲疾差率，十萬三千九百四十七。
The lunar anomaly difference rate: 103,947.
39
甲戌紀第二：
Era series II: Jiaxu.
40
紀首合朔，月在日道裏。
At this era's starting conjunction the moon is inside the solar track.
41
交會差率，五十一萬六千五百二十九。
The conjunction-node difference rate: 516,529.
42
遲疾差率，七萬三千七百六十七。
The anomaly difference rate: 73,767.
43
甲申紀第三：
Era series III: Jiashen.
44
紀首合朔，月在日道裏。
At era head the moon lies inside the ecliptic at conjunction.
45
交會差率，六十二萬一百三十九。
The conjunction-node difference rate: 621,139.
46
遲疾差率，四萬三千五百八十七。
The anomaly difference rate: 43,587.
47
甲午紀第四：
Era series IV: Jiawu.
48
紀首合朔，月在日道裏。
At the era-head conjunction the moon remains inside the solar path.
49
交會差率，七十二萬三千七百四十九。
The conjunction-node difference rate: 723,749.
50
遲疾差率，一萬三千四百七。
The anomaly difference rate: 13,407.
51
甲辰紀第五：
Era series V: Jiachen.
52
紀首合朔，月在日道裏。 [8]
At the era-head conjunction the moon is on the inner side of the sun's path. Editorial note 8.
53
交會差率，三萬七千二百四十九。
The conjunction-node difference rate: 37,249.
54
遲疾差率，一十萬八千八百四十八。
The anomaly difference rate: 108,848.
55
甲寅紀第六：
Era series VI: Jiayin.
56
紀首合朔，月在日道裏。
At the era-head conjunction the moon lies inside the solar track.
57
交會差率，十四萬八百五十九。
The conjunction-node difference rate: 140,859.
58
遲疾差率，七萬八千六百六十八。
The anomaly difference rate: 78,668.
59
交會紀差，十萬三千六百一十。 求其數之所生者，置一紀積月以通數乘之，會通去之，所去之餘，紀差之數也。 以之轉加前紀，則得後紀。 加之未滿會通者，則紀首之歲天正合朔，月在日道裏。 滿去之，則月在日道表。 加表滿在裏，加裏滿在表。
The step from one era's conjunction-node rate to the next: 103,610. To derive it: take the months in one era, multiply by the communication number, discard full multiples of the conjunction communication constant; what remains defines the era increment. Add that increment to the previous era's rate to get the next era's rate. If the sum stays below the conjunction communication constant, the new moon at the era's first civil year lies inside the solar track. If it crosses and you subtract the product, the moon at conjunction lies outside the solar track. Stepping from outside, a full increment places the moon inside; stepping from inside, a full increment places it outside.
60
遲疾紀差，三萬一百八十。 求其數之所生者，置一紀積月，以通數乘之，通周去之，餘以減通周，所減之餘，紀差之數也。 以之轉減前紀，則得後紀。 不足減者，加通周。
The slow-fast era difference: 30,180. To derive it: multiply one era's months by the communication number, reduce modulo the communication circuit constant, subtract the remainder from that constant; what is left is the era increment. Subtract that decrement from the prior era's anomaly rate to obtain the next. If you cannot subtract, add the communication circuit constant first.
61
求次元紀差率，轉減前元甲寅紀差率，餘則次元甲子紀差率也。 求次紀，如上法也。
For the next grand origin, subtract the previous Jiayin-era rate from the Jiazi-era rate as prescribed to get the new Jiazi increment. To advance to the following era, repeat the same rule.
62
推朔積月術曰：置壬辰元以來，盡所求年，外所求，以紀法除之，所得算外，所入紀第也，餘則入紀年數。 年以章月乘之，如章歲而一為積月，不盡為閏餘。 閏餘十二以上，其年有閏。 閏月以無中氣為正。
To count lunations from the Renchen epoch up to but not including the target year, divide by 1,843: the quotient plus one gives the era index, the remainder the year within that era. Multiply that year-count by 235 and divide by 19 for total months; the remainder is the leap fraction. A leap year occurs when the intercalary remainder reaches twelve or higher. The leap month is the lunar month that contains no major solar term.
63
推朔術曰：以通數乘積月，為朔積分，如日法而一為積日，不盡為小餘。 以六十去積日，餘為大餘。 大餘命以紀，算外，所求年天正十一月朔日也。 求次月，加大餘二十九，小餘二千四百一十九，小餘滿日法從大餘，命如前，次月朔日也。 小餘二千一百四十以上，其月大也。
To compute conjunction instants, multiply accumulated months by 134,630; divide by 4,559 for whole days; the residue is the fractional day. Reduce the day count modulo sixty for the sexagenary stem-branch index. Index that remainder against the sexagenary cycle to name the day of the eleventh month's conjunction in the target civil year. For the following month add 29 to the day count and 2,419 to the fraction, carrying overflows by 4,559 into the day count, then read off the next conjunction. If the fractional part is 2,140 or greater, the month is a long thirty-day month.
64
推弦望，加朔大餘七，小餘千七百四十四，小分一，小分滿二從小餘，小餘滿日法從大餘，大餘滿六十去之，餘命以紀，算外，上弦日也。 又加得望、下弦、後月朔。 其月蝕望者，定小餘，如所近中節間限，限數以下者，[9]算上為日。 望在中節前後各四日以還者，視限數； 望在中節前後各五日以上者，視間限。
For quarters: add 7 days, 1,744 parts, and one small part to the new-moon line, propagating carries through the small-part, fractional-day, and sexagenary levels; the result names the first-quarter day. Repeat the same addition chain to reach full moon, third quarter, and the next conjunction. When a lunar eclipse falls on the full moon, adjust the fractional day by the gap limit of the nearest major term; if the value lies at or under the limit number, per note [9], round upward to fix the day. When full moon lies within four days before or after the major term, use the standard limit figure. When full moon sits five or more days away from the major term, switch to the wider gap-limit criterion instead.
65
推二十四氣術曰：置所入紀年，外所求，以餘數乘之，滿紀法為大餘，不盡為小餘。 大餘滿六十去之，餘命以紀，算外，天正十一月冬至日也。 求次氣，加大餘十五，小餘四百二，小分十一，小分滿氣法從小餘，小餘滿紀法從大餘，[10]命如前，次氣日也。
To place the twenty-four solar terms: take years elapsed within the current era, not counting the target year, multiply by 9,670, divide by 1,843 for the day count, and keep the fractional remainder. Reduce the day line modulo sixty and read it against the sexagenary cycle to name the winter solstice of the eleventh month. For each successive term add 15 days, 402 parts, and 11 minute-fractions, carrying overflows as prescribed; per note [10], index the result as before to name the next solar term.
66
推閏月術曰：以閏餘減章歲，餘以歲中乘之，滿章閏得一月，餘滿半法以上亦得一月。 數從天正十一月起，算外，閏月也。 閏有進退，以無中氣御之。
To find leap months: subtract the leap fraction from 19, multiply the residue by 12, divide by 7, and add an extra leap month whenever the remainder reaches half the divisor. Count forward from the eleventh month; the slot so reached is the intercalary month. Leap placement can shift forward or back, but the governing rule remains that no major solar term may fall within the leap month.
67
大雪，十一月節。 〈限數千二百四十二。 間限千二百四十八。〉
Major Snow, eleventh month, minor term. 〈Limit number: 1,242. Gap limit: 1,248.〉
68
冬至，十一月中。 〈限數千二百五十四。 間限千二百四十五。〉
Winter solstice, eleventh month, major term. 〈Limit number: 1,254. Gap limit: 1,245.〉
69
小寒，十二月節。 〈限數千二百三十五。 間限千二百二十四。〉
Minor Cold, twelfth month, minor term. 〈Limit number: 1,235. Gap limit: 1,224.〉
70
大寒，十二月中。 〈限數千二百一十三。 間限千一百九十二。〉
Major Cold, twelfth month, major term. 〈Limit number: 1,213. Gap limit: 1,192.〉
71
立春，正月節。 〈限數千一百七十二。 間限千一百四十七。〉 [11]
Beginning of spring, first month, minor term. 〈Limit number: 1,172. Gap limit: 1,147.〉 Editorial note 11.
72
雨水，正月中。 〈限數千一百二十二。 [12]間限千九十三。〉
Rain water, first month, major term. 〈Limit number: 1,122. See note [12]. Gap limit: 1,093.〉
73
驚蟄，二月節。 〈限數千六十五。 間限千三十六。〉 [13]
Waking of insects, second month, minor term. 〈Limit number: 1,065. Gap limit: 1,036.〉 Editorial note 13.
74
春分，二月中。 〈限數千八。 間限九百七十九。〉
Spring equinox, second month, major term. 〈Limit number: 1,008. Gap limit: 979.〉
75
清明，三月節。 〈限數九百五十一。 間限九百二十五。〉
Clear and Bright, third month, minor term. 〈Limit number: 951. Gap limit: 925.〉
76
穀雨，三月中。 〈限數九百。 間限八百七十九。〉
Grain Rain, third month, major term. 〈Limit number: 900. Gap limit: 879.〉
77
立夏，四月節。 〈限數八百五十七。 間限八百四十。〉
Summer begins, fourth month, minor term. 〈Limit number: 857. Gap limit: 840.〉
78
小滿，四月中。 〈限數八百二十三。 [14]間限八百一十二。〉 [15]
Lesser Fullness, fourth month, major term. 〈Limit number: 823. See note [14]. Gap limit: 812.〉 Editorial note 15.
79
芒種，五月節。 〈限數八百。 間限七百九十九。〉
Grain in Ear, fifth month, minor term. 〈Limit number: 800. Gap limit: 799.〉
80
夏至，五月中。 〈限數七百九十八。 間限八百一。〉 [16]
Summer solstice, fifth month, major term. 〈Limit number: 798. Gap limit: 801.〉 Editorial note 16.
81
小暑，六月節。 〈限數八百五。 間限八百一十五。〉
Lesser Heat, sixth month, minor term. 〈Limit number: 805. Gap limit: 815.〉
82
大暑，六月中。 〈限數八百二十五。 間限八百四十二。〉
Greater Heat, sixth month, major term. 〈Limit number: 825. Gap limit: 842.〉
83
立秋，七月節。 〈限數八百五十九。 間限八百八十三。〉
Autumn begins, seventh month, minor term. 〈Limit number: 859. Gap limit: 883.〉
84
處暑，七月中。 〈限數九百七。 間限九百三十五。〉
End of Heat, seventh month, major term. 〈Limit number: 907. Gap limit: 935.〉
85
白露，八月節。 〈限數九百六十二。 間限九百九十二。〉
White Dew, eighth month, minor term. 〈Limit number: 962. Gap limit: 992.〉
86
秋分，八月中。 〈限數千二十一。 間限千五十一。〉
Autumn equinox, eighth month, major term. 〈Limit number: 1,021. Gap limit: 1,051.〉
87
寒露，九月節。 〈限數千八十。 間限千一百七。〉
Cold Dew, ninth month, minor term. 〈Limit number: 1,080. Gap limit: 1,107.〉
88
霜降，九月中。 〈限數千一百三十三。 間限千一百五十七。〉 [17]
Frost Descends, ninth month, major term. 〈Limit number: 1,133. Gap limit: 1,157.〉 Editorial note 17.
89
立冬，十月節。 〈限數千一百八十一。 [18]間限千一百九十八。〉
Winter begins, tenth month, minor term. 〈Limit number: 1,181. Per note [18], gap limit: 1,198.〉
90
小雪，十月中。 〈限數千二百一十五。 間限千二百二十九。〉
Lesser Snow, tenth month, major term. 〈Limit number: 1,215. Gap limit: 1,229.〉
91
推沒滅術曰：因冬至積日有小餘者，加積一，以沒分乘之，以沒法除之，所得為大餘，不盡為小餘。 大餘滿六十去之，餘命以紀，算外，即去年冬至後沒日也。
To find extinction and disappearance days: when the winter-solstice day count carries a fractional remainder, increment the day total by one, multiply by 67,315, divide by 967 for the sexagenary day line, and keep the residue as the fractional part. Reduce the day line modulo sixty, index it against the sexagenary cycle, and the result names the first extinction day after the prior year's winter solstice.
92
求次沒，加大餘六十九，小餘五百九十二，小餘滿沒法得一，從大餘，命如前。 小餘盡，為滅也。
For each subsequent extinction day add 69 to the day count and 592 to the fraction, carrying overflow by 967 into the day count, then read off the date as before. When the fractional part reaches zero, the day is a full extinction day rather than a mere disappearance day.
93
推五行用事日：立春、立夏、立秋、立冬者，即木、火、金、水始用事日也。 各減其大餘十八，小餘四百八十三，小分六，餘命以紀，算外，各四立之前土用事日也。 大餘不足減者，加六十； 小餘不足減者，減大餘一，加紀法； 小分不足減者，減小餘一，加氣法。
The days when Wood, Fire, Metal, and Water begin to govern are found at Establishing Spring, Establishing Summer, Establishing Autumn, and Establishing Winter. Subtract 18 days, 483 parts, and 6 minute-fractions from each of those four dates, then index the result to obtain Earth's governing day immediately before each seasonal establishment. If the day count is too small, add 60 before subtracting. If the fractional part is too small, borrow one day and add 1,843 to the fraction. If the minute-fraction falls short, borrow one fractional unit and add 12 to the small remainder.
94
推卦用事日：因冬至大餘，六其小餘，坎卦用事日也。 加小餘萬九十一，滿元法從大餘，即中孚用事日也。
For hexagram rulership days, take the winter-solstice sexagenary line and multiply its fractional part by six to fix the day when Kan governs. Add 10,091 to the fractional part, carrying overflow by 11,058 into the day count, to reach the day when Zhong Fu governs.
95
求次卦，各加大餘六，小餘九百六十七。 其四正各因其中日，六其小餘。
Advance each subsequent hexagram by adding 6 to the day count and 967 to the fractional part. For the four cardinal hexagrams, use each one's midpoint day and multiply its fractional part by six.
96
推日度術曰：以紀法乘朔積日，滿周天去之，餘以紀法除之，所得為度，不盡為分。 命度從牛前五起，宿次除之，不滿宿，則天正十一月朔夜半日所在度及分也。 求次日，日加一度，分不加，經斗除斗分，分少退一度。
To find the sun's ecliptic longitude, multiply accumulated conjunction days by 1,843, reduce modulo 673,150, then divide by 1,843 for whole degrees and keep the residue as arc-minutes. Index the result from five lodges before Ox through the twenty-eight lodges; the remainder gives the sun's longitude at midnight on the eleventh month's conjunction. For each following day add one degree without adding fractional parts; when crossing the Dipper lodge subtract its special fraction, and if the fraction is too small, step back one degree.
97
推月度術曰：以月周乘朔積日，滿周天去之，餘以紀法除之，所得為度，不盡為分，命如上法，則天正十一月朔夜半月所在度及分也。
To find the moon's longitude at the eleventh month's conjunction midnight, multiply accumulated days by 24,638, reduce modulo 673,150, divide by 1,843 for degrees and fractional parts, and index through the lodges as above.
98
求次月，小月加度二十二，分八百六； 大月又加一日，度十三，分六百七十九； 分滿紀法得一度，則次月朔夜半月所在度及分也。 其冬下旬，月在張心署之。 [19]
For a short month add 22 degrees and 806 fractional parts to the moon's position. For a long month add one day, 13 degrees, and 679 fractional parts. Carry every 1,843 fractional parts into one degree to obtain the moon's longitude at the next conjunction midnight. During the last ten days of winter, make a special notation whenever the moon stands in Zhang or Heart. Editorial note 19.
99
推合朔度術曰：以章歲乘朔小餘，滿通法為大分，不盡為小分。 以大分從朔夜半日度分，分滿紀法從度，[20]命如前，則天正十一月合朔日月所共合度也。
To find the longitude of syzygy, multiply the conjunction fractional day by 19; divide by 47 for large fractional parts and keep the remainder as small fractional parts. Add the large fractional parts to the sun's midnight longitude on conjunction day, carrying 1,843 fractional parts per degree; per note [20], index as before to fix the shared solar-lunar longitude for the eleventh month's syzygy.
100
求次月，加度二十九，大分九百七十七，小分四十二，小分滿通法從大分，大分滿紀法從度。 經斗除其分，則次月合朔日月所共合度也。
For the following month add 29 degrees, 977 large fractional parts, and 42 small fractional parts, propagating carries through the small-fraction, large-fraction, and degree levels. Subtract the Dipper lodge's special fraction when crossing it to obtain the next month's syzygy longitude.
101
推弦望日所在度：加合朔度七，大分七百五，小分十，微分一，微分滿二從小分，小分滿通法從大分，大分滿紀法從度，命如前，則上弦日所在度也。 又加得望、下弦、後月合也。
For the sun at first quarter, add 7 degrees, 705 large parts, 10 small parts, and 1 minute-fraction to the syzygy longitude, carrying overflows at each level, then index the result to name first quarter. Repeat the same addition chain to reach full moon, third quarter, and the next syzygy.
102
推弦望月所在度：加合朔度九十八，大分千二百七十九，小分三十四，數滿命如前，即上弦月所在度也。 又加得望下弦後月合也。
For the moon at first quarter, add 98 degrees, 1,279 large parts, and 34 small parts to the syzygy longitude, carrying overflows and indexing as before. Repeat the addition to reach full moon, third quarter, and the next syzygy.
103
推日月昏明度術曰：日以紀法，月以月周，乘所近節氣夜漏，二百而一，為明分。 日以減紀法，月以減月周，餘為昏分。 各以加夜半，如法為度。
To find dusk and dawn longitudes, multiply the era divisor for the sun or the lunar circuit for the moon by the nearest term's night water-clock reading, then divide by 200 for the dawn arc. Subtract that dawn arc from the era divisor for the sun or from the lunar circuit for the moon to obtain the dusk arc. Add each arc to the midnight longitude and convert by the appropriate divisor to degrees.
104
推合朔交會月蝕術曰：置所入紀朔積分，以所入紀交會差率之數加之，以會通去之，餘則所求年天正十一月合朔去交度分也。 以通數加之，滿會通去之，餘則次月合朔去交度分也。 以朔望合數各加其月合朔去交度分，滿會通去之，餘則各其月望去交度分也。 朔望去交分如朔望合數以下，[21]入交限數以上者，朔則交會，望則月蝕。
To find nodal distance at syzygy, take the era's accumulated conjunction fraction, add its eclipse-node difference rate, reduce modulo 790,110, and the remainder is the eleventh-month conjunction's distance from the node. Add 134,630 and reduce modulo 790,110 to obtain the next month's nodal distance at conjunction. Add 67,315 to each month's conjunction nodal distance and reduce modulo 790,110 to fix the full-moon nodal distance for that month. When the nodal distance at syzygy or opposition falls between 67,315 and 722,795 inclusive, per note [21], the new moon crosses the node and the full moon brings a lunar eclipse.
105
推合朔交會月蝕月在日道表裏術曰：置所入紀朔積分，以所入紀下交會差率之數加之，倍會通去之，餘不滿會通者，紀首表，天正合朔月在表，紀首裏，天正合朔月在裏。 滿會通去之，表在裏，裏在表。
To determine whether the moon lies north or south of the ecliptic, add the era's lower eclipse-node difference rate to its conjunction fraction, reduce modulo twice 790,110, and compare against 790,110: the era's opening polarity fixes whether the eleventh-month conjunction moon is north or south of the path. When the remainder crosses 790,110, north and south reverse.
106
求次月，以通數加之，滿會通去之，加裏滿在表，加表滿在裏。 先交會後月蝕者，朔在表則望在表，朔在裏則望在裏。 先月蝕後交會者，看食月朔在裏則望在表，朔在表則望在裏。 交會月蝕如朔望合數以下，[22]則前交後會； 如入交限數以上，則前會後交。 其前交後會近於限數者，則豫伺之前月； 前會後交近於限數者，則後伺之後月。
For each following month add 134,630, reduce modulo 790,110, and flip polarity whenever the count crosses the nodal communication constant. If the node is crossed before the eclipse, new moon and full moon share the same side of the ecliptic. If the eclipse comes before the nodal crossing, the full moon lies on the opposite side of the path from the new moon. When the nodal-eclipse value is at or below 67,315, per note [22], the node is crossed before the eclipse. When the value reaches 722,795 or above, the eclipse precedes the nodal crossing. If the node is crossed first and the value lies near the threshold, inspect the previous month as well. If the eclipse comes first and the value lies near the threshold, inspect the following month as well.
107
求去交度術曰：其前交後會者，今去交度分如日法而一，[23]所得則却去交度分也。 [24]其前會後交者，以去交度分減會通，餘如日法而一，所得則前去交度，餘皆度分也。 去交度十五以上，雖交不蝕也。 十以下是蝕，十以上虧蝕微少，光晷相及而已。 虧之多少，以十五為法。
To convert nodal distance when the node is crossed first, divide the current nodal distance by 4,559, per note [23], to obtain the distance measured back from the node. [24] When the eclipse precedes the crossing, subtract the nodal distance from 790,110 and divide by 4,559; the quotient is the forward nodal distance, and any remainder stays in degrees and parts. At a nodal distance of fifteen degrees or more, the bodies may cross the node but no eclipse occurs. At ten degrees or less a true eclipse occurs; above ten the obscuration is slight, with only the penumbra touching. Express the depth of obscuration as a fraction of fifteen.
108
求日蝕虧起角術曰：其月在外道，先交後會者，虧蝕西南角起； 先會後交者，虧蝕東南角起。 其月在內道，先交後會者，虧食西北角起； 先會後交者，虧食東北角起。 虧食分多少，如上以十五為法。 會交中者，蝕盡。 月蝕在日之衝，虧角與上反也。
To find where a solar eclipse first bites: on the outer path with node crossed before eclipse, obscuration begins at the southwest limb. If the eclipse precedes the crossing, obscuration begins at the southeast limb. On the inner path with crossing before eclipse, obscuration begins at the northwest limb. If the eclipse precedes the crossing, obscuration begins at the northeast limb. Measure the extent of obscuration as above, using fifteen as the divisor. When the node lies at the center, the eclipse is complete. A lunar eclipse occurs at opposition to the sun, and the corner of first obscuration is the mirror image of the solar case.
109
月行遲疾度損益率盈縮積分月行分
Lunar speed table: degree, excess-deficit rate, accumulated surplus-deficit, motion parts.
110
一日十四度 〈十四分〉 益二十六盈初[25]二百八十
Day 1: 14 degrees. 〈14 fractional parts.〉 Excess rate 26; initial surplus per note [25]; motion parts 280.
111
二日十四度 〈十一分〉 益二十三盈積分一十一萬八千五百三十四二百七十七
Day 2: 14 degrees. 〈11 fractional parts.〉 Excess rate 23; accumulated surplus 118,534; motion parts 277.
112
三日十四度 〈八分〉 益二十盈積分二十二萬三千三百九十一二百七十四
Day 3: 14 degrees. 〈8 fractional parts.〉 Excess rate 20; accumulated surplus 223,391; motion parts 274.
113
四日十四度 〈五分〉 益十七盈積分三十一萬四千五百七十一二百七十一[26]
Day 4: 14 degrees. 〈5 fractional parts.〉 Excess rate 17; accumulated surplus 314,571; motion parts 271, per note [26].
114
五日十四度 〈一分〉 益十三盈積分三十九萬二千七十四二百六十七
Day 5: 14 degrees. 〈1 fractional part.〉 Excess rate 13; accumulated surplus 392,074; motion parts 267.
115
六日十三度 〈十四分〉 益七盈積分四十五萬一千三百四十一二百六十一
Day 6: 13 degrees. 〈14 fractional parts.〉 Excess rate 7; accumulated surplus 451,341; motion parts 261.
116
七日十三度 〈七分〉 損盈積分四十八萬三千二百五十四二百五十四
Day 7: 13 degrees. 〈7 fractional parts.〉 Deficit rate zero; accumulated surplus 483,254; motion parts 254.
117
八日十三度 〈一分〉 損六盈積分四十八萬三千二百五十四二百四十八
Day 8: 13 degrees. 〈1 fractional part.〉 Deficit rate 6; accumulated surplus 483,254; motion parts 248.
118
九日十二度 〈十六分〉 損十盈積分四十五萬五千九百二百四十四
Day 9: 12 degrees. 〈16 fractional parts.〉 Deficit rate 10; accumulated surplus 455,900; motion parts 244.
119
十日十二度 〈十三分〉 損十三盈積分四十一萬三百一十二百四十一
Day 10: 12 degrees. 〈13 fractional parts.〉 Deficit rate 13; accumulated surplus 410,310; motion parts 241.
120
十一日十二度 〈十一分〉 損十五盈積分三十五萬一千四十三二百三十九
Day 11: 12 degrees. 〈11 fractional parts.〉 Deficit rate 15; accumulated surplus 351,043; motion parts 239.
121
十二日十二度 〈八分〉 損十八盈積分二十八萬二千六百五十八二百三十六
Day 12: 12 degrees. 〈8 fractional parts.〉 Deficit rate 18; accumulated surplus 282,658; motion parts 236.
122
十三日十二度 〈五分〉 損二十一盈積分二十萬五百九十六二百三十三
Day 13: 12 degrees. 〈5 fractional parts.〉 Deficit rate 21; accumulated surplus 200,596; motion parts 233.
123
十四日十二度 〈三分〉 損二十三盈積分十萬四千八百五十七二百三十一
Day 14: 12 degrees. 〈3 fractional parts.〉 Deficit rate 23; accumulated surplus 104,857; motion parts 231.
124
十五日十二度 〈五分〉 益二十一縮初二百三十三
Day 15: 12 degrees. 〈5 fractional parts.〉 Excess rate 21; initial shrinkage; motion parts 233.
125
十六日十二度 〈七分〉 益十九縮積分九萬五千七百三十九二百三十五
Day 16: 12 degrees. 〈7 fractional parts.〉 Excess rate 19; accumulated shrinkage 95,739; motion parts 235.
126
十七日十二度 〈九分〉 益十七縮積分十八萬二千三百六十二百三十七
Day 17: 12 degrees. 〈9 fractional parts.〉 Excess rate 17; accumulated shrinkage 182,360; motion parts 237.
127
十八日十二度 〈十二分〉 益十四縮積分二十五萬九千八百六十三二百四十
Day 18: 12 degrees. 〈12 fractional parts.〉 Excess rate 14; accumulated shrinkage 259,863; motion parts 240.
128
十九日十二度 〈十五分〉 益十一縮積分三十二萬三千六百八十九二百四十三
Day 19: 12 degrees. 〈15 fractional parts.〉 Excess rate 11; accumulated shrinkage 323,689; motion parts 243.
129
二十日十二度 〈十八分〉 益八縮積分三十七萬三千八百三十八二百四十六
Day 20: 12 degrees. 〈18 fractional parts.〉 Excess rate 8; accumulated shrinkage 373,838; motion parts 246.
130
二十一日十三度 〈三分〉 益四縮積分四十一萬三百一十二百五十
Day 21: 13 degrees. 〈3 fractional parts.〉 Excess rate 4; accumulated shrinkage 410,310; motion parts 250.
131
二十二日十三度 〈七分〉 損縮積分四十二萬八千五百四十六二百五十四
Day 22: 13 degrees. 〈7 fractional parts.〉 Deficit rate zero; accumulated shrinkage 428,546; motion parts 254.
132
二十三日十三度 〈十二分〉 損五縮積分四十二萬八千五百四十六二百五十九
Day 23: 13 degrees. 〈12 fractional parts.〉 Deficit rate 5; accumulated shrinkage 428,546; motion parts 259.
133
二十四日十三度 〈十八分〉 損十一縮積分四十萬五千七百五十一二百六十五
Day 24: 13 degrees. 〈18 fractional parts.〉 Deficit rate 11; accumulated shrinkage 405,751; motion parts 265.
134
二十五日十四度 〈五分〉 損十七縮積分三十五萬五千六百二二百七十一
Day 25: 14 degrees. 〈5 fractional parts.〉 Deficit rate 17; accumulated shrinkage 355,602; motion parts 271.
135
二十六日十四度 〈十一分〉 損二十三縮積分二十七萬八千九十九[27]二百七十七
Day 26: 14 degrees. 〈11 fractional parts.〉 Deficit rate 23; accumulated shrinkage 278,099, per note [27]; motion parts 277.
136
二十七日十四度 〈十一分〉 損二十四縮積分十七萬三千二百四十二二百七十八
Day 27: 14 degrees. 〈11 fractional parts.〉 Deficit rate 24; accumulated shrinkage 173,242; motion parts 278.
137
周日十四度 〈十三分有小分六百二十六〉 損二十五 〈有小分六百二十六〉 縮積分六萬三千八百二十六二百七十九 〈有小分二百二十六〉
Anomalistic week day: 14 degrees. 〈13 fractional parts with micro-fraction 626.〉 Deficit rate 25. 〈Micro-fraction component 626.〉 Accumulated shrinkage 63,826; motion parts 279. 〈Micro-fraction component 226.〉
138
推合朔交會月蝕入遲疾曆術曰：置所入紀朔積分，以所入紀下遲疾差率之數加之，以通周去之，餘滿日法得一日，不盡為日餘，命日算外，則所求年天正十一月合朔入曆日也。
To place a syzygy inside the lunar speed table: add the tabulated anomaly increment to the lunation fraction, reduce modulo 125,621, divide by 4,559 for whole days plus remainder, and index the result to name the anomaly day for the eleventh month's conjunction.
139
求次月，加一日，日餘四千四百五十。 [28]求望，加十四日，日餘三千四百八十九。 日餘滿日法成日，日滿二十七去之。 又除餘如周日餘，日餘不足除者，減一日，加周虛。
For the following month add one day and 4,450 day-fraction parts. Per note [28], for opposition add fourteen days and 3,489 fractional parts. Carry fractions into days at 4,559, then reduce full weeks of 27 days. If the fractional division underflows at the week boundary, borrow one day and add the weekly complement 2,031.
140
推合朔交會月蝕定大小餘：以入曆日餘，[29]乘所入曆損益率，以損益盈縮積分為定積分。 以章歲減所入曆月行分，餘以除之，所得以盈減縮加本小餘。 加之滿日法者，交會加時在後日； 減之，不足者，交會加時在前日。 月蝕者，隨定大小餘為日加時。 入曆在周日者，以周日日餘乘縮積分，為定積分。 以損率乘入曆日餘，[30]又以周日日餘乘之，以周日日度小分并之，以損定積分，餘為後定積分。 以章歲減周日月行分，餘以周日日餘乘之，以周日度小分并之，以除後定積分，所得以加本小餘，如上法。
Per note [29], to fix syzygy instants multiply the anomalistic day fraction by the tabulated rate and apply it to the surplus-deficit column for the corrected integral. Divide the adjusted integral by the difference between 19 and the tabulated motion parts, then subtract surplus or add shrinkage to the base fractional day. A carry past the day divisor pushes the conjunction instant into the next civil day. A borrow moves the corrected instant onto the previous day. Lunar eclipses take their clock time directly from the corrected remainder line. At the anomalistic week boundary, multiply the deficit column by the weekly fractional day for the corrected integral. Per note [30], combine the rate product with weekly micro-parts to obtain the post-boundary correction stack. Finish the week-crossing correction by dividing with the prescribed divisor mix, then add to the base fraction as above.
141
推加時：以十二乘定小餘，滿日法得一辰，數從子起，算外，則朔望加時所在辰也。 有餘不盡者四之，如日法而一為少，二為半，三為太。 又有餘者三之，如日法而一為強，半法以上排成之，不滿半法廢棄之。 以強并少為少強，并半為半強，并太為太強。 得二強者為少弱，以之并少為半弱，以之并半為太弱，以之并太為一辰弱。 以所在辰命之，則各得其少、太、半及強、弱也。 其月蝕望在中節前後四日以還者，視限數； 五日以上者，視間限。 定小餘如間限、限數以下者，以算上為日。
Multiply the corrected fraction by twelve and divide by 4,559 to name the twelve double-hours from midnight. Split the leftover into quarters of the divisor for the shao, ban, and tai fine subdivisions. Further triple the tail to reach the strong step, rounding up at half-divisor. Add strong units to weak, half, or full subdivisions per the classical clepsydra notation. Two strongs collapse into a weak grade, stepping through the ladder to a full weak double-hour mark. Read the final label against the stem hour to recover shao, tai, ban, qiang, and ruo. When an eclipse full moon lies within four days of a major term, consult the tight limit number. Beyond five days from the major term, apply the wider gap-limit rule. When the corrected fraction falls under both margin thresholds, promote the count to the next day.
142
斗二十六 〈分四百五十五〉 牛八女十二虛十危十七室十六壁九
Dipper: 26 degrees. 〈455 fractional parts.〉 Ox 8°, Maiden 12°, Emptiness 10°, Rooftop 17°, House 16°, Wall 9°.
143
北方九十八度 〈分四百五十五〉
Northern quadrant: 98°. 〈455 fractional parts.〉
144
奎十六婁十二胃十四昴十一畢十六觜二參九
Straddles 16°, Harvest 12°, Stomach 14°, Hairy Head 11°, Net 16°, Turtle Beak 2°, Triaster 9°.
145
西方八十度
Western quadrant: 80°.
146
井三十三鬼四柳十五星七張十八翼十八軫十七
Well 33°, Ghost 4°, Willow 15°, Star 7°, Extended Net 18°, Wings 18°, Chariot Platform 17°.
147
南方百一十二度
Southern quadrant: 112°.
148
角十二亢九氐十五房五心五尾十八箕十一
Horn 12°, Gullet 9°, Base 15°, Chamber 5°, Heart 5°, Tail 18°, Winnowing Basket 11°.
149
東方七十五度
Eastern quadrant: 75°.
150
中節日所在度日行黃道去極度日中晷景晝漏刻夜漏刻昏中星明中星
Table columns: major solar term; solar lodge longitude; solar declination from the ecliptic pole; noon shadow length; day clepsydra marks; night clepsydra marks; dusk culmination; dawn culmination.
151
冬至 〈十一月中〉 斗二十一 〈少〉 百一十五度丈三尺四十五五十五奎六 〈弱〉 亢二 〈少強〉
Winter solstice. 〈Eleventh month, major term.〉 Dipper 21. 〈Shao.〉 Declination 115°; shadow 1 zhang 3 chi; day clepsydra 45; night clepsydra 55; dusk star Straddles 6. 〈Weak.〉 Dawn star Gullet 2. 〈Shao-strong.〉
152
小寒 〈十二月節〉 女二 〈少〉 百一十三 〈強〉 丈二尺三寸四十五 〈八分〉 五十四 〈二分〉 婁六 〈半強〉
Minor Cold. 〈Twelfth month, minor term.〉 Maiden 2. 〈Shao.〉 Declination 113°. 〈Strong.〉 Shadow 1 zhang 2 chi 3 cun; day clepsydra 45. 〈8 fractional parts.〉 Night clepsydra 54. 〈2 fractional parts.〉 Dusk star Harvest 6. 〈Half-strong.〉
153
氐七 〈強〉
Dawn star Base 7. 〈Strong.〉
154
大寒 〈十二月中〉 虛五 〈半弱〉
Major Cold. 〈Twelfth month, major term.〉 Emptiness 5. 〈Half-weak.〉
155
百一十 〈太弱〉 丈一尺四十六 〈八分〉 五十三 〈二分〉 胃十一 〈太強〉 [33]心 〈半〉
Declination 110°. 〈Very weak.〉 Shadow 1 zhang 1 chi; day clepsydra 46. 〈8 fractional parts.〉 Night clepsydra 53. 〈2 fractional parts.〉 Dusk star Stomach 11. 〈Very strong.〉 Editorial note 33: Heart. 〈Half.〉
156
立春 〈正月節〉 危十 〈太弱〉 百六 〈少弱〉 九尺六寸四十八 〈六分〉 五十一 〈四分〉 畢五 〈少弱〉 尾七 〈半弱〉
Establishment of Spring. 〈First month, minor term.〉 Rooftop 10. 〈Very weak.〉 Declination 106°. 〈Shao-weak.〉 Shadow 9 chi 6 cun; day clepsydra 48. 〈6 fractional parts.〉 Night clepsydra 51. 〈4 fractional parts.〉 Dusk star Net 5. 〈Shao-weak.〉 Dawn star Tail 7. 〈Half-weak.〉
157
雨水 〈正月中〉 室八 〈太強〉 百一 〈強〉 七尺九寸 〈五分〉 五十 〈八分〉 四十九 〈二分〉 參六 〈半弱〉 箕 〈半弱〉 [34]
Rain Water. 〈First month, major term.〉 House 8. 〈Very strong.〉 Declination 101°. 〈Strong.〉 Shadow 7 chi 9 cun. 〈5 fractional parts.〉 Day clepsydra 50. 〈8 fractional parts.〉 Night clepsydra 49. 〈2 fractional parts.〉 Dusk star Triaster 6. 〈Half-weak.〉 Dawn star Winnowing Basket. 〈Half-weak.〉 Editorial note 34.
158
驚蟄 〈二月節〉 壁八 〈強〉 九十五 〈強〉 六尺五寸五十三 〈三分〉 四十六 〈七分〉 井十七 〈少弱〉 斗初 〈少〉
Awakening of Insects. 〈Second month, minor term.〉 Wall 8. 〈Strong.〉 Declination 95°. 〈Strong.〉 Shadow 6 chi 5 cun; day clepsydra 53. 〈3 fractional parts.〉 Night clepsydra 46. 〈7 fractional parts.〉 Dusk star Well 17. 〈Shao-weak.〉 Dawn star Dipper beginning. 〈Shao.〉
159
春分 〈二月中〉 奎十四 〈少強〉 八十九 〈少強〉 五尺二寸 〈五分〉 五十五 〈八分〉 四十四 〈二分〉 鬼四斗十一 〈弱〉
Spring equinox. 〈Second month, major term.〉 Straddles 14. 〈Shao-strong.〉 Declination 89°. 〈Shao-strong.〉 Shadow 5 chi 2 cun. 〈5 fractional parts.〉 Day clepsydra 55. 〈8 fractional parts.〉 Night clepsydra 44. 〈2 fractional parts.〉 Dusk star Ghost 4; dawn star Dipper 11. 〈Weak.〉
160
清明 〈三月節〉 胃一 〈半〉 八十三 〈少弱〉 四尺一寸 〈五分〉 五十八 〈三分〉 四十一 〈七分〉 星四 〈太〉 斗二十一 〈半〉
Pure Brightness. 〈Third month, minor term.〉 Stomach 1. 〈Half.〉 Declination 83°. 〈Shao-weak.〉 Shadow 4 chi 1 cun. 〈5 fractional parts.〉 Day clepsydra 58. 〈3 fractional parts.〉 Night clepsydra 41. 〈7 fractional parts.〉 Dusk star Star 4. 〈Very.〉 Dawn star Dipper 21. 〈Half.〉
161
穀雨 〈三月中〉 昴二 〈太〉 七十七 〈太強〉 三尺二寸六十 〈五分〉 三十九 〈五分〉 張十七牛六 〈半〉
Grain Rain. 〈Third month, major term.〉 Hairy Head 2. 〈Very.〉 Declination 77°. 〈Very strong.〉 Shadow 3 chi 2 cun; day clepsydra 60. 〈5 fractional parts.〉 Night clepsydra 39. 〈5 fractional parts.〉 Dusk star Extended Net 17; dawn star Ox 6. 〈Half.〉
162
立夏 〈四月節〉 畢六 〈太〉 [35]七十三 〈少弱〉 二尺五寸 〈二分〉 六十二 〈四分〉 三十七 〈六分〉 翼十七 〈太〉 女十 〈少弱〉
Start of Summer. 〈Fourth month, minor term.〉 Net 6. 〈Very.〉 Editorial note 35: declination 73°. 〈Shao-weak.〉 Shadow 2 chi 5 cun. 〈2 fractional parts.〉 Day clepsydra 62. 〈4 fractional parts.〉 Night clepsydra 37. 〈6 fractional parts.〉 Dusk star Wings 17. 〈Very.〉 Dawn star Maiden 10. 〈Shao-weak.〉
163
小滿 〈四月中〉 參四 〈少弱〉 六十九 〈太〉 尺九寸 〈八分〉 六十三 〈九分〉 三十六 〈一分〉 角 〈太弱〉 危 〈太弱〉
Lesser Fullness. 〈Fourth month, major term.〉 Triaster 4. 〈Shao-weak.〉 Declination 69°. 〈Very.〉 Shadow 1 chi 9 cun. 〈8 fractional parts.〉 Day clepsydra 63. 〈9 fractional parts.〉 Night clepsydra 36. 〈1 fractional part.〉 Dusk star Horn. 〈Very weak.〉 Dawn star Rooftop. 〈Very weak.〉
164
芒種 〈五月節〉 井十 〈半弱〉 六十七 〈少弱〉 尺六寸 〈八分〉 六十四 〈九分〉 三十五 〈一分〉 亢五 〈太〉 危十四 〈強〉
Grain in Ear. 〈Fifth month, minor term.〉 Well 10. 〈Half-weak.〉 Declination 67°. 〈Shao-weak.〉 Shadow 1 chi 6 cun. 〈8 fractional parts.〉 Day clepsydra 64. 〈9 fractional parts.〉 Night clepsydra 35. 〈1 fractional part.〉 Dusk star Gullet 5. 〈Very.〉 Dawn star Rooftop 14. 〈Strong.〉
165
夏至 〈五月中〉 井二十五 〈半強〉 六十七 〈強〉 尺五寸六十五三十五氐十二 〈少弱〉 室十二 〈強〉
Summer solstice. 〈Fifth month, major term.〉 Well 25. 〈Half-strong.〉 Declination 67°. 〈Strong.〉 Shadow 1 chi 5 cun; day clepsydra 65; night clepsydra 35; dawn star Base 12. 〈Shao-weak.〉 Dusk star House 12. 〈Strong.〉
166
小暑 〈六月節〉 柳三 〈太強〉 六十七 〈太強〉 尺七寸六十四 〈七分〉 三十五 〈三分〉 尾一 〈太強〉 奎二 〈太強〉
Lesser Heat. 〈Sixth month, minor term.〉 Willow 3. 〈Very strong.〉 Declination 67°. 〈Very strong.〉 Shadow 1 chi 7 cun; day clepsydra 64. 〈7 fractional parts.〉 Night clepsydra 35. 〈3 fractional parts.〉 Dusk star Tail 1. 〈Very strong.〉 Dawn star Straddles 2. 〈Very strong.〉
167
大暑 〈六月中〉 星四 〈強〉 七十二尺六十三 〈八分〉 三十六 〈二分〉 尾十五 〈半強〉 [36]婁三 〈太〉
Greater Heat. 〈Sixth month, major term.〉 Star 4. 〈Strong.〉 Declination 70°; shadow 2 chi; day clepsydra 63. 〈8 fractional parts.〉 Night clepsydra 36. 〈2 fractional parts.〉 Dusk star Tail 15. 〈Half-strong.〉 Editorial note 36: Harvest 3. 〈Very.〉
168
立秋 〈七月節〉 張十二 〈少〉 七十三 〈半強〉 二尺五寸 〈五分〉 六十二 〈三分〉 三十七 〈七分〉 箕九 〈太強〉 胃九 〈太弱〉
Start of Autumn. 〈Seventh month, minor term.〉 Extended Net 12. 〈Shao.〉 Declination 73°. 〈Half-strong.〉 Shadow 2 chi 5 cun. 〈5 fractional parts.〉 Day clepsydra 62. 〈3 fractional parts.〉 Night clepsydra 37. 〈7 fractional parts.〉 Dusk star Winnowing Basket 9. 〈Tai-strong.〉 Dawn star Stomach 9. 〈Tai-weak.〉
169
處暑 〈七月中〉 翼九 〈半〉 七十八 〈半強〉 三尺三寸 〈三分〉 六十 〈二分〉 三十九 〈八分〉 斗十 〈少〉 畢三 〈太〉
End of Heat. 〈Seventh month, major term.〉 Solar longitude Wings 9. 〈Half.〉 Declination 78°. 〈Half-strong.〉 Shadow 3 chi 3 cun. 〈3 fractional parts.〉 Day clepsydra 60. 〈2 fractional parts.〉 Night clepsydra 39. 〈8 fractional parts.〉 Dusk star Dipper 10. 〈Shao.〉 Dawn star Net 3. 〈Tai.〉
170
白露 〈八月節〉 軫六 〈太〉 八十四 〈少強〉 四尺三寸 〈五分〉 五十七 〈八分〉 四十二 〈二分〉 斗二十一 〈強〉 參五 〈少強〉
White Dew. 〈Eighth month, minor term.〉 Solar longitude Chariot Platform 6. 〈Tai.〉 Declination 84°. 〈Shao-strong.〉 Shadow 4 chi 3 cun. 〈5 fractional parts.〉 Day clepsydra 57. 〈8 fractional parts.〉 Night clepsydra 42. 〈2 fractional parts.〉 Dusk star Dipper 21. 〈Strong.〉 Dawn star Triaster 5. 〈Shao-strong.〉
171
秋分 〈八月中〉 角五 〈弱〉 九十 〈半強〉 五尺五寸[37]五十五 〈二分〉 四十四 〈八分〉 牛五 〈少〉 井十六 〈少強〉
Autumn equinox. 〈Eighth month, major term.〉 Solar longitude Horn 5. 〈Weak.〉 Declination 90°. 〈Half-strong.〉 Shadow 5 chi 5 cun, per note [37]; night clepsydra 55. 〈2 fractional parts.〉 Day clepsydra 44. 〈8 fractional parts.〉 Dusk star Ox 5. 〈Shao.〉 Dawn star Well 16. 〈Shao-strong.〉
172
寒露 〈九月節〉 亢八 〈半弱〉 [38]九十六 〈太強〉 六尺八寸 〈五分〉 五十二 〈六分〉 四十七 〈四分〉 女七 〈太〉 鬼三 〈少強〉
Cold Dew. 〈Ninth month, minor term.〉 Solar longitude Gullet 8. 〈Half-weak.〉 Per note [38], declination 96°. 〈Tai-strong.〉 Shadow 6 chi 8 cun. 〈5 fractional parts.〉 Day clepsydra 52. 〈6 fractional parts.〉 Night clepsydra 47. 〈4 fractional parts.〉 Dusk star Maiden 7. 〈Tai.〉 Dawn star Ghost 3. 〈Shao-strong.〉
173
霜降 〈九月中〉 氐十四 〈少強〉 百二 〈少強〉 八尺四寸五十 〈三分〉 四十九 〈七分〉 虛六 〈太〉 星三 〈太〉
Frost Descends. 〈Ninth month, major term.〉 Solar longitude Base 14. 〈Shao-strong.〉 Declination 102°. 〈Shao-strong.〉 Shadow 8 chi 4 cun; day clepsydra 50. 〈3 fractional parts.〉 Night clepsydra 49. 〈7 fractional parts.〉 Dusk star Emptiness 6. 〈Tai.〉 Dawn star Star 3. 〈Tai.〉
174
立冬 〈十月節〉 尾四 〈半強〉 百七 〈少強〉 丈[39]四十八 〈二分〉 五十一 〈八分〉 危八 〈強〉 張十五 〈太強〉
Winter begins. 〈Tenth month, minor term.〉 Solar longitude Tail 4. 〈Half-strong.〉 Declination 107°. 〈Shao-strong.〉 Shadow 1 zhang, per note [39]; day clepsydra 48. 〈2 fractional parts.〉 Night clepsydra 51. 〈8 fractional parts.〉 Dusk star Rooftop 8. 〈Strong.〉 Dawn star Extended Net 15. 〈Tai-strong.〉
175
小雪 〈十月中〉 箕一 〈太強〉 百一十一 〈弱〉 丈一尺四寸四十六 〈七分〉 五十三 〈三分〉 室三 〈半強〉 [40]翼十五 〈太〉 [41]
Lesser Snow. 〈Tenth month, major term.〉 Solar longitude Winnowing Basket 1. 〈Tai-strong.〉 Declination 111°. 〈Weak.〉 Shadow 1 zhang 1 chi 4 cun; night clepsydra 46. 〈7 fractional parts.〉 Day clepsydra 53. 〈3 fractional parts.〉 Dusk star House 3. 〈Half-strong.〉 Per note [40], dawn star Wings 15. 〈Tai.〉 Editorial note 41.
176
大雪 〈十一月節〉 斗六百一十三 〈太強〉 丈二尺五寸 〈六分〉 四十五 〈五分〉 [42]五十四 〈五分〉 壁 〈半強〉 軫十五 〈少強〉 [43]
Greater Snow. 〈Eleventh month, minor term.〉 Solar longitude Dipper 6; declination 113°. 〈Tai-strong.〉 Shadow 1 zhang 2 chi 5 cun. 〈6 fractional parts.〉 Day clepsydra 45. 〈5 fractional parts.〉 Per note [42], night clepsydra 54. 〈5 fractional parts.〉 Dusk star Wall. 〈Half-strong.〉 Dawn star Chariot Platform 15. 〈Shao-strong.〉 Editorial note 43.
177
右中節二十四氣，如術求之，得冬至十一月中也。 加之得次月節，加節得其月中。 中星以日所在為正。 置所求年二十四氣小餘四之，如法得一為少，不盡少三之，如法為強。 所得以減其節氣昏明中星各定。 [44]
The median terms listed at right are found by the procedure above; the first result is the winter solstice in the middle of the eleventh month. Adding one step yields the next minor term; adding a minor term in turn yields the corresponding major term. Culmination stars are reckoned from the sun's lodge position on the given day. For the target year, take each solar term's fractional remainder, multiply by four and divide by the day divisor for shao units; triple the leftover and divide again for qiang units. Subtract those shao and qiang amounts from the tabulated dusk and dawn culmination stars to obtain the corrected values. Editorial note 44.
178
推五星術：
Procedure for the five planets:
179
五星者，木曰歲星，火曰熒惑，土曰填星，金曰太白，水曰辰星。 凡五星之行，有遲有疾，有留有逆。 曩自開闢，清濁始分，則日月五星聚于星紀。 發自星紀，並而行天，遲疾留逆，互相逮及。 星與日會，同宿共度，則謂之合。 從合至合之日，則謂之終。 各以一終之日與一歲之日，通分相約，終而率之，歲數歲則謂之合終歲數，歲終則謂之合終合數。 [45]」。 二率既定，則法數生焉。 以章歲乘合數為合月法，以紀法乘合數為日度法，以章月乘歲數為合月分，如合月法為合月數，合月之餘為月餘。 以通數乘合月數，如日法而一為大餘，以六十去大餘，餘為星合朔大餘。 大餘之餘為朔小餘。 [46]以通數乘月餘，以合月法乘朔小餘，并之，以日法乘合月法除之，所得星合入月日數也。 餘以通法約之，[47]為入月日餘。 [48]以朔小餘減日法，餘為朔虛分。 以曆斗分乘合數，為星度斗分。 木、火、土各以合數減歲數，餘以周天乘之，如日度法而一，所得則行星度數也，餘則度餘。 金、水以周天乘歲數，如日度法而一，所得則行星度數也，餘則度餘。
The five planets are Jupiter (Year Star), Mars (Sparkling Deluder), Saturn (Reimbursing Star), Venus (Great White), and Mercury (Chronogram Star). Each planet moves now slowly, now swiftly, sometimes halting and sometimes moving backward. Since the world's beginning, when yin and yang first divided, the sun, moon, and five planets all clustered in the Star Chronogram lodge. Starting from Star Chronogram they travel the sky in concert, their slow, fast, stationary, and retrograde motions continually catching and passing one another. When a planet and the sun share the same lodge and degree, the event is termed a conjunction. The interval from one conjunction day to the next is one complete planetary cycle. Reduce each planet's cycle length and the civil year to a common fractional base; the resulting year ratio is the total cycle year-count and the conjunction ratio is the total cycle conjunction-count. Editorial note 45. With both rates established, the computational divisors follow. Form the conjunction-month divisor by rule years times conjunction-count, the day-degree divisor by era divisor times conjunction-count, and conjunction-month parts by rule months times year-count; divide for whole conjunction months and keep the remainder as month remainder. Multiply conjunction months by the communication number, divide by the day divisor for the sexagenary line, reduce modulo sixty, and the result is the star-conjunction new-moon large remainder. What remains after the sexagenary reduction is the new-moon small remainder. Per note [46], combine month remainder and new-moon fraction with the prescribed divisors; the quotient is the day-count on which the planetary conjunction enters the lunar month. Reduce the leftover by the communication divisor; per note [47], the result is the fractional day remainder within the month. Per note [48], subtract the new-moon small remainder from the day divisor; the difference is the new-moon void fraction. Multiply the calendar's dipper fraction by the conjunction-count to obtain the planetary dipper fraction. For Jupiter, Mars, and Saturn, subtract the conjunction-count from the year-count, multiply by the circuit-of-heaven constant, divide by the day-degree divisor for whole degrees, and keep the remainder as degree remainder. For Venus and Mercury, multiply the year-count by the circuit-of-heaven constant and divide by the day-degree divisor for degrees and degree remainder.
180
木：合終歲數，千二百五十五。
Jupiter: total cycle year-count, 1,255.
181
合終合數，千一百四十九。
Total cycle conjunction-count, 1,149.
182
合月法，二萬一千八百三十一。
Conjunction-month divisor, 21,831.
183
日度法，二百一十一萬七千六百七。
Day-degree divisor, 2,117,607.
184
合月數，十三。
Conjunction-month count, 13.
185
月餘，萬一千一百二十二。
Month remainder, 11,122.
186
朔大餘，二十三。
New-moon large remainder, 23.
187
朔小餘，四千九十三。
New-moon small remainder, 4,093.
188
入月日，十五。
Day entered in month, 15.
189
日餘，百九十九萬五千六百六十四。
Day remainder, 1,995,664.
190
朔虛分，四百六十六。
New-moon void fraction, 466.
191
斗分，五十二萬二千七百九十五。
Dipper fraction, 522,795.
192
行星度，三十三。
Planetary degree, 33.
193
度餘，百四十七萬二千八百六十九。 [49]
Degree remainder, 1,472,869. Editorial note 49.
194
火：合終歲數，五千一百五。
Mars: total cycle year-count, 5,105.
195
合終合數，二千三百八十八。
Total cycle conjunction-count, 2,388.
196
合月法，四萬五千三百七十二。
Conjunction-month divisor, 45,372.
197
日度法，四百四十萬一千八十四。
Day-degree divisor, 4,401,084.
198
合月數，二十六。
Conjunction-month count, 26.
199
月餘，二萬三。
Month remainder, 20,003.
200
朔大餘，四十七。
New-moon large remainder, 47.
201
朔小餘，三千六百二十七。
New-moon small remainder, 3,627.
202
入月日，十三。
Day entered in month, 13.
203
日餘，三百五十八萬五千二百三十。
Day remainder, 3,585,230.
204
朔虛分，九百三十二。
New-moon void fraction, 932.
205
斗分，百八萬六千五百四十。
Dipper fraction, 1,086,540.
206
行星度，五十。
Planetary degree, 50.
207
度餘，百四十一萬二千一百五十。
Degree remainder, 1,412,150.
208
土：合終歲數，三千九百四十三。
Saturn: total cycle year-count, 3,943.
209
合終合數，三千八百九。
Total cycle conjunction-count, 3,809.
210
合月法，七萬二千三百七十一。
Conjunction-month divisor, 72,371.
211
日度法，七百一萬九千九百八十七。
Day-degree divisor, 7,019,987.
212
合月數，十二。
Conjunction-month count, 12.
213
月餘，五萬八千一百五十三。
Month remainder, 58,153.
214
朔大餘，五十四。
New-moon large remainder, 54.
215
朔小餘，千六百七十四。
New-moon small remainder, 1,674.
216
入月日，二十四。
Day entered in month, 24.
217
日餘，六十七萬五千三百六十四。
Day remainder, 675,364.
218
朔虛分，二千八百八十五。
New-moon void fraction, 2,885.
219
斗分，百七十三萬三千九十五。
Dipper fraction, 1,733,095.
220
行星度，十二。
Planetary degree, 12.
221
度餘，五百九十六萬二千二百五十六。
Degree remainder, 5,962,256.
222
金：合終歲數，千九百七。
Venus: total cycle year-count, 1,907.
223
合終合數，二千三百八十五。
Total cycle conjunction-count, 2,385.
224
合月法，四萬五千三百一十五。
Conjunction-month divisor, 45,315.
225
日度法，四百三十九萬五千五百五十五。
Day-degree divisor, 4,395,555.
226
合月數，九。
Conjunction-month count, 9.
227
月餘，四萬三百一十。
Month remainder, 40,310.
228
朔大餘，二十五。
New-moon large remainder, 25.
229
朔小餘，三千五百三十五。
New-moon small remainder, 3,535.
230
入月日，二十七。
Day entered in month, 27.
231
日餘，十九萬四千九百九十。
Day remainder, 194,990.
232
朔虛分，千二十四。
New-moon void fraction, 1,024.
233
斗分，百八萬五千一百七十五。
Dipper fraction, 1,851,175.
234
行星度，二百九十二。
Planetary degree, 292.
235
度餘，十九萬四千九百九十。
Degree remainder, 194,990.
236
水：合終歲數，一千八百七十。
Mercury: total cycle year-count, 1,870.
237
合終合數，萬一千七百八十九。
Total cycle conjunction-count, 11,789.
238
合月法，二十二萬三千九百九十一。
Conjunction-month divisor, 223,991.
239
日度法，二千一百七十二萬七千一百二十七。
Day-degree divisor, 21,721,727.
240
合月數，一。
Conjunction-month count, 1.
241
月餘，二十一萬五千四百五十九。
Month remainder, 215,459.
242
朔大餘，二十九。
New-moon large remainder, 29.
243
朔小餘，二千四百一十九。
New-moon small remainder, 2,419.
244
入月日，二十八。
Day entered in month, 28.
245
日餘，二千三十四萬四千二百六十一。 [50]
Day remainder, 20,344,261. Editorial note 50.
246
朔虛分，二千一百四十。
New-moon void fraction, 2,140.
247
斗分，五百三十六萬三千九百九十五。
Dipper fraction, 5,363,995.
248
行星度，五十七。
Planetary degree, 57.
249
度餘，二千三十四萬四千二百六十一。
Degree remainder, 20,344,261.
250
推五星術曰：置壬辰元以來盡所求年，以合終合數乘之，滿合終歲數得一，名積合，不盡名合餘。 以合終合數減合餘，得一者星合往年，得二者合前往年，無所得，合其年。 餘以減合終合數，為度分。 金、水積合，偶為晨，奇為夕。
The procedure for the five planets runs thus: take the years since the Renchen origin, multiply by the total cycle conjunction-count, and divide by the total cycle year-count; the quotient is the accumulated conjunction, the remainder the conjunction surplus. Subtract the conjunction remainder from the total cycle conjunction-count: a result of one places the conjunction in the prior year, of two in the year before that; if nothing remains, the conjunction falls in the year sought. Subtract that remainder from the total cycle conjunction-count to obtain the degree fraction. For Venus and Mercury, an even accumulated conjunction marks a morning appearance, an odd one an evening appearance.
251
推五星合月：以月數月餘各乘積合，餘滿合月法從月，為積月，不盡為月餘。 以紀月除積月，所得算外，所入紀也，餘為入紀月。 副以章閏乘之，[51]滿章月得一為閏，以減入紀月，餘以歲中去之，餘為入歲月，命以天正起，算外，星合月也。 其在閏交際，以朔御之。
To find the conjunction month, multiply both the month count and month remainder by the accumulated conjunction; carry into whole months whenever the remainder fills the conjunction-month divisor, and keep the leftover as month remainder. Divide accumulated months by the era month-total; the quotient names the era entered and the remainder the month within that era. Then multiply the auxiliary by rule intercalations; per note [51], divide by rule months for whole intercalary months, subtract them from the era month, remove full years from what remains, and index from the celestial first month—the result is the star-conjunction month. If the date falls at an intercalary boundary, govern it by the nearest new moon.
252
推合月朔：以通數乘入紀月，滿日法得一為積日，不盡為小餘。 以六十去積日，餘為大餘，命以所入紀，算外，星合朔日也。
To obtain the conjunction new moon, multiply the era month by the communication number, divide by the day divisor for accumulated days, and keep the remainder as the small fraction. Reduce accumulated days modulo sixty for the sexagenary remainder, index from the era entered, and the result is the star-conjunction new-moon day.
253
推入月日：以通數乘月餘，合月法乘朔小餘，并之，通法約之，所得滿日度法得一，則星合入月日也，不滿為日餘。 命日以朔，算外，入月日也。
For the day within the month, combine month remainder and new-moon fraction through the communication and conjunction divisors; divide by the day-degree divisor for the entry day, keeping any remainder as day surplus. Index that day from the new moon outside the reckoning to name the day entered in the month.
254
推星合度：以周天乘度分，滿日度法得一為度，不盡為餘，命以牛前五度起，算外，星所合度也。
For the conjunction degree, multiply the circuit of heaven by the degree fraction and divide by the day-degree divisor; index from five degrees before Ox to obtain the lodge and degree of conjunction.
255
求後合月，以月數加入歲月，以餘加月餘，餘滿合月法得一月，月不滿歲中，即在其年； 滿去之，有閏計焉，餘為後年； 再滿，在後二年。 金、水加晨得夕，加夕得晨也。
To find the next conjunction month, add the tabulated month count and remainder to the current year-month; carry a month whenever the remainder fills the conjunction-month divisor—if the month still lies within the year, the conjunction falls in that year; otherwise remove full years, accounting for intercalation, and the remainder points to a later year; if it fills once more, the conjunction lies two years ahead. For Venus and Mercury, adding to a morning date yields an evening one, and adding to an evening date yields a morning one.
256
求後合朔，以朔大小餘數加合朔月大小餘，其月餘上成月者，又加大餘二十九，小餘二千四百一十九。 [52]小餘滿日法從大餘，命如前法。
For the next conjunction new moon, add the tabulated new-moon remainders to those of the conjunction month, and when the month fraction carries a month, also add twenty-nine to the large remainder and 2,419 to the small. Per note [52], carry from small to large remainder when the small fraction fills the day divisor, then index as before.
257
求後入月日，[53]以入月日、日餘加入月日及餘，[54]餘滿日度法得一。 其前合朔小餘滿其虛分者，去一日； 後小餘滿二千四百一十九以上，去二十九日； 不滿，去三十日，其餘則後合入月日，命以朔。 求後合度，以度數及分，如前合宿次命之。
Per note [53], add the tabulated entry day and day remainder to the current ones; per note [54], carry a day whenever the remainder fills the day-degree divisor. If the preceding conjunction new-moon small remainder fills its void fraction, subtract one day; if the later small remainder reaches 2,419 or above, subtract twenty-nine days; otherwise subtract thirty days; what remains is the next conjunction day within the month, indexed from the new moon. For the next conjunction degree, add the tabulated degree and fraction and index the lodge sequence as before.
258
木：晨與日合，伏，順，十六日九十九萬七千八百三十二分，行星二度百七十九萬五千二百三十八分，而晨見東方，在日後。 順，疾，日行五十七分之十一，五十七日行十一度。 順，遲，日行九分，五十七日行九度而留。 不行，二十七日而旋。 逆，日行七分之一，八十四日退十二度，而復留二十七日。 復遲，日行九分，五十七日行九度而復順。 疾，日行十一分，五十七日行十一度，在日前，夕伏西方。 順，十六日九十九萬七千八百三十二分，行星二度百七十九萬五千二百三十八分，而與日合。 凡一終，三百九十八日百九十九萬五千六百六十四分，行星三十三度百四十七萬二千八百六十九分。
Jupiter: conjoined with the sun at dawn, it disappears; moving prograde for sixteen days plus 997,832 fractional parts and two degrees plus 1,795,238 fractional parts, it reappears in the east at dawn, behind the sun. Prograde and swift: it advances eleven parts per fifty-seven each day, eleven degrees in fifty-seven days. Prograde and slow: nine parts per day, nine degrees in fifty-seven days, then it stops. It remains motionless for twenty-seven days, then reverses. Retrograde: one part in seven per day, twelve degrees back in eighty-four days, then another twenty-seven-day halt. Slow again at nine parts per day, nine degrees in fifty-seven days, then prograde once more. Swift motion at eleven parts per day, eleven degrees in fifty-seven days; standing ahead of the sun, it sets in the west at dusk. Prograde for sixteen days plus 997,832 fractional parts and two degrees plus 1,795,238 fractional parts, then it conjoins with the sun. One complete cycle: 398 days plus 1,995,664 fractional parts; planetary travel 33 degrees plus 1,472,869 fractional parts.
259
火：晨與日合，伏，七十二日百七十九萬二千六百一十五分，行星五十六度百二十四萬九千三百四十五分，而晨見東方，在日後。 順，日行二十三分之十四，百八十四日行百一十二度。 更順，遲，日行十二分，九十二日行四十八度而留。 不行，十一日而旋。 逆，日行六十二分之十七，六十二日退十七度，而復留十一日。 復順，遲，日行十二分，九十二日，行四十八度而復疾。 日行十四分，百八十四日行百一十二度，在日前，夕伏西方。 順，七十二日百七十九萬二千六百一十五分，行星五十六度百二十四萬九千三百四十五分，而與日合。 凡一終，七百八十日三百五十八萬五千二百三十分，行星四百一十五度二百四十九萬八千六百九十分。
Mars: conjoined with the sun at dawn, it disappears; after seventy-two days plus 1,792,615 fractional parts and fifty-six degrees plus 1,249,345 fractional parts, it reappears in the east at dawn, behind the sun. Prograde: fourteen parts in twenty-three per day, one hundred twelve degrees in one hundred eighty-four days. Prograde and slow again: twelve parts per day, forty-eight degrees in ninety-two days, then it stops. Motionless for eleven days, then it reverses. Retrograde: seventeen parts in sixty-two per day, seventeen degrees back in sixty-two days, then another eleven-day halt. Prograde and slow again: twelve parts per day, forty-eight degrees in ninety-two days, then swift motion resumes. Fourteen parts per day, one hundred twelve degrees in one hundred eighty-four days; ahead of the sun, it sets in the west at dusk. Prograde for seventy-two days plus 1,792,615 fractional parts and fifty-six degrees plus 1,249,345 fractional parts, then it conjoins with the sun. One complete cycle: 780 days plus 3,585,230 fractional parts; planetary travel 415 degrees plus 2,498,690 fractional parts.
260
土：晨與日合，伏，十九日三百八十四萬七千六百七十五分半，行星二度六百四十九萬一千一百二十一分半，而晨見東方，在日後。 順，行百七十二分之十三，八十六日行六度半而留。 不行，三十二日半而旋。 逆，日行十七分之一，百二日退六度而復留。 不行，三十二日半復順，日行十三分，八十六日行六度半，在日前，夕伏西方。 順，十九日三百八十四萬七千六百七十五分半，行星二度六百四十九萬一千一百二十一分半，而與日合。 凡一終，三百七十八日六十七萬五千三百六十四分，行星十二度五百九十六萬二千二百五十六分。
Saturn: conjoined with the sun at dawn, it disappears; after nineteen days plus 3,847,675½ fractional parts and two degrees plus 6,491,121½ fractional parts, it reappears in the east at dawn, behind the sun. Prograde: thirteen parts in 172 per day, six and a half degrees in eighty-six days, then it stops. Motionless for thirty-two and a half days, then it reverses. Retrograde: one part in seventeen per day, six degrees back in one hundred two days, then another halt. After thirty-two and a half motionless days it turns prograde again at thirteen parts per day, six and a half degrees in eighty-six days; ahead of the sun, it sets in the west at dusk. Prograde for nineteen days plus 3,847,675½ fractional parts and two degrees plus 6,491,121½ fractional parts, then it conjoins with the sun. One complete cycle: 378 days plus 675,364 fractional parts; planetary travel 12 degrees plus 5,962,256 fractional parts.
261
金：晨與日合，伏，六日退四度，而晨見東方，在日後而逆。 遲，日行五分之三，十日退六度。 留，不行，七日而旋。 順，遲，日行四十五分之三十三，四十五日行三十三度而順。 疾，日行一度九十一分之十四，九十一日行百五度而順。 益疾，日行一度九十一分之二十一，九十一日行百一十二度，在日後，而晨伏東方。 順，四十二日十九萬四千九百九十分，行星五十二度十九萬四千九百九十分，而與日合。 一合，二百九十二日十九萬四千九百九十分，行星如之。
Venus: conjoined with the sun at dawn, it disappears; after six days retreating four degrees it reappears in the east at dawn, behind the sun and moving backward. Slow and retrograde: three parts in five per day, six degrees back in ten days. Motionless for seven days, then it reverses. Prograde and slow: thirty-three parts in forty-five per day, thirty-three degrees in forty-five days, then faster prograde. Swift: one degree plus fourteen parts in ninety-one per day, one hundred five degrees in ninety-one days, still prograde. Still faster: one degree plus twenty-one parts in ninety-one per day, one hundred twelve degrees in ninety-one days; behind the sun, it vanishes in the east at dawn. Prograde for forty-two days plus 194,990 fractional parts and fifty-two degrees plus 194,990 fractional parts, then it conjoins with the sun. One conjunction: 292 days plus 194,990 fractional parts; planetary travel the same.
262
金：夕與日合，伏，順，四十二日十九萬四千九百九十分，行星五十二度十九萬四千九百九十分，而夕見西方，在日前。 順，疾，日行一度九十一分之二十一，九十一日行百一十二度而更順。 遲，日行一度十四分，九十一日行百五度而順。 益遲，日行四十五分之三十三，四十五日行三十三度而留。 不行，七日而旋。 逆，日行五分之三，十日退六度，在日前，夕伏西方。 逆，六日，退四度，而與日合。 凡再合一終，五百八十四日三十八萬九千九百八十分，行星如之。
Venus: conjoined with the sun at dusk, it disappears; prograde for forty-two days plus 194,990 fractional parts and fifty-two degrees plus 194,990 fractional parts, then reappears in the west at dusk, ahead of the sun. Prograde and swift: one degree plus twenty-one parts in ninety-one per day, one hundred twelve degrees in ninety-one days, then slower prograde. Slow: one degree plus fourteen parts per day, one hundred five degrees in ninety-one days, still prograde. Still slower: thirty-three parts in forty-five per day, thirty-three degrees in forty-five days, then it stops. Motionless for seven days, then it reverses. Retrograde: three parts in five per day, six degrees back in ten days; ahead of the sun, it sets in the west at dusk. Retrograde for six days, four degrees back, then it conjoins with the sun. Two conjunctions complete one cycle: 584 days plus 389,980 fractional parts; planetary travel the same.
263
水：晨與日合，伏，十一日退七度，而晨見東方，在日後。 逆，疾，一日退一度而留。 不行，一日而旋。 順，遲，日行八分之七，八日行七度而順。 疾，日行一度十八分之四，十八日行二十二度，在日後，晨伏東方。 順，十八日二千三十四萬四千二百六十一分，行星三十六度二千三十四萬四千二百六十一分，而與日合。 凡一合，五十七日二千三十四萬四千二百六十一分，行星如之。
Mercury: conjoined with the sun at dawn, it disappears; after eleven days retreating seven degrees it reappears in the east at dawn, behind the sun. Retrograde and swift: one degree back in one day, then it stops. Motionless for one day, then it reverses. Prograde and slow: seven parts in eight per day, seven degrees in eight days, then faster prograde. Swift: one degree plus four parts in eighteen per day, twenty-two degrees in eighteen days; behind the sun, it vanishes in the east at dawn. Prograde for eighteen days plus 20,344,261 fractional parts and thirty-six degrees plus 20,344,261 fractional parts, then it conjoins with the sun. One conjunction: 57 days plus 20,344,261 fractional parts; planetary travel the same.
264
水：夕與日合，伏，十八日二千三十四萬四千二百六十一分，行星三十六度二千三十四萬四千二百六十一分，而夕見西方，在日前。 順，疾，日行一度十八分之四，十八日行二十二度而更順。 遲，日行八分之七，八日行七度而留。 不行，一日而旋。 逆，一日退一度，在日前，夕伏西方。 逆，十一日退七度，而與日合。 凡再合一終，百一十五日千八百九十六萬一千三百九十五分，行星如之。
Mercury: conjoined with the sun at dusk, it disappears; after eighteen days plus 20,344,261 fractional parts and thirty-six degrees plus 20,344,261 fractional parts it reappears in the west at dusk, ahead of the sun. Prograde and swift: one degree plus four parts in eighteen per day, twenty-two degrees in eighteen days, then slower prograde. Slow: seven parts in eight per day, seven degrees in eight days, then it stops. Motionless for one day, then it reverses. Retrograde: one degree back in one day; ahead of the sun, it sets in the west at dusk. Retrograde for eleven days, seven degrees back, then it conjoins with the sun. Two conjunctions complete one cycle: 115 days plus 18,961,395 fractional parts; planetary travel the same.
265
五星曆步術：以法伏日度餘，加星合日度餘，餘滿日度法得一從全，命之如前，得星見日及度餘也。 以星行分母乘見度分，如日度法得一，分不盡，半法以上，亦得一，而日加所行分，分滿其母得一度。 逆順母不同，以當行之母乘故分，如故母而一，當行分也。 留者承前，逆則減之，伏不書度，除斗分，[55]以行母為率。 分有損益，前後相御。
Five-star step procedure: add the tabulated hidden-day and conjunction degree remainders; carry a whole unit when the sum fills the day-degree divisor, then index as before for the star's appearance day and fractional degree. Multiply the appearance degree fraction by the motion denominator, divide by the day-degree divisor rounding up at half, then add the daily motion fraction until a full degree is reached. When retrograde and prograde use different denominators, convert the prior fraction by the ratio of denominators to obtain the motion fraction for the current phase. Stationary phases carry forward the prior value, retrograde phases subtract, hidden phases omit degrees, dipper fractions are cleared; per note [55], use the motion denominator as the rate. When fractional parts increase or decrease, successive phases adjust one against the other.
266
凡五星行天，遲疾留逆，雖大率有常，至犯守逆順，難以術推。 月之行天，猶有遲疾，況五星乎。 唯日之行天有常，進退有率，不遲不疾，不外不內，人君德也。
Though the five planets generally follow broad patterns of slow, swift, stationary, and retrograde motion, their trespasses, halts, and reversals cannot always be forced by computation alone. If even the moon shows slow and fast intervals in its course, the five planets do so all the more. Only the sun moves through Heaven at a fixed pace, advancing and retreating by a steady rate—neither lagging nor rushing, neither straying outward nor inward. That is the virtue of the sovereign.
267
求木合終歲數法，以木日度法乘一木終之日，內分，周天除之，即得也。
To derive Jupiter's total cycle year-count, multiply its day-degree divisor by one complete cycle in days and fractional parts, then divide by the circuit of heaven.
268
求木合終合數法，以木日度法乘周天，滿紀法，所得復以周天除之，即得。 五星皆放此也。
For Jupiter's total cycle conjunction-count, multiply the day-degree divisor by the circuit of heaven, reduce by the era divisor, then divide again by the circuit of heaven. The same method applies to all five planets.
269
魏黃初元年十一月小，己卯蔀首，己亥歲，十一月己卯朔旦冬至，臣偉上。」
In the first year of Huangchu of Wei, the eleventh month was short; the obscuration commenced on Jimao in a Jihai year; on the eleventh month, Jimao day, at new moon and winter solstice, Wei submitted this memorial."
270
劉氏在蜀，不見改曆，當是仍用漢四分法。 吳中書令闞澤受劉洪乾象法於東萊徐岳字公河。 故孫氏用乾象曆，至于吳亡。
Shu under the Liu clan never adopted a new calendar and presumably kept the Han Quarter-Remainder system. Kan Ze, Director of the Secretariat in Wu, learned Liu Hong's Supernal Icon calendar from Xu Yue of Donglai, known as Gonghe. The Sun regime therefore employed the Supernal Icon calendar until Wu fell.
271
史臣按鄒衍五德，周為火行。 衍生在周時，不容不知周氏行運。 且周之為曆年八百，秦氏即有周之建國也。 周之火木，其事易詳。 且五德更王，唯有二家之說。 鄒衍以相勝立體，劉向以相生為義。 據以為言，不得出此二家者。 假使即劉向之說，周為木行，秦氏代周，改其行運。 若不相勝，則克木者金； 相生則木實生火。 秦氏乃稱水德，理非謬然。 斯則劉氏所證為不值矣。 臣以為張蒼雖是漢臣，生與周接，司秦柱下，備覩圖書。 且秦雖滅學，不廢術數，則有周遺文雖不畢在，據漢水行，事非虛作。 賈誼取秦云：「漢土德。」 蓋以是漢代秦。 詳論二說，各有其義。 張蒼則以漢水勝周火，廢秦不班五德。 賈誼則以漢土勝秦水，以秦為一代。 論秦、漢雖殊，而周為火一也。 然則相勝之義，於事為長。 若同蒼黜秦，則漢水、魏土、晉木、宋金； 若同賈誼取秦，則漢土、魏木、晉金、宋火也。 難者云：「漢高斷蛇而神母夜哭，云赤帝子殺白帝子，然則漢非火而何？」 斯又不然矣。 漢若為火，則當云赤帝，不宜云赤帝子也。 白帝子又何義況乎？ 蓋由漢是土德，土生乎火，秦是水德，水生乎金，斯則漢以土為赤帝子，秦以水德為白帝子也。 難者又曰：「向云五德相勝，今復云土為赤帝子，何也？」 答曰：「五行自有相勝之義，自有相生之義。 不得以相勝廢相生，相生廢相勝也。 相勝者，以土勝水耳； 相生者，土自火子，義豈相關。」
The historiographer observes that in Zou Yan's scheme of the Five Phases, Zhou belonged to Fire. Zou Yan lived during the Zhou age; he can hardly have been unaware of Zhou's assigned phase. Moreover, the Zhou calendar ran eight hundred years, and the Qin house already held the record of Zhou's founding. The question of Zhou as Fire or Wood is easily settled on the evidence. On the rotation of dynastic phases, only two major schools of interpretation exist. Zou Yan built his theory on mutual conquest; Liu Xiang on mutual generation. On these grounds, no serious account can escape one of the two schools. Suppose one accepts Liu Xiang's claim that Zhou was Wood and that Qin, replacing Zhou, altered the phase succession. On the conquest theory, Wood should yield to Metal; on the generation theory, Wood ought to produce Fire. Yet Qin proclaimed itself Water—a claim that fits neither scheme cleanly. Liu Xiang's argument therefore fails to convince. Although Zhang Cang was a Han official, he had lived into Zhou times, served Qin as Keeper of the Pillars below, and had access to the full archive. Qin suppressed the classics but not calendrical science; enough Zhou material survived to show that Han's claim to Water was not fabricated from nothing. Jia Yi cites Qin as saying, "Han belongs to Earth." —meaning that Han supplanted Qin. Each of the two theories, examined closely, has its own internal logic. Zhang Cang argued that Han Water conquered Zhou Fire and therefore excluded Qin from the phase sequence. Jia Yi held that Han Earth conquered Qin Water and counted Qin as a legitimate dynasty. However they differ on Qin and Han, both agree that Zhou was Fire. On balance, the conquest theory fits the historical record better. Following Zhang Cang and demoting Qin yields Han Water, Wei Earth, Jin Wood, and Song Metal; following Jia Yi and retaining Qin yields Han Earth, Wei Wood, Jin Metal, and Song Fire. One critic asks: "When Han Gaozu cut the serpent, the divine mother wept in the night that the Red Emperor's son had slain the White Emperor's son—does this not prove Han was Fire?" That objection, too, misses the point. If Han were truly Fire, the omen should have named the Red Emperor himself, not his son. And what, in any case, is meant by the White Emperor's son? The answer is that Han was Earth—Earth born of Fire—and Qin was Water—Water born of Metal. Han as Earth was therefore the Red Emperor's son, and Qin as Water the White Emperor's son. The critic adds: "Liu Xiang taught that the Five Phases mutually conquer; how then can Earth also be the Red Emperor's son?" The reply is that the Five Phases admit both a logic of conquest and a logic of generation. Neither principle cancels the other. In conquest, Earth overcomes Water; in generation, Earth is born of Fire—the two relationships are simply different."
272
崔寔四民月令曰：祖者，道神。 黃帝之子曰累祖，好遠遊，死道路，故祀以為道神。 嵇含祖道賦序曰：[56]漢用丙午，魏用丁未，晉用孟月之酉。 曰莫識祖之所由。 說者云祈請道神，謂之祖有事於道者，君子行役，則列之於中路，喪者將遷，則稱名於階庭。 或云，百代遠祖，名諡彫滅，墳塋不復存於銘表，游魂不得託於廟祧，故以初歲良辰，建華蓋，揚綵旌，將以招靈爽，庶眾祖之來憑云爾。 [57]
Cui Shi's Monthly Ordinances for the Four Peoples states: "Zu is the spirit of the road." The Yellow Emperor's son Leizu loved distant journeys and died on the road; for that reason he is honored as the road spirit. Ji Han's Preface to the Fu on the Road Spirit records: [56] Han used Bingwu, Wei Dingwei, and Jin the you day of the first month. No one can say with certainty when the cult of Zu began. Some hold that travelers pray to the road spirit and call the rite Zu: a gentleman departing on duty sets out the offering along the route; a family about to move a coffin invokes the name on the courtyard steps. Others say that after many generations names and tombs are lost and spirits have no temple to receive them; at the year's first auspicious day people therefore raise decorated canopies and bright banners to summon the dead, that all the ancestors may draw near. Editorial note 57.
273
晉武帝時，侍中平原劉智，[58]推三百年斗曆改憲，以為四分法三百年而減一日，以百五十為度法，三十七為斗分。 飾以浮說，以扶其理。 江左中領軍琅邪王朔之以其上元歲在甲子，善其術，欲以九萬七千歲之甲子為開闢之始，何承天云「悼於立意」者也。 景初日中晷景，即用漢四分法，是以漸就乖差。 其推五星，則甚疏闊。 晉江左以來，更用乾象五星法以代之，猶有前却。
Under Emperor Wu of Jin, Palace Attendant Liu Zhi of Pingyuan [58] proposed a three-hundred-year Dipper-calendar reform, arguing that the Quarter-Remainder system drops one day every three centuries, with 150 as degree divisor and 37 as dipper fraction. He padded the proposal with flattering rhetoric to bolster his case. Wang Shuo, Central Commander of the Left in the Eastern Jin, admired the method because its upper origin fell in a jiazi year and wanted to treat a jiazi ninety-seven thousand years from creation as the world's beginning—what He Chengtian dismissed as a flawed foundation. The Jingchu calendar's noon shadow measurements relied on the Han Quarter-Remainder method and therefore drifted steadily out of true. Its planetary calculations were especially coarse. After the Eastern Jin the Supernal Icon planetary method replaced it, yet still showed forward and backward error.
274
宋太祖頗好曆數，太子率更令何承天私撰新法。 元嘉二十年，上表曰：
Emperor Wu of Song took a keen interest in calendrical science, and the Crown Prince's Director of Standards He Chengtian privately composed a new system. In the twentieth year of Yuanjia he submitted a memorial stating:
275
臣授性頑惰，少所關解。 自昔幼年，頗好曆數，耽情注意，迄于白首。 臣亡舅故祕書監徐廣，素善其事，有既往七曜曆，每記其得失。 自太和至太元之末，四十許年。 臣因比歲考校，至今又四十載。 故其疏密差會，皆可知也。
Your subject is dull by nature and understood little in his youth. From boyhood I have loved calendrical work and have pursued it with single-minded devotion into old age. My late uncle, former Director of the Secretariat Xu Guang, was expert in these matters and kept a running ledger of the seven-luminaries calendar, noting every hit and miss. From the Taihe era to the end of Taiyuan spans roughly forty years. I have tested and corrected that record in recent years, adding another forty years to the present. Its errors and accuracies can therefore be known in detail.
276
夫圓極常動，七曜運行，離合去來，雖有定勢，以新故相涉，自然有毫末之差，連日累歲，積微成著。 是以虞書著欽若之典，周易明治曆之訓，言當順天以求合，非為合以驗天也。 漢代雜候清臺，以昏明中星，課日所在，雖不可見，月盈則蝕，必當其衝，以月推日，則躔次可知焉。 捨易而不為，役心於難事，此臣所不解也。
Heaven's pivot turns without rest and the seven luminaries run appointed courses; though their meetings and partings follow broad regularities, the interplay of old and new cycles produces tiny discrepancies that, day by day and year by year, accumulate into plain error. Hence the Yu Documents teaches reverent accord with Heaven, and the Changes instructs us to regulate the calendar—meaning that the calendar should follow Heaven, not force Heaven to follow the calendar. Han observers mixed measurements at the Clear Terrace, using dusk, dawn, and meridian stars to test the sun's place; though the sun itself is invisible, a full moon must eclipse at opposition, and from the moon one can infer the sun's lodge. To leave the easy path and exhaust oneself over the difficult—this your subject cannot fathom.
277
堯典云「日永星火，以正仲夏」。 今季夏則火中。 又「宵中星虛，以殷仲秋」。 今季秋則虛中。 爾來二千七百餘年，以中星檢之，所差二十七八度。 則堯令冬至，日在須女十度左右也。 漢之太初曆，冬至在牽牛初，後漢四分及魏景初法，同在斗二十一。 臣以月蝕檢之，則景初今之冬至，應在斗十七。 又史官受詔，以土圭測景，考校二至，差三日有餘。 從來積歲及交州所上，檢其增減，亦相符驗。 然則今之二至，非天之二至也。 天之南至，[59]日在斗十三四矣。 此則十九年七閏，數微多差。 復改法易章，則用算滋繁，宜當隨時遷革，以取其合。 案後漢志，春分日長，秋分日短，差過半刻。 尋二分在二至之間，而有長短，因識春分近夏至，故長； 秋分近冬至，故短也。 楊偉不悟，即用之，上曆表云：「自古及今，凡諸曆數，皆未能並己之妙。」 何此不曉，亦何以云。 是故臣更建元嘉曆，以六百八為一紀，半之為度法，七十五為室分，以建寅之月為歲首，雨水為氣初，以諸法閏餘一之歲為章首。 冬至從上三日五時。 日之所在，移舊四度。 又月有遲疾，合朔月蝕，不在朔望，亦非曆意也。 故元嘉皆以盈縮定其小餘，以正朔望之日。
The Canon of Yao says, "The day is longest and the star is Fire—thereby fix midsummer." Today, at the last month of summer, Fire crosses the meridian. It also says, "At midnight the star is Emptiness—thereby mark mid-autumn." Today, at the last month of autumn, Emptiness crosses the meridian. Across the twenty-seven centuries since, meridian-star tests show a drift of twenty-seven or twenty-eight degrees. At the time of Yao's ordinance, winter solstice would have placed the sun near ten degrees of Maiden. The Han Taichu calendar placed winter solstice at the start of Ox; the Later Han Quarter-Remainder and Wei Jingchu systems both placed it at Dipper 21. Testing by lunar eclipse, I find that Jingchu's winter solstice today should fall at Dipper 17. Historiographical officers, under edict, measured solstice shadows with the earth-standard and found a discrepancy of more than three days. Cross-checking accumulated years and reports from Jiaozhou confirms the same drift. Today's solstices are therefore not Heaven's true solstices. Heaven's southern extreme: [59] the sun now stands at Dipper 13 or 14. This implies that the nineteen-year, seven-intercalation rule runs slightly high. Constantly rewriting rules only multiplies computation; the calendar should be revised when needed to recover agreement with Heaven. The Later Han Treatise notes that the spring equinox day is longer and the autumn equinox day shorter, by more than half a quarter. Since the equinoxes fall between the solstices yet differ in length, one sees that the spring equinox lies nearer summer and therefore yields a longer day; while the autumn equinox lies nearer winter and therefore yields a shorter one. Yang Wei failed to grasp this and applied the values anyway; his memorial on the superior calendar boasts, "From antiquity to the present no calendar has matched its own perfection." Yet how could he fail to see the problem and still make such a claim? Your subject therefore proposes the Yuanjia calendar: era length 608, degree divisor 304, lodge fraction 75, year beginning at the month of Establishment of Spring, first qi at Rain Water, and era heads in years whose intercalary remainder equals one. Winter solstice is placed at the third day, fifth watch, before the upper day. The sun's position is moved four degrees from the old placement. The moon, moreover, has slow and fast intervals; syzygy and eclipse need not fall exactly on new or full moon—another point the old methods mishandle. The Yuanjia system therefore uses surplus-deficit correction on the small remainder to fix the days of new and full moon.
278
伏惟陛下允迪聖哲，先天不違，劬勞庶政，寅亮鴻業，究淵思於往籍，探妙旨於未聞，窮神知化，罔不該覽。 是以愚臣欣遇盛明，効其管穴。 伏願以臣所上元嘉法下史官考其疏密。 若謬有可採，庶或補正闕謬，以備萬分。
Your Majesty, extending the Way of the sages and anticipating Heaven without offense, labors over the realm's affairs and illuminates the great enterprise; probing former texts and seeking principles not yet understood, you exhaust spirit and comprehend change—nothing lies beyond your view. Your humble subject, meeting such enlightened times, offers what little a narrow tube or needle's eye can contribute. I respectfully ask that the Yuanjia method I submit be referred to the historiographical officers for examination of its accuracy. If any part proves usable despite its flaws, it may yet help correct errors and fill gaps.
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詔曰：「何承天所陳，殊有理據。 可付外詳之。」
Edict: "He Chengtian's presentation is exceptionally well founded. Send it out for detailed review."
280
太史令錢樂之、兼丞嚴粲奏曰：
Grand Astrologer Qian Lezhi and Assistant Director Yan Can submitted:
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太子率更令領國子博士何承天表更改元嘉曆法，以月蝕檢今冬至日在斗十七，以土圭測影，知冬至已差三日。 詔使付外檢署。 以元嘉十一年被勑，使考月蝕，土圭測影，檢署由來用偉景初法，冬至之日，日在斗二十一度少。 檢十一年七月十六日望月蝕，加時在卯，到十五日四更二唱丑初始蝕，到四唱蝕既，在營室十五度末。 景初其日日在軫三度。 以月蝕所衝考之，其日日應在翼十五度半。 [60]又到十三年十二月十六日望月蝕，加時在酉，到亥初始食，到一更三唱蝕既，在鬼四度。 景初其日日在女三。 以衝考之，其日日應在牛六度半。 又到十四年十二月十六日望月蝕，[61]加時在戌之半，到二更四唱亥末始蝕，到三更一唱食既，在井三十八度。 [62]景初其日日在斗二十五。 以衝考之，其日日應在斗二十二度半。 [63]到十五年五月十五日望月蝕，加時在戌，其日月始生而已，蝕光已生四分之一格，在斗十六度許。 景初其日日在井二十四。 考取其衝，其日日應在井二十。 又到十七年九月十六日望月蝕，加時在子之少，到十五日未二更一唱始蝕，到三唱蝕十五分之十二格，在昴一度半。 景初其日在房二。 以衝考之，則其日日在氐十三度半。 凡此五蝕，以月衝一百八十二度半考之，冬至之日，日並不在斗二十一度少，並在斗十七度半間，悉如承天所上。
He Chengtian, Crown Prince's Director of Standards and Director of the Imperial Academy, petitioned to revise the Yuanjia calendar; lunar-eclipse tests place today's winter solstice at Dipper 17, and earth-standard shadow measurements show winter solstice already three days off. An edict ordered the proposal sent out for verification. In the eleventh year of Yuanjia we were ordered to test lunar eclipses and earth-standard shadows; under the Jingchu method then in use, winter solstice placed the sun slightly past Dipper 21. For the full-moon eclipse on day 16 of month 7, year 11, the appointed hour was mao; the eclipse began on the 15th at the start of the second watch, fourth chou drum, and ended at the fourth drum in the last degree of Encampment 15. Jingchu placed the sun that day at Chariot 3. By opposition, the sun should have stood at Wings 15½. [60] Again, for the full-moon eclipse on day 16 of month 12, year 13, the appointed hour was you; the eclipse began at the start of hai and ended at the first watch, third drum, in Ghost 4. Jingchu placed the sun that day at Woman 3. By opposition, the sun should have stood at Ox 6½. Again, for the full-moon eclipse on day 16 of month 12, year 14, [61] the appointed hour was mid-xu; the eclipse began at the second watch, fourth drum, near the end of hai, and ended at the third watch, first drum, in Well 38. [62] Jingchu placed the sun that day at Dipper 25. By opposition, the sun should have stood at Dipper 22½. [63] For the full-moon eclipse on day 15 of month 5, year 15, the appointed hour was xu; the moon had barely risen when a quarter of the disk was already eclipsed, near Dipper 16. Jingchu placed the sun that day at Well 24. By opposition, the sun should have stood at Well 20. Again, for the full-moon eclipse on day 16 of month 9, year 17, the appointed hour was just past zi; on the 15th, before the second watch, first drum, the eclipse began; at the third drum twelve-fifteenths of the disk was eclipsed in Hairy Head 1½. Jingchu placed the sun that day at Room 2. By opposition, the sun should have stood at Base 13½. Across all five eclipses, opposition at 182½ degrees shows winter solstice not at Jingchu's Dipper 21+ but consistently near Dipper 17½—exactly as He Chengtian claimed.
282
又去十一年起，以土圭測影。 其年景初法十一月七日冬至，前後陰不見影。 到十二年十一月十八日冬至，其十五日影極長。 到十三年十一月二十九日冬至，其二十六日影極長。 到十四年十一月十一日冬至，其前後並陰不見。 [64]到十五年十一月二十一日冬至，十八日影極長。 到十六年十一月二日冬至，其十月二十九日影極長。 到十七年十一月十三日冬至，其十日影極長。 到十八年十一月二十五日冬至，二十一日影極長。 [65]到十九年十一月六日冬至，其三日影極長。 到二十年十一月十六日冬至，其前後陰不見影。 尋校前後，以影極長為冬至，並差三日。 以月蝕檢日所在，已差四度。 土圭測影，冬至又差三日。 今之冬至，乃在斗十四間，又如承天所上。
From year 11 onward we also measured shadows with the earth-standard. That year Jingchu placed winter solstice on month 11, day 7; clouds before and after hid the shadow. Winter solstice of year 12 fell on month 11, day 18; the longest shadow came on day 15. Winter solstice of year 13 fell on month 11, day 29; the longest shadow came on day 26. Winter solstice of year 14 fell on month 11, day 11; clouds before and after hid the shadow. [64] Winter solstice of year 15 fell on month 11, day 21; the longest shadow came on day 18. Winter solstice of year 16 fell on month 11, day 2; the longest shadow came on month 10, day 29. Winter solstice of year 17 fell on month 11, day 13; the longest shadow came on day 10. Winter solstice of year 18 fell on month 11, day 25; the longest shadow came on day 21. [65] Winter solstice of year 19 fell on month 11, day 6; the longest shadow came on day 3. Winter solstice of year 20 fell on month 11, day 16; clouds before and after hid the shadow. Comparing all years, the longest shadow consistently precedes the computed winter solstice by three days. Lunar-eclipse tests already show the sun four degrees off. Earth-standard shadows show winter solstice three days off as well. Today's true winter solstice falls near Dipper 14—again matching He Chengtian.
283
又承天法，每月朔望及弦，皆定大小餘，於推交會時刻雖審，皆用盈縮，則月有頻三大、頻二小，比舊法殊為異。 舊日蝕不唯在朔，亦有在晦及二日。 公羊傳所謂「或失之前，或失之後」。 愚謂此一條自宜仍舊。
He Chengtian's method also fixes large and small remainders for every new moon, full moon, and quarter; though syzygy times are carefully computed, the surplus-deficit correction produces runs of three long or two short months—quite unlike the old system. Under the old method, solar eclipses occurred not only at new moon but also at month-end and on the second day. This is what the Gongyang Commentary means by "sometimes early, sometimes late." I hold that this one rule should remain unchanged.
284
員外散騎郎皮延宗又難承天：「若晦朔定大小餘，紀首值盈，則退一日，便應以故歲之晦，為新紀之首。」 承天乃改新法依舊術，不復每月定大小餘，如延宗所難，太史所上。
Pi Yanzong, Staff Master for Miscellaneous Cavalry, also challenged He Chengtian: "If new and full moons fix the remainders and the era head lands in surplus, one day is dropped and the old year's last day should become the new era head." He Chengtian therefore revised the new method back toward the old procedure, no longer fixing monthly remainders, as Yanzong urged and the Grand Astrologer reported.
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有司奏：「治曆改憲，經國盛典，爰及漢、魏，屢有變革。 良由術無常是，取協當時。 方今皇猷載暉，舊域光被，誠應綜覈晷度，以播維新。 承天曆術，合可施用。 宋二十二年，普用元嘉曆。」 詔可。
The relevant offices memorialized: "Calendar reform is a great state rite; since Han and Wei it has been revised again and again. No method stays correct forever; one adopts what fits the age. Now that the imperial design shines forth and the realm is restored, the gnomon and celestial degrees should be verified to proclaim renewal. He Chengtian's calendar is fit for adoption. In Song year 22 the Yuanjia calendar was adopted throughout the realm." The edict approved.
286
校勘記
Collation Notes
287
劾壽王逆天地「劾」字上各本並有「效」字，據漢志刪。 「天地」，漢志作「天道」。
On "Impeaching Shouwang for opposing Heaven and Earth": all earlier editions read the cited text above the cited text; deleted per the Han Treatise. The Han Treatise reads "Heaven's Way" for "Heaven and Earth."
288
章帝召治曆編訢李梵等綜校其狀「綜校其狀」各本並作「綜核意狀」，據續漢志改。
On Emperor Zhang summoning Li Fan and fellow calendar compilers: all editions read the cited text; changed to the cited text per the Continuation of the Han Treatise.
289
日在斗二十一度「二十一度」各本並作「二十二度」，據後漢書集解引盧文弨說改。
On "the sun at twenty-one degrees of Dipper": all editions read the cited text; changed to the cited text per Lu Wenchao in the Collected Commentaries on the Later Han.
290
安帝延光三年「三年」續漢志作「二年」。
On Yanguang year 3 of Emperor An: the Continuation of the Han Treatise reads "year 2."
291
累載相襲各本並脫「襲」字，據晉志補。
On "accumulated over many years in mutual succession": all editions omit the cited text; supplied per the Jin Treatise.
292
歲中「歲中」上各本並衍「紀日」二字。 按古曆無「紀日歲中」之名，開元占經一０五景初曆條下即作「歲中」，無「紀日」，今據刪。
On "mid-year": all editions also have the cited text above the cited text. Ancient calendars had no term "era-day mid-year"; under the Jingchu calendar entry in Kaiyuan Occupations Classic 105 it reads simply "mid-year" without "era-day"—deleted accordingly.
293
會通七十九萬一百一十「一十」各本作「二十」，據局本及晉志改。
Communication total 790,110: all editions read the cited text for the cited text; changed per the Bureau edition and Jin Treatise.
294
紀首合朔月在日道裏按下術文謂，以交會紀差轉加前紀，得後紀交會差率。 加之滿會通者去之，則月在日道表。 本紀及下甲寅紀交會率皆滿會通去之後所得之數。 故此紀首合朔俱應作「月在日道表」。
On "at era head, syzygy moon inside the sun's path": the procedure below adds nodal era difference forward to obtain the next era's nodal rate. When the sum fills the communication total, subtract it; the moon is then outside the sun's path. This era and the following Jiayin era nodal rates are all remainders after removing full communication totals. Every syzygy at this era head should therefore read "moon outside the sun's path."
295
如所近中節間限限數以下者按文義，「如」下應有「在」字。
On "if below the nearest median qi the interval limit is less than the limit number": the text requires the cited text below the cited text.
296
小分滿氣法從小餘小餘滿紀法從大餘「小餘」下各本脫「小餘」二字，據晉志補。
On carrying from small fraction and small remainder: all editions omit the cited text below the cited text; supplied per the Jin Treatise.
297
間限千一百四十七「四十七」各本作「三十七」，據局本及晉志改。
Interval limit 1,147: all editions read the cited text; changed to the cited text per the Bureau edition and Jin Treatise.
298
限數千一百二十二「二十二」各本並作「一十二」，今從局本。
Limit number 1,122: all editions read the cited text; now following the Bureau edition.
299
間限千三十六「三十六」各本並作「二十五」，晉志則誤作「四十六」，今從局本。
Interval limit 1,036: all editions read the cited text; the Jin Treatise wrongly reads the cited text; now following the Bureau edition.
300
限數八百二十三「二十三」各本並作「二十二」，據局本及晉志改。
Limit number 823: all editions read the cited text; changed to the cited text per the Bureau edition and Jin Treatise.
301
間限八百一十二「一十二」各本及晉志並作「一十三」，今從局本。
Interval limit 812: all editions and the Jin Treatise read the cited text; now following the Bureau edition.
302
間限八百一各本並脫「一」字，據局本及晉志補。
Interval limit 801: all editions omit the cited text; supplied per the Bureau edition and Jin Treatise.
303
間限千一百五十七各本並脫「千」字，據局本及晉志補。
Interval limit 1,157: all editions omit the cited text; supplied per the Bureau edition and Jin Treatise.
304
限數千一百八十一「八十一」各本並作「八十」，據局本及晉志補。
Limit number 1,181: all editions read the cited text; supplied as the cited text per the Bureau edition and Jin Treatise.
305
其冬下旬月在張心署之「月」各本作「夕」； 「之」除局本外，各本作「也」，今據續漢志及錢大昕廿二史考異說改。
On "in the last ten days of winter, the moon at Heart of Net": all editions read the cited text for the cited text; except the Bureau edition, all read the cited text for the cited text; now changed per the Continuation of the Han Treatise and Qian Daxin.
306
以大分從朔夜半日度分分滿紀法從度各本「度分」下脫「分」字，今據文義補。
On carrying large fraction from new-moon midnight sun-degree: all editions omit the cited text below the cited text; supplied per the sense of the text.
307
朔望去交分如朔望合數以下「如」各本作「加」，據晉志改。
On nodal departure fraction like nodal syzygy total or below: all editions read the cited text for the cited text; changed per the Jin Treatise.
308
交會月蝕如朔望合數以下「合」各本作「會」，據晉志改。
On nodal syzygy and lunar eclipse like nodal syzygy total or below: all editions read the cited text for the cited text; changed per the Jin Treatise.
309
今去交度分如日法而一「今」依文義當作「令」。
On dividing nodal departure degree by the day divisor: the cited text should read the cited text by the sense of the text.
310
所得則却去交度分也各本脫去「分」字，據晉志補。
On removing nodal departure degree fraction: all editions omit the cited text; supplied per the Jin Treatise.
311
盈初「盈初」各本作「盈一初」，據局本及晉志改。
Surplus initial: all editions read the cited text; changed to the cited text per the Bureau edition and Jin Treatise.
312
二百七十一各本脫「一」字，據局本及晉志補。
271: all editions omit the cited text; supplied per the Bureau edition and Jin Treatise.
313
縮積分二十七萬八千九十九「九十九」各本並作「六十九」。 據局本及晉志改。
Deficit accumulated parts 278,099: all editions read the cited text for the cited text. Changed per the Bureau edition and Jin Treatise.
314
日餘四千四百五十「日」字各本並脫，今從局本補。
Day remainder 4,450: all editions omit the cited text; now following the Bureau edition.
315
以入曆日餘各本並脫「餘」字，據晉志補。
On the day remainder entered in the calendar: all editions omit the cited text; supplied per the Jin Treatise.
316
以損率乘入曆日餘「損率」各本作「率損」，據局本及晉志乙正。
On multiplying deficit rate by calendar day remainder: all editions read the cited text for the cited text; transposed per the Bureau edition and Jin Treatise.
317
婁六 〈半強〉 各本作「婁五 〈半強〉」 ，誤，今改正。 按景初曆二十四氣各數，基本上沿用四分曆，數字雖間有出入，則由於兩曆斗分微有差異所致。 本表數字均據李銳四分術注所述方法，加以推算。 以下凡差異較大者，加以改正。 如僅尾數有出入，則指出正確之數，不加改正。
Bond 6 〈Half-strong.〉 All editions read "Bond 5 〈Half-strong.〉" ; erroneous—now corrected. The Jingchu calendar's twenty-four qi values largely follow the Quarter-Remainder system; minor numerical differences reflect slight dipper-fraction variation between the two. Table figures are computed by the method in Li Rui's commentary on the Quarter-Remainder procedure. Large discrepancies below are corrected in the text. Minor tail-digit differences are noted without changing the received text.
318
虛五 〈半弱〉 各本並作「虛女 〈半強〉」 ，誤，今改正。
Emptiness 5 〈Half-weak.〉 All editions read "Maiden 〈Half-strong.〉" ; erroneous—now corrected.
319
胃十一 〈太強〉 按當作「胃十一 〈半強〉」。
Stomach 11 〈Very strong.〉 Should read "Stomach 11 〈Half-strong.〉"
320
箕 〈半弱〉 按當作「箕 〈半強〉」。
Winnowing Basket 〈Half-weak.〉 Should read "Winnowing Basket 〈Half-strong.〉"
321
畢六 〈太〉 按當作「畢七」。
Net 6 〈Very.〉 Should read "Net 7."
322
尾十五 〈半強〉 按當作「尾十五 〈半弱〉」。
Tail 15 〈Half-strong.〉 Should read "Tail 15 〈Half-weak.〉"
323
五尺五寸各本並作「五尺五寸 〈二分〉」 ，誤，今刪正。
Five chi five cun: all editions read "5 chi 5 cun 〈2 fractional parts.〉" ; erroneous—now deleted and corrected.
324
亢八 〈半弱〉 按當作「亢八 〈少弱〉」。
Neck 8 〈Half-weak.〉 Should read "Neck 8 〈Shao-weak.〉"
325
丈各本並作「丈八寸 〈三分〉」 ，誤，今訂正。
1 zhang: all editions read "1 zhang 8 cun 〈3 fractional parts.〉" ; erroneous—now corrected.
326
室三 〈半強〉 按當作「室三 〈太強〉」。
Room 3 〈Half-strong.〉 Should read "Room 3 〈Very strong.〉"
327
翼十五 〈太〉 按當作「翼十五 〈太弱〉」。
Wings 15 〈Very.〉 Should read "Wings 15 〈Very weak.〉"
328
四十五 〈五分〉 三朝本作「五分」是。 各本並誤作「三分」。
45 〈5 fractional parts.〉 The Sanchao edition correctly reads "5 fractional parts." All editions wrongly read "3 fractional parts."
329
軫十五 〈少強〉 按當作「軫十五 〈少〉」。
Chariot 15 〈Shao-strong.〉 Should read "Chariot 15 〈Shao.〉"
330
所得以減其節氣昏明中星各定「所」下各本並脫「得」字，據晉志補。 「定」下疑脫「數」字。
On subtracting from each qi's meridian stars: all editions omit the cited text below the cited text; supplied per the Jin Treatise. The character the cited text is probably omitted below the cited text.
331
歲數歲則謂之合終歲數歲終則謂之合終合數依文義當作「歲則謂之合終歲數，終則謂之合終合數
Year-count year then is called total cycle year-count; year end then is called total cycle conjunction-count: by the sense of the text it should read "year then is called total cycle year-count, end then is called total cycle conjunction-count.
332
大餘之餘為朔小餘各本並脫「大餘」二字，據晉志補。
On the new-moon small remainder: all editions omit the cited text; supplied per the Jin Treatise.
333
餘以通法約之「以」下，各本並衍「朔」字，據晉志刪。
On reducing by the communication divisor: all editions add superfluous the cited text below the cited text; deleted per the Jin Treatise.
334
為入月日餘「日」下各本並脫「餘」字，今從局本補。
On day entered in month remainder: all editions omit the cited text below the cited text; now following the Bureau edition.
335
度餘百四十七萬二千八百六十九各本脫去「六十九」三字，據局本及晉志補。
Degree remainder 1,472,869: all editions omit the cited text; supplied per the Bureau edition and Jin Treatise.
336
日餘二千三十四萬四千二百六十一各本「千」上脫「四」字，今補。
Day remainder 20,344,261: all editions omit the cited text above the cited text; now supplied.
337
副以章閏乘之依文義，「副」字疑衍。
On "auxiliary, multiply by rule intercalations": the cited text is probably superfluous.
338
小餘二千四百一十九「二千」各本並作「一千」，據局本及晉志改。
Small remainder 2,419: all editions read the cited text for the cited text; changed per the Bureau edition and Jin Treatise.
339
求後入月日按所求者為後合入月日，「後」下當有「合」字。
On seeking the next day entered in month: the cited text should follow the cited text.
340
以入月日日餘加入月日及餘按所加者為一合的入月日及餘，「加」下當有「合」字。
On adding day entered in month and remainder: the cited text should follow the cited text.
341
除斗分按依義當作「經斗除斗分」。
Remove dipper fraction: by the sense it should read "passing the Dipper, remove dipper fraction."
342
嵇含祖道賦序曰各本並作「合祖賦序曰」，據沈濤說改。 沈濤銅熨斗齋隨筆云：「此乃初學記卷十三禮部嵇含祖道賦序文。 『合』乃『含』字之譌，傳寫又奪『嵇』字『道』字耳。」
On Ji Han's Preface to the Fu on the Road Spirit: all editions read the cited text; changed per Shen Tao. Shen Tao's Random Notes from the Bronze Iron notes: "This is Ji Han's Preface to the Fu on the Road Spirit in Initial Learning Record 13, Ritual Section. "the cited text" corrupts "the cited text", and "the cited text" and "the cited text" were lost in transmission."
343
崔寔四民月令曰 〈至〉 庶眾祖之來憑云爾張元濟曰：「與上文不接，是禮志錯簡。」 孫虨宋書考論：「此節論祖道，不當入之曆志。」 又「四民月令」原作「四人月令」，蓋後人避唐諱追改，今改正。
Cui Shi's Monthly Ordinances for the Four Peoples says 〈To.〉 On "that the multitude of ancestors may come and draw near": Zhang Yuanji says it does not connect with the preceding text and is a displaced passage from the Treatise on Rites. Sun Biao's Study of the Song History argues that this road-spirit section does not belong in the calendar treatise. Also, "Four Peoples" originally read "Four Men"—presumably altered to avoid Tang taboo; now corrected.
344
晉武帝時侍中平原劉智「晉武帝時」各本並作「晉江左時」，據晉志改正。 錢大昕廿二史考異云：「劉智字子房，司空寔之弟也。 仕武帝朝，非江左時，志誤。」
On Liu Zhi of Pingyuan under Emperor Wu of Jin: all editions read "Eastern Jin time"; changed per the Jin Treatise. Qian Daxin notes: "Liu Zhi, styled Zifang, was younger brother of Minister of Works Shi. He served Emperor Wu, not Eastern Jin—the Treatise errs."
345
天之南至「至」字各本並脫，據通鑑宋文帝元嘉二十一年補。
On Heaven's southern extreme: all editions omit the cited text; supplied per the Comprehensive Mirror, the twenty-first year of Yuanjia.
346
其日日應在翼十五度半各本並作「翼十五度半」。 按蝕既在營室十五度末，以月衝一百八十二度考之，其日日應在「翼十六度半」。
The sun that day should have stood at fifteen and a half degrees of Wings: all editions read "Wings fifteen and a half degrees." Since the eclipse ended in Encampment 15, opposition at 182° places the sun at Wings 16½, not 15½.
347
又到十四年十二月十六日望月蝕各本並作「十二月」。 按元嘉十三年十二月望月蝕，至元嘉十四年十二月望，已超過一蝕年，不當有月蝕。 今推是年十一月丁亥望 (十六日) 月蝕，原文有誤。
On the full-moon eclipse of month 12, day 16, year 14: all editions read "twelfth month." From the thirteenth year of Yuanjia month 12 full moon to the fourteenth year of Yuanjia month 12 full moon exceeds one eclipse year; a lunar eclipse is impossible. Recalculation shows a dinghai full moon in month 11 of that year (day 16) lunar eclipse; the received text is wrong.
348
在井三十八度各本並作「三十八度」。 按井僅有三十三度，原數顯誤。 今推元嘉十四年十一月望月蝕應在「井二十六度」。
On "at Well 38": all editions read "38 degrees" without naming Well. Well has only 33 degrees; the received figure is clearly wrong. Recalculation places the full-moon eclipse in the eleventh month of the fourteenth year of Yuanjia at Well 26.
349
其日日應在斗二十二度半各本並作「二十二度半」。 按以此處所述各月蝕檢日所在，與景初曆所推者實差三度半，今景初其日日在斗二十五，則實際應在斗二十一度半。 今推是月望月蝕在井二十六度，以衝考之，亦與此數相合。 故應作「二十一度半」。
The sun that day should have stood at twenty-two and a half degrees of Dipper: all editions read "twenty-two and a half degrees." Testing these eclipses against Jingchu shows a 3½° error; Jingchu places the sun at Dipper 25 that day, but the true position is Dipper 21½. Recalculation places that month's full-moon eclipse at Well 26; opposition agrees. The text should therefore read "21½ degrees."
350
其前後並陰不見按依上下文例，「見」下應有「影」字。
On "overcast skies showed none": following parallel passages, the cited text should follow the cited text.
351
二十一日影極長各本並作「二十一日」。 按上下各例，以土圭測影，冬至各差三日，二十五日冬至，則應「二十二日影極長」。
On "twenty-first day the shadow was longest": all editions read only "twenty-first day." Following parallel cases, earth-standard shadows place the longest shadow three days before computed winter solstice; with solstice on day 25, the longest shadow should fall on day 22.