1
律曆志中
Treatise on Pitch-pipes and the Calendar, Part Two.
2
昔者聖人擬宸極以運璿璣，揆天行而序景曜，分辰野，辨躔曆，敬農時，興物利，皆以系順兩儀，紀綱萬物者也。 然則觀象設卦，扐閏成爻，歷數之原，存乎此也。 逮乎炎帝，分八節以始農功，軒轅紀三綱而闡書契，乃使羲和占日，常儀占月，臾區占星氣，伶倫造律呂，大撓造甲子，隸首作算數。 容成綜斯六術，考定氣象，建五行，察發斂，起消息，正閏餘，述而著焉，謂之《調曆》。 洎於少昊則鳳鳥司曆，顓頊則南正司天，陶唐則分命羲和，虞舜則因循堯法。 及夏殷承運，周氏應期，正朔既殊，創法斯異。 《傳》曰：「火出，于夏為三月，于商為四月，于周為五月。」 是故天子置日官，諸侯有日禦，以和萬國，以協三辰。 至乎寒暑晦明之徵，陰陽生殺之數，啟閉升降之紀，消息盈虛之節，皆應躔次而無淫流，故能該浹生靈，堪輿天地。 周德既衰，史官失職，疇人分散，禨祥不理。 秦並天下，頗推五勝，自以獲水德之瑞，用十月為正。 漢氏初興，多所未暇，百有餘載，襲秦正朔。 爰及武帝，始詔司馬遷等議造《漢曆》，乃行夏正。 其後劉歆更造夏《三統》，以說《左傳》，辯而非實，班固惑之，采以為志。 逮光武中興，太僕硃浮數言曆有乖謬，于時天下初定，未能詳考。 至永平之末，改行《四分》，七十餘年，儀式乃備。 及光和中，乃命劉洪、蔡邕共修律曆，其後司馬彪因之，以繼班史。 今采魏文黃初已後言歷數行事者，以續司馬彪雲。
Long ago the sages took the celestial pole as their model and set the armillary sphere turning; they measured heaven’s motion and set the lights of sun, moon, and stars in their sequence; they mapped the lodges against the regions of the earth, fixed the calendar of planetary paths, honored the farmer’s seasons, and fostered the useful yield of the world—all to align the twin poles of heaven and earth and give good order to every creature under them. From this same foundation come the observation of celestial signs, the laying out of the hexagrams, the intercalation that completes the year, and the lines of change; the root of all calendar mathematics lies here. Under the Flame Emperor the eight seasonal divisions were fixed to open the farming year; under the Yellow Emperor the three great norms were set down and writing took shape. Xi He was charged with the sun, Chang Yi with the moon, Yu Qu with the stars and weather signs, Ling Lun with the twelve pitch-pipes, Da Nao with the stem-and-branch cycle, and Li Shou with the art of reckoning. Rong Cheng wove these six skills into one system: he tested the seasonal qi against observation, framed the five phases, traced growth and decline, marked increase and decrease, and straightened out intercalation, then committed the whole to writing under the title Harmonizing Calendar. Under Shaohao the “phoenix officials” kept the calendar; under Zhuanxu the office of the southern rectifier watched the sky; under Yao the duties of Xi He were split among officers; Shun simply continued the institutions he had inherited from Yao. When Xia and Shang took the throne in turn, and Zhou followed at its allotted hour, each dynasty chose a different first month of the year and framed its calendar in its own way. The Zuo commentary records: “When the Fire asterism rises heliacally, Xia reckoned it the third month of summer, Shang the fourth, and Zhou the fifth.” On that account the Son of Heaven appointed officers of the sundial, and each feudal state kept its own day-keeper, so that all the realms might stay in step and sun, moon, and stars might run true to their courses. Cold and heat, light and dark, the tallies by which yin and yang foster life or cut it short, the rhythm of the seasons’ opening and closing, rising and falling, waxing and waning—all of these answer to the true places of the constellations and never wander off course. That is how the calendar can embrace every living thing and stand level with heaven and earth. When Zhou’s moral authority faded, the court astronomers abandoned their posts, private specialists drifted apart, and no one any longer kept orderly track of omens and portents. Once Qin had conquered the empire, it pressed the doctrine of the five conquering phases, claimed the black omen of Water as its own, and took the tenth lunar month as the first month of the year. The early Han had too many crises on its hands; for well over a century it simply kept Qin’s calendar unchanged. Not until Emperor Wu did the court commission Sima Qian and his colleagues to draft a Han dynasty calendar and adopt the Xia-style first month. Liu Xin later recast the Triple Concordance calendar to force a fit with the Zuo Commentary—ingenious argument, shaky astronomy—and Ban Gu, taken in by it, built his monograph around that scheme. After Guangwu restored Han, Grand Coachman Zhu Fu kept protesting that the official calendar was wrong, but the empire was only just settling and no full review could be mounted. Near the close of Yongping the court switched to the Quarter Remainder system; only after seventy-odd years did instruments and ritual procedure catch up. During Guanghe, Liu Hong and Cai Yong were told to overhaul harmonics and calendrics together; Sima Biao later took their work as the basis for the treatise that follows Ban Gu’s Han shu. What follows collects, from Wei Wendi’s Huangchu era on, every substantive discussion of calendar theory and practice, extending the thread that Sima Biao began.
3
漢靈帝時，會稽東部尉劉洪，考史官自古迄今曆注，原其進退之行，察其出入之驗，視其往來，度其終始，始悟《四分》於天疏闊，皆斗分太多故也。 更以五百八十九為紀法，百四十五為斗分，作《乾象法》，冬至日日在鬥二十二度，以術追日、月、五星之行，推而上則合于古，引而下則應於今。 其為之也，依《易》立數，遁行相號，潛處相求，名為《乾象曆》。 又創制日行遲速，兼考月行，陰陽交錯于黃道表裏，日行黃道，於赤道宿度復有進退。 方於前法，轉為精密矣。 獻帝建安元年，鄭玄受其法，以為窮幽極微，又加注釋焉。
Under Emperor Ling, Liu Hong of Kuaiji, while serving as eastern commandery captain, combed the court’s calendar notes from the oldest records down to his own day, tracking how the bodies sped up or slowed, checking predicted risings and settings against fact, and mapping their whole cycle—until he saw that the Quarter Remainder calendar simply did not hug the sky closely enough, and that the fault lay in making the “dipper fraction” too generous. He set the cycle divisor at 589 and the dipper fraction at 145, producing what is called the Qianxiang system: at winter solstice the sun stood at 22° within the Dipper lodge. His algorithms for the sun, moon, and five planets match ancient records when projected backward and track the sky we see today when carried forward. The structure rests on numerology drawn from the Book of Changes: hidden motions call to one another across the sky, and latent positions are solved for one another—the whole was published as the Qianxiang calendar. He was the first to model the sun’s unequal motion and at the same time to refine the moon’s path, tracing how lunar and solar motion weave above and below the ecliptic while the sun itself slides along the ecliptic and still gains or loses ground against the equatorial longitudes. Beside the older procedures, this was a marked gain in precision. In the first Jian’an year the great scholar Zheng Xuan studied Liu Hong’s system, pronounced it the last word in subtlety, and furnished it with a full commentary.
4
魏文帝黃初中，太史令高堂隆復詳議歷數，更有改革。 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，千二百五為斗分。
Early in Wei’s Huangchu reign, Grand Astrologer Gao Tanglong reopened the debate on the calendar and further changes were proposed. 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 a cycle divisor of 4883 and a dipper fraction of 1250.
5
其後尚書令陳群奏，以為：「歷數難明，前代通儒多共紛爭。 《黃初》之元以《四分曆》久遠疏闊，大魏受命，宜改曆明時，韓翊首建，猶鞏不審，故以《乾象》互相參校。 其所校日月行度，弦望朔晦，曆三年，更相是非，無時而決。 案三公議皆綜盡典理，殊塗同歸，欲使效之璿璣，各盡其法，一年之間，得失足定。」 奏可。
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. They compared solar and lunar positions, first and last quarters, full and new moons, for three years, trading contradictory verdicts without ever reaching 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.” The throne approved the memorial.
6
太史令許芝云：「劉洪月行術用以來且四十餘年，以復覺失一辰有奇。」
Grand Astrologer Xu Zhi observed: “We have relied on Liu Hong’s lunar theory for the better part of forty years, and it is now drifting off by well over one-twelfth of a day.”
7
孫飲議：「史遷造《太初》，其後劉歆以為疏，復為《三統》。 章和中，改為《四分》，以儀天度，考合符應，時有差跌，日蝕覺過半日。 至熹平中，劉洪改為《乾象》，推天七曜之符，與天地合其敘。」
Sun Yin argued: “Sima Qian framed the Taichu calendar; Liu Xin later judged it coarse and replaced it with the Triple Concordance. Under Zhanghe the court adopted the Quarter Remainder and checked it with instruments against heaven; slips still occurred—solar eclipses were sometimes half a day out. In the Xiping years Liu Hong introduced Qianxiang, which lines up the seven lights with celestial signs and keeps heaven and earth in the same story.”
8
董巴議云：「聖人跡太陽于晷景，效太陰於弦望，明五星於見伏，正是非於晦朔。 弦望伏見者，歷數之綱紀，檢驗之明者也。」
Dong Ba said: “The sages ground the sun in the noon shadow, tested the moon against quarters and full moons, fixed the five planets by their appearances and disappearances, and settled true from false at new and full moon. First quarter, full moon, first visibility, and last sight—these are the backbone of any calendar and the cleanest tests we have.”
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徐嶽議：「劉洪以曆後天，潛精內思二十餘載，參校漢家《太初》、《三統》、《四分》曆術，課弦望于兩儀郭間。 而月行九歲一終，謂之九道； 九章，百七十一歲，九道小終； 九九八十一章，五百六十七分而九終，進退牛前四度五分。 學者務追合《四分》，但減一道六十三分，分不下通，是以疏闊，皆由斗分多故也。 課弦望當以昏明度月所在，則知加時先後之意，不宜用兩儀郭間。 洪加《太初》元十二紀，減十斗下分，元起己丑，又為月行遲疾交會及黃道去極度、五星術，理實粹密，信可長行。 今韓翊所造，皆用洪法，小益鬥下分，所錯無幾。 翊所增減，致亦留思，然十術新立，猶未就悉，至於日蝕，有不盡效。 效曆之要，要在日蝕。 熹平之際，時洪為郎，欲改《四分》，先上驗日蝕：日蝕在晏，加時在辰，蝕從下上，三分侵二。 事禦之後如洪言，海內識真，莫不聞見，劉歆以來，未有洪比。 夫以黃初二年六月二十九日戊辰加時未日蝕，《乾象術》加時申半強，於消息就加未，《黃初》以為加辛強，《乾象》後天一辰半強為近，《黃初》二辰半為遠，消息與天近。 三年正月丙寅朔加時申北日蝕，《黃初》加酉弱，《乾象》加午少，消息加未，《黃初》後天半辰近，《乾象》先天二辰少弱，於消息先天一辰強，為遠天。 三年十一月二十九日庚申加時西南維日蝕，《乾象》加未初，消息加申，《黃初》加未強，《乾象》先天一辰遠，《黃初》先天半辰近，消息《乾象》近中天。 二年七月十五日癸未，日加壬月加丙蝕，《乾象》月加申，消息加未，《黃初》月加子強，入甲申日，《乾象》後天二辰，消息後一辰為近，《黃初》後天六辰遠。 三年十一月十五日乙巳，日加丑月加未蝕，《乾象》月加巳半，於消息加午，《黃初》以丙午月加酉強，《乾象》先天二辰近，《黃初》後天二辰強為遠，于消息於《乾象》先一辰。 凡課日月蝕五事，《乾象》四遠，《黃初》一近。」
Xu Yue replied: “Seeing that the official calendar fell behind the sky, Liu Hong brooded for twenty years over the Taichu, Triple Concordance, and Quarter Remainder systems, checking quarter- and full-moon predictions against the paired armillary circles. He showed that the moon completes a nine-year cycle of paths—the so-called nine roads; nine rounds make 171 years, the minor closure of the nine paths; nine times nine, or eighty-one chapters, span 567 fractional parts to finish the great ninefold cycle, with residual motion of four and five-fifths degrees short of the Ox lodge. Later scholars tried to patch the Quarter Remainder by shaving one path sixty-three parts, but the fractions would not reconcile downward, so the fit stayed loose—the dipper fraction was simply too big. Quarter- and full-moon tests should fix the moon’s place at dusk and dawn so you can see whether the predicted hour runs early or late; the gap between the two armillary rings is the wrong place to measure. Liu Hong extended the Taichu epoch by twelve cycles, trimmed the lower dipper fraction by ten parts, began the count at the jichou year, and added models for the moon’s speed, syzygies, ecliptic latitude, and the five planets—the theory is tight and durable enough to serve for generations. Han Yi’s calendar is Liu Hong’s with a slightly larger lower dipper fraction—the deviation is tiny. His adjustments show care, but the ten new procedures are not fully debugged; on solar eclipses they still misfire sometimes. If you want to prove a calendar, solar eclipses are the decisive test. During Xiping, Liu Hong—then a court gentleman—petitioned to replace the Quarter Remainder and led with a solar-eclipse check: the eclipse fell late in the day, the true hour landed in the fifth double-hour, shadow crept upward from the lower limb, and two-thirds of the disk was eaten. When the event was over, observation matched Hong’s prediction; everyone who understood astronomy saw it, and since Liu Xin’s day no one has rivaled him. Take the eclipse of wuchen, the twenty-ninth of the sixth month, Huangchu 2, predicted for the wei double-hour: Qianxiang placed it in shen, a bit past the midpoint; the waxing-waning adjustment still favored wei; Huangchu insisted on xin. Qianxiang ran about one and a half double-hours late—closest to the sky—while Huangchu missed by two and a half; the waxing-waning correction hugged observation best. On new moon bingyin, first month, Huangchu 3, the eclipse should center north of due south in the shen hour: Huangchu predicts you, slightly early; Qianxiang predicts late wu; waxing-waning picks wei. Huangchu ends half a double-hour late—fairly close; Qianxiang jumps more than two double-hours early; against the waxing-waning correction it is still more than one double-hour ahead—clearly farthest from heaven. The gengshen eclipse on the twenty-ninth of the eleventh month, Huangchu 3, toward the southwest: Qianxiang opens wei, waxing-waning favors shen, Huangchu a strong wei. Qianxiang is a full double-hour early; Huangchu half a double-hour early is closer; waxing-waning and Qianxiang both sit nearer the actual meridian crossing. Lunar eclipse guiwei, fifteenth of the seventh month, Huangchu 2, with sun at ren and moon at bing: Qianxiang puts the moon at shen, waxing-waning at wei, Huangchu forcefully at zi, bleeding into jiashen day. Qianxiang is two double-hours late; waxing-waning one double-hour late is closest; Huangchu lags six double-hours—hopelessly wide of the mark. Lunar eclipse yisi, fifteenth of month eleven, Huangchu 3: Qianxiang sets the moon halfway through si, waxing-waning prefers wu, Huangchu on bingwu drives the moon to a forced you. Qianxiang’s two-double-hour lead is the tighter fit; Huangchu’s two-double-hour lag is worst; waxing-waning splits the difference, about one double-hour ahead of Qianxiang. Of the five eclipse tests, Qianxiang misses badly on four counts and Huangchu on only one.”
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翊于課難徐嶽：「《乾象》消息但可減，不可加。 加之無可說，不可用。」 嶽云：本術自有消息，受師法，以消息為奇，辭不能改，故列之正法消息。 翊術自疏。
Han Yi cross-examined Xu Yue: “In Qianxiang the waxing-waning term may be subtracted, never added. If you add it, the theory gives no account of itself and the correction is unusable.” Xu Yue answered: “The canonical method already embeds a waxing-waning correction; I received it from my teacher and cannot rewrite the formula, so I list the orthodox waxing-waning alongside the rest.” Han Yi’s own procedure is the loose one.
11
木以三年五月二十四日丁亥晨見； 《黃初》五月十七日庚辰見，先七日； 《乾象》五月十五日戊寅見，先九日。
Jupiter: third year, fifth month, day dinghai, first sighting at dawn; Huangchu gives gengchen, the seventeenth—seven days too soon; Qianxiang gives wuyin, the fifteenth—nine days too soon.
12
土以二年十一月二十六日壬辰見； 《乾象》十一月二十一日丁亥見，先五日； 《黃初》十一月十八日甲申見，先八日。
Saturn: second year, eleventh month, day renchen, morning appearance; Qianxiang predicts dinghai, the twenty-first—five days early; Huangchu predicts jiashen, the eighteenth—eight days early.
13
土以三年十月十一日壬申伏； 《乾象》同，壬申伏； 《黃初》已下十月七日戊辰伏，先四日。
Saturn: third year, tenth month, day renshen, evening disappearance; Qianxiang agrees—renshen; Huangchu puts it on wuchen, the seventh—four days early.
14
土以三年十一月二十二日壬子見； 《乾象》十一月十五日乙巳見，先七日； 《黃初》十一月十二日壬寅見，先十日。
Saturn: third year, eleventh month, day renzi, reappearance; Qianxiang gives yisi, the fifteenth—seven days early; Huangchu gives renyin, the twelfth—ten days early.
15
金以三年閏六月十五日丁丑晨伏； 《乾象》六月二十五日戊午伏，先十九日； 《黃初》六月二十二日乙卯伏，先二十三日。
Venus: third year, intercalary sixth month, day dingchou, lost in dawn glare; Qianxiang picks wuwu, the twenty-fifth of the regular sixth month—nineteen days early; Huangchu picks yimao, the twenty-second—twenty-three days early.
16
金以三年九月十一日壬寅見； 《乾象》以八月十八日庚辰見，先二十三日； 《黃初》八月十五日丁丑見，先二十五日。
Venus: third year, ninth month, day renyin, evening star; Qianxiang uses gengchen, the eighteenth of the eighth month—twenty-three days early; Huangchu uses dingchou, the fifteenth—twenty-five days early.
17
水以二年十一月十七日癸未晨見； 《乾象》十一月十三日己卯見，先四日； 《黃初》十一月十二日戊寅見，先五日。
Mercury: second year, eleventh month, day guiwei, dawn sighting; Qianxiang gives jimao, the thirteenth—four days early; Huangchu gives wuyin, the twelfth—five days early.
18
水以二年十二月十三日己酉晨伏； 《乾象》十二月十五日辛亥伏，後二日； 《黃初》十二月十四日庚戌伏，後一日。
Mercury: second year, twelfth month, day jiyou, lost at dawn; Qianxiang delays to xinhai, the fifteenth—two days late; Huangchu uses gengxu, the fourteenth—one day late.
19
水以三年五月十八日辛巳夕見； 《乾象》亦以五月十八日見； 《黃初》五月十七日庚辰見，先一日。
Mercury: third year, fifth month, day xinsi, evening apparition; Qianxiang also picks the eighteenth of the fifth month; Huangchu uses gengchen, the seventeenth—one day early.
20
水以三年六月十三日丙午伏； 《乾象》六月二十日癸丑伏，後七日； 《黃初》六月十九日壬子伏，後六日。
Mercury: third year, sixth month, day bingwu, conjunction; Qianxiang slips to guichou, the twentieth—seven days late; Huangchu uses renzi, the nineteenth—six days late.
21
水以三年閏六月二十五日丁亥晨見； 《乾象》以閏月九日辛未見，先十六日； 《黃初》閏月八日庚午見，先十七日。
Mercury: third year, intercalary sixth month, day dinghai, morning visibility; Qianxiang chooses xinwei, the ninth of the intercalary month—sixteen days early; Huangchu chooses gengwu, the eighth—seventeen days early.
22
水以三年七月七日己亥伏； 《乾象》七月十一日癸卯伏，後四日； 《黃初》以七月十日壬寅伏，後三日。
Mercury: third year, seventh month, day jihai, lost to the sun; Qianxiang postpones to guimao, the eleventh—four days late; Huangchu uses renyin, the tenth—three days late.
23
水以三年十一月日於晷度十四日甲辰伏； 《乾象》以十一月九日己亥伏，先五日； 《黃初》十一月八日戊戌伏，先六日。
Mercury: third year, eleventh month, on jiachen (the text ties the day to the fourteenth graduation on the sundial) evening disappearance; Qianxiang uses jihai, the ninth—five days early; Huangchu uses wuxu, the eighth—six days early.
24
水以三年十二月二十八日戊子夕見； 二曆同以十二月壬申見，俱先十六日。
Mercury: third year, twelfth month, day wuzi, sighted after sunset. Both calendars place the observation on renshen in the twelfth month, each sixteen days ahead of the fact.
25
凡四星見伏十五； 《乾象》七近二中，《黃初》五近一中。
Counting the four planets, there are fifteen heliacal risings and settings to compare; Qianxiang hits seven predictions closely and two middlingly; Huangchu hits five closely and one middlingly.
26
郎中李恩議：「乙太史天度與相覆校，二年七月、三年十一月望與天度日皆差異，月蝕加時乃後天六時半，非從三度之謂，定為後天過半日也。」
Li En of the Langzhong bureau testified: “When we stacked our figures against the official sky-longitudes, the full moons of the seventh month (year two) and the eleventh month (year three) both failed to line up with the solar count; the lunar eclipse’s predicted moment ran six and a half double-hours behind the heavens—far more than a three-degree slip—so we are dealing with a solid half-day lag.”
27
董巴議曰：「昔伏羲始造八卦，作三畫，以象二十四氣。 黃帝因之，初作《調曆》。 歷代十一，更年五千，凡有七曆。 顓頊以今之孟春正月為元，其時正月朔旦立春，五星會于廟，營室也，冰凍始泮，蟄蟲始發，雞始三號，天曰作時，地曰作昌，人曰作樂，鳥獸萬物莫不應和，故顓頊聖人為曆宗也。 湯作《殷曆》弗復以正月朔旦立春為節也，更以十一月朔旦冬至為元首，下至周魯及漢，皆從其節，據正四時。 夏為得天，以承堯舜，從顓頊故也。 《禮記》大戴曰虞夏之曆，建正于孟春，此之謂也。」
Dong Ba continued: “Legend says Fuxi first drew the eight trigrams in three lines each, emblematizing the twenty-four seasonal nodes. The Yellow Emperor built on that foundation and produced the earliest Harmonizing Calendar. Across eleven royal houses and some five thousand counted years, seven distinct calendrical systems appeared. Zhuanxu fixed his era to what we now call the first month of early spring: new moon and dawn coincided with Beginning of Spring, the five planets gathered over the ritual hall in the Encampment lodge, ice broke, dormant creatures stirred, and cocks crowed thrice—heaven was said to “open the seasons,” earth to “flourish,” and mankind to “take joy,” while every bird and beast fell into step. That is why Zhuanxu is remembered as the founding sage of calendrics.” King Tang’s Yin calendar abandoned the old rule that new year begin with new moon and Beginning of Spring; instead it enthroned the eleventh month’s new moon at winter solstice as the year’s pivot. Zhou, Lu, and Han all kept that hinge and used it to true the four seasons. The Xia calendar was said to “accord with heaven” because it continued the practice Yao and Shun had taken over from Zhuanxu.” The Da Dai chapter of the Record of Rites notes that the Yu and Xia calendars set their first month at the opening of spring—exactly this principle.”
28
楊偉請：「六十日中疏密可知，不待十年。 若不從法，是校方員棄規矩，考輕重背權衡，課長短廢尺寸，論是非違分理。 若不先定校曆之本法，而懸聽棄法之末爭，則孟軻所謂'方寸之基，可使高於岑樓'者也。 今韓翊據劉洪術者，知貴其術，珍其法。 而棄其論，背其術，廢其言，違其事，是非必使洪奇妙之式不傳來世。 若知而違之，是挾故而背師也； 若不知而據之，是為挾不知而罔知也。」 校議未定，會帝崩而寢。
Yang Wei urged: “Give the systems sixty days—tightness or slack will show; we do not need a decade. To ignore the proper formulas would be like squaring a circle without rule and compass, weighing goods without steelyard and beam, or measuring cloth without yardstick—pure confusion of true and false. Unless we nail down the fundamental rules of calendar testing first, we are only chasing the noisy fringe of people who want to scrap the law—just the folly Mencius mocked when he said a handspan of ground could be piled higher than a mountain villa. Han Yi claims to stand on Liu Hong’s work—he professes to prize Hong’s skill and cherish his formulas. Yet in practice he jettisons Hong’s reasoning, contradicts his algorithms, silences his text, and breaks his procedures—ensuring that Hong’s elegant system never reaches posterity. To understand the teaching and still rebel is wilful betrayal of one’s master; To cling to it in ignorance is to parade false learning as real.” The panel never finished its verdict; the emperor died and the whole matter was tabled.
29
至明帝景初元年，尚書郎楊偉造《景初曆》。 表上，帝遂改正朔，施行偉曆，以建丑之月為正，改其年三月為孟夏，其孟、仲、季月雖與夏正不同，至於郊祀蒐狩，班宣時令，皆以建寅為正。 三年正月帝崩，復用夏正。
In the first Jingchu year of Emperor Ming, Yang Wei of the secretariat produced the Jingchu calendar. The throne accepted his memorial, reset the civil year, and adopted Yang Wei’s calendar: the chou month became month one, and the old third month was relabeled early summer. Seasonal names no longer matched the Xia calendar, yet for suburban rites, royal hunts, and publishing the 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.
30
其劉氏在蜀，仍漢《四分曆》。 吳中書令闞澤受劉洪《乾象法》于東萊徐嶽，又加解注。 中常待王蕃以洪術精妙，用推渾天之理，以制儀象及論，故孫氏用《乾象曆》，至吳亡。
The Liu house in Shu simply kept Han’s Quarter Remainder calendar. Kan Ze, Wu’s chief secretary, studied Liu Hong’s Qianxiang method under Xu Yue of Donglai and supplied his own commentary. Palace attendant Wang Fan, impressed by the finesse of Hong’s mathematics, applied it to spherical-heaven theory and built instruments and treatises on that basis, so the Sun regime employed the Qianxiang calendar until Wu collapsed.
31
武帝踐阼，泰始元年，因魏之《景初曆》，改名《泰始曆》。 楊偉推五星尤疏闊，故元帝渡江左以後，更以《乾象》五星法代偉曆。 自黃初已後，改作曆術，皆斟酌《乾象》所減斗分、朔餘、月行陰陽遲疾，以求折衷。 洪術為後代推步之師表，故先列之雲。
Emperor Wu’s accession year Taishi simply inherited Wei’s Jingchu system and rechristened it the Taishi calendar. Yang Wei’s planetary tables were particularly slack, so once Emperor Yuan moved the court south of the Yangzi, the Qianxiang planetary rules supplanted Wei’s calendar. Every calendar reform after Huangchu tinkered with Liu Hong’s trimmed dipper fraction, lunation remainder, and model of the moon’s speeding and slowing in order to strike a middle path. Because Liu Hong’s work became the classroom standard for later astronomers, it is presented here first.
32
乾象曆
The Qianxiang Calendar.
33
上元己丑以來，至建安十一年丙戌，歲積七千三百七十八年。
From the epoch year jichou to bingxu, the eleventh year of Jian’an, the accumulated year count is 7378.
34
乾法，千一百七十八。
Qian divisor: 1178.
35
會通，七千一百七十一。
Conjunction cycle constant: 7171.
36
紀法，五百八十九。
Era divisor: 589.
37
周天，二十一萬五千一百三十。
Full celestial circle, in parts: 215,130.
38
通法，四萬三千二十六。
Common divisor: 43,026.
39
通數，三十一。
Common factor: 31.
40
日法，千四百五十七。
Day denominator: 1457.
41
歲中，十二。
Months per solar year: 12.
42
餘數，三千九十。
Solar remainder constant: 3090.
43
章歲，十九。
Nineteen-year metonic cycle.
44
沒法，百三。
Intercalary extinction divisor: 103.
45
章閏，七。
Intercalary months per metonic cycle: 7.
46
會數，四十七。
Lunation factor: 47.
47
會歲，八百九十三。
Great conjunction period: 893 years.
48
章月，二百三十五。
Months in a metonic cycle: 235.
49
會率，千八百八十二。
Conjunction modulus: 1882.
50
朔望合數，九百四十一。
New- and full-moon combination number: 941.
51
會月，萬一千四十五。
Months in the great conjunction cycle: 11,045.
52
紀月，七千二百八十五。
Months per era unit: 7285.
53
元月，一萬四千五百七十。
Origin-cycle month count: 14,570.
54
月周，七千八百七十四。
Lunar revolution constant: 7874.
55
小周，二百五十四。
Minor lunar cycle: 254.
56
推入紀
Procedure: find the position within the era.
57
置上元盡所求年，以乾法除之，不滿乾法，以紀法除之，余不滿紀法者，入內紀甲子年也。 滿法去之，入外紀甲午年也。
Take the years from the high origin to the target year, divide by the Qian divisor; with the remainder divide by the era divisor; any remainder still smaller than the era divisor lands you in the inner era beginning with jiazi. If the remainder fills the divisor, strip it out and you are in the outer era that starts with jiawu.
58
置入紀年，外所求，以章月乘之，章歲而一，所得為定積月，不盡為閏餘。 閏餘十二以上，歲有閏。 以通法乘定積月，為假積日，滿日法為定積日，不盡為小餘。 以六旬去積日為大餘，命以所入紀，算外，所求年天正十一月朔日也。
Multiply the years since era entry by 235 and divide by 19; the quotient is the accumulated month count and the remainder is the intercalary residue. If the intercalary remainder reaches twelve or more, insert a leap month that year. Multiply the accumulated months by the tong divisor to get provisional day-count; divide by the day denominator for whole days, leaving the fractional day as the small remainder. Cast out multiples of sixty from the day total for the stem-branch remainder; count it off within the current era cycle—what lies beyond the count is new moon of the eleventh civil month of the standard year.
59
求次月，加大餘二十九，小餘七百七十三，小餘滿日法從大餘。 小餘六百八十四已上，其月大。
For the following month add 29 to the large remainder and 773 to the small; whenever the small part overflows the day denominator, carry into the large remainder. A small remainder of 684 or more marks a long month.
60
推冬至
Procedure: winter solstice.
61
置入紀年，外所求，以餘數乘之，滿紀法為大餘，不盡為小餘。 以六旬去之，命以紀，算外，天正冬至日也。
Multiply the years since era entry by the solar remainder constant; divide by the era divisor—the quotient is the large remainder, the residue the small. Drop sixties, assign within the era count, and the remainder beyond the tally is winter solstice of the standard year.
62
求二十四氣
Procedure: the twenty-four solar terms.
63
置冬至小餘，加大餘十五，小餘五百一十五，滿二千三百五十六從大餘，命如法。
From the winter solstice fractions add 15 to the large part and 515 to the small; carry 2356 from the small into the large; continue the count by the usual rule.
64
推閏月
Procedure: intercalary month.
65
以閏餘減章歲，餘以歲中乘之，滿章閏為一月。 不盡，半法己上亦一，有進退，以無中月。
Subtract the intercalary residue from 19, multiply the difference by 12, and divide by 7—the quotient tells how many leap months to insert. If there is a leftover half the divisor or more, add another month; adjust forward or backward so the leap month lacks a mid-season qi.
66
推弦望
Procedure: first quarter, full moon, last quarter.
67
加大餘七，小餘五百五十七半，小餘如日法從大餘，餘命如前，得上弦。 又加得望，又加得下弦，又加得後月朔。 其弦望定小餘四百一以下，以百刻乘之，滿日法得一刻，不盡什之，求分，以課所近節氣夜漏未盡，以算上為日。
Add 7 to the large remainder and 557½ to the small; carry overflows of the day denominator; count off as before to reach first quarter. Repeat the step for full moon, again for last quarter, and again for the next new moon. If the fractional part of a quarter or full moon is 401 or less, multiply by 100 ke, divide by the day denominator for whole ke, subdivide the remainder by ten for finer parts, and compare with the night run of the neighboring qi—if the water-clock has not run out, count it to the previous day.
68
置入紀年，外所求，以餘數乘之，滿紀法為積沒，有餘加盡積為一。 以會通乘之，滿沒法為大餘，不盡為小餘。 大餘命以紀，算外，冬至後沒日。
Multiply years since era entry by the solar remainder; divide by the era divisor for accumulated extinction days; if anything remains, add a full cycle. Multiply by the conjunction constant; divide by 103 for the large remainder, the residue is the small. Count the large remainder within the era; the tally after it marks the first “extinction” day following winter solstice.
69
求次沒，加大餘六十九，小餘六十四，滿其法從大餘，無分為滅。
For the next extinction add 69 and 64; carry when full; zero fraction signals complete extinction.
70
推日度
Procedure: solar longitude.
71
以紀法乘積日，滿周天去之，余以紀法除之，所得為度。 命度以牛前五度起，宿次除之，不滿宿，即天正朔夜半日所在。
Multiply elapsed days by 589, discard multiples of the celestial circumference, divide the remainder by 589—the quotient is degrees of motion. Begin counting from five degrees before the Ox lodge, step through successive mansions, and the fractional lodge at standard new-moon midnight is the sun’s place.
72
求次日，加一度，經鬥除分； 分少，損一度為紀法，加焉。
For the next day add one degree; when you cross through Dipper, drop the fractional part; if the fraction is too small, borrow one degree as 589 parts and add.
73
推月度
Procedure: lunar longitude.
74
以月周乘積日，滿周天去之，余滿紀法為度，不盡為分，命如上，則天正朔夜半月所在度。
Multiply days by the lunar circuit constant, reduce modulo the sky circle, divide by 589 for whole degrees with remainder as minutes, and assign mansions as for the sun—this is the moon’s longitude at standard new-moon midnight.
75
求次月，小月加度二十二，分二百五十八。 大月又加一日，度十三，分二百一十七，滿法得一度。 其冬下旬，月在張、心署之。
After a short month add 22°258′ to the moon’s position. After a long month add another day’s worth—13°217′—and carry 589 parts to one degree. In the last third of winter, note the moon against Zhang and Heart.
76
推合朔度
Procedure: syzygy longitude.
77
以章歲乘朔小餘，滿會數為大分； 不盡，小分。 以大分從朔夜半日分，滿紀法從度，命如前，天正合朔日月所共會也。
Multiply the lunation fraction by 19; divide by 47 for the major fraction; the remainder is the minor fraction. Add the major fraction to the sun’s midnight position on new moon, carry 589 into degrees, and count mansions as before—this locates the true conjunction of sun and moon for the standard month.
78
求次月，加度二十九，大分三百一十二，小分滿會數從大分，大分滿紀法後度，經鬥除大分。
For the next syzygy add 29°312′ plus fractional carries through 47 and 589, and strip the large fraction when crossing Dipper.
79
求弦望日所在度，加合朔度七，分二百二十五，小分十七半，大小分及度命如前，則上弦日所在度。 又加得望、下弦、後月合。
To place the sun at quarter or full phase, add 7°225′17½″ to the conjunction longitude, carrying and assigning as before to obtain first quarter. Repeat the increment for full moon, last quarter, and the next conjunction.
80
求弦望月行所在度，加合朔度九十八，大分四百八，小分四十一，大小分及度命如前合朔，則上弦月所在。 又加得望、下弦、後月合。
For the moon at quarters and full, add 98°408′41″ to the conjunction longitude, using the same carrying rules, to fix first quarter. Iterate again for full moon, last quarter, and following new moon.
81
求日月昏明度，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。
For dawn and dusk positions, multiply the night run of the neighboring qi by 589 for the sun or by the lunar circuit for the moon, then divide by 200 to get the bright fraction. Subtract that from 589 for the sun or from the lunar circuit for the moon to obtain the dusk fraction. Add each fraction to the midnight longitude and reduce by the usual mansion rule.
82
推月蝕
Procedure: lunar eclipse prediction.
83
置上元年，外所求，以會歲去之，其餘年以會率乘之，如會歲為積蝕，有餘加積一。 會月乘之，如會率為積月，不盡為月餘。 以章閏乘餘年，滿章歲為積閏，以減積月，餘以歲中去之，不盡，數起天正。
From the epoch to the target year, take the remainder modulo 893, multiply by 1882, divide by 893—the quotient counts eclipse cycles, with any leftover incrementing the tally by one. Multiply that count by 11045 and divide by 1882 for whole months; the residue is the fractional month part. Multiply leftover years by 7, divide by 19 for leap months, subtract from the month total, then divide by 12; the remainder starts the count from the standard first month.
84
求次蝕，加五月，月餘千六百三十五，滿會率得一月，月以望。
For the next eclipse add five months plus remainder 1635; carry 1882 into another month; the eclipse falls at full moon.
85
推卦用事日
Procedure: find the day each hexagram governs.
86
因冬至大餘，倍其小餘，坎用事日也。 加小餘千七十五，滿乾法從大餘，中孚用事日也。
Take the winter solstice stem-branch remainder and double the fractional part—that fixes the day when the Kan trigram takes charge. Add 1075 to the fraction, carry 1178 into the day count, and you have the Zhongfu governing day.
87
求次卦，各加大餘六，小餘百三。 其四正各因共中日，而倍其小餘。
For each following hexagram add six to the large part and 103 to the small. For the four cardinal hexagrams, start from the solstice or equinox day and double the fractional part.
88
推五行用事
Procedure: days when wood, fire, earth, metal, and water rule.
89
置冬至大小餘，加大餘二十七，小餘九百二十七，滿二千三百五十六從大餘，得土用事日也。 加大餘十八，小餘六百一十八，得立春木用事日。 加大餘七十三，小餘百一十六，復得土。 又加土如得其火，金、水放此。
From winter solstice add 27 days and 927 fractional parts, carrying 2356 into the day column, to reach the day earth governs. Add 18 and 618 more to locate the day wood takes charge at Beginning of Spring. Another 73 days and 116 parts brings earth back into authority. Repeat the same step spacing for fire, then by analogy for metal and water.
90
推加時
Procedure: convert fractions into the twelve double-hours.
91
以十二乘小餘，滿其法得一辰，數從子起，算外，朔、弦、望以定小餘。
Multiply the fractional day by 12 and divide by the day denominator to get the double-hour, counting from midnight zi; the remainder beyond the tally gives the exact moment for new, quarter, or full moon.
92
推漏刻
Procedure: convert day fractions into water-clock marks.
93
以百乘小餘，滿其法得一刻，不盡什之，求分，課所近節氣，起夜分盡； 夜上水未盡，以所近言之。
Scale the fraction by 100, divide by the day law for whole ke, subdivide the tail by ten, and compare with the night run of the neighboring qi until the night watch runs out; if the clepsydra has not finished its night ascent, describe the time by the nearer boundary.
94
推有進退，進加退減所得也。 進退有差，起二分度後，率四度轉增少，少每半者，三而轉之，差滿三止，曆五度而減如初。
Where the algorithm says advance, add the correction; where it says retreat, subtract it. After the equinox points the correction steps change: roughly every four degrees the increment shrinks, halving each third step until the adjustment reaches three units, then after five degrees of travel the decrease resets like the start.
95
月行三道術
Theory of the moon moving along three paths.
96
月行遲疾，周進有恆。 會數從天地凡數，乘餘率自乘，如會數而一，為過周分。 以從周天，月周除之，曆日數也。 遲疾有衰，其變者勢也。 以衰減加月行率，為日轉度分。 衰左右相加，為損益率。 益轉相益，損轉相損，盈縮積也。 半小周乘通法，如通數而一，以曆周減焉，為朔行分也。 日轉度分列衰損益率盈縮積月行分一日十四度十分一退減益二十二盈初二百七十六二日十四度九分二退減益二十一盈二十二二百七十五三日十四度七分三退減益十九盈四十三二百七十三四日十四度四分四退減益十六盈六十二二百七十五日十四度四退減益十二盈七十八二百六十六六日十三度十五分四退減益八盈九十二百六十二七日十三度十一分四退減益四盈九十八二百五十八八日十三度七分四退減損盈百二二百五十四九日十三度三分四退加損四盈百二二百五十十日十二度十八分三退加損八盈九十八二百四十六十一日十二度十五分四退加損十一盈九十二百四十三十二日十二度十一分三退加損十五盈七十九二百三十九十三日十二度八分二退加損十八盈六十四二百三十六十四日十二度六分一退加損二十盈四十六二百三十四十五日十二度五分一進減損二十一盈二十六二百三十三十六日十二度六分二進減損二十損不足反減五為益，盈有五謂益
The moon speeds and slows, yet its cyclic advance follows a fixed law. Take the universal factor, multiply the residual rate by itself, and divide by 47 to obtain the over-cycle fraction. Add this to the celestial circle and divide by the lunar circuit constant to get the tabulated day index. The speed variation itself ebbs and flows—that gradient is what shifts. Apply the decay correction to the mean lunar rate to yield the day’s sidereal motion in degrees and parts. Sum the left and right decay terms to form the increase-or-decrease rate. Successive increases compound the surplus; successive decreases compound the shrinkage—those running totals are the waxing and waning integrals. Take half of 254, multiply by the tong divisor, divide by 31, subtract from the calendar circumference—this yields the lunation opening fraction. Tabulated ephemeris (first half-month): each line states the moon’s daily travel, the retreat-or-decrease flag, the additive correction, the running surplus tally (starting at 276), and the motion remainder; from day 8 the corrections switch from surplus to shrinkage, and the note explains how to flip a failed decrease into an increase once surplus hits five.
97
而損縮初二十，故不足。
Because the shrinkage column opens at twenty, early steps can fall short.
98
盈五縮初二百三十四十七日十二度八分三進減益十八縮十五二百三十六十八日十二度十一分四進減益十五縮二十三二百三十九十九日十二度十五分三進減益十一縮四十八二百四十三二十日十二度十八分四進減益八縮五十九二百四十六二十一日十三度三分四進減益四縮六十七二百五十二十二日十三度七分四進加損縮七十一二百五十四二十三日十三度十一分四進加損四縮七十一二百五十八二十四日十三度十五分四進加損八縮六十七二百六十二二十五日十四度四進加損十二縮五十九二百六十六二十六日十四度四分三進加損十六縮四十七二百七十二十七日十四度七分三曆初進加損十九縮三十一二百七十三
Continuation of the lunar ephemeris from day 17 through day 27: surplus has dropped to five while shrinkage begins at 234, then each row advances the moon’s speed, adjusts the correction, and updates the shrinkage balance until the cycle nears its end at day 27.
99
三大周日周日十四度九分少進加損二十一縮十二二百七十五
Final row of the third anomalistic week: fourteen degrees nine parts (slightly less), advance correction adding a loss of twenty-one, shrinkage twelve, balance 275.
100
周日分，三千三百三。
Daily remainder numerator: 3303.
101
周虛，二千六百六十六。
Complement to complete the week denominator: 2666.
102
周日法，五千九百六十九。
Week-day denominator: 5969.
103
通周，十八萬五千三十九。
Full-cycle common numerator: 185,039.
104
曆周，十六萬四千四百六十六。
Anomalistic circumference constant: 164,466.
105
少大法，一千一百一。
Minor-difference master divisor: 1111.
106
朔行大分，萬一千八百一。
Lunation opening major fraction: 11,801.
107
小分，二十五。
Minor fraction: 25.
108
周半，一百二十七。
Half-week index: 127.
109
推合朔入曆
Procedure: place the syzygy inside the anomalistic month.
110
以上元積月乘朔行大小分，小分滿通數三十一從大分，大分滿曆周去之，余滿周法得一日，不盡為日餘。 日餘命算外，所求合朔入曆也。
Multiply epoch months by the lunation fraction, carry 31 from the small into the large part, reduce modulo the anomalistic circumference, then divide by the week divisor for whole days and keep the tail as the day remainder. Count beyond the day remainder and you have the syzygy’s position within the tabulated month.
111
求次月，加一日，日餘五千八百三十二，小分二十五。
For the following lunation add one day plus remainder 5832 and minor part 25.
112
求弦望，各加七日，日餘二千二百八十三，小分二十九半，分各如法成日，日滿二十七日去之。 余如周分。 不足除，減一日，加周虛。
For quarters and full moons add seven days, remainder 2283, and fraction 29½, carrying by the usual rules; drop multiples of 27 from the day count. Reduce the remainder against the week fractional part. If subtraction borrows, drop one day and add the week void constant.
113
求弦望定大小餘
Procedure: true fractional instants of quarters and full moons.
114
置所入曆盈縮積，以通周乘之為實。 令通數乘日餘分，以乘損益率，以損益實，為加時盈縮也。 章歲減月行分，乘周半為差法，以除之，所得盈減縮加大小餘，如日法盈不足，朔加時在前後日。 弦望進退大餘，為定小餘。
Take the tabulated surplus or shrinkage integral for the current day and multiply by 185,039 to form the dividend. Multiply the day remainder by 31, then by the increase-decrease rate, and apply that to the dividend to get the double-hour correction for surplus or shrinkage. Form the difference divisor from (19 minus the lunar rate) times 127, divide the correction by it, then subtract or add to the day fractions per surplus or shrinkage; if the day fraction overflows, roll the syzygy to the adjacent day. The adjusted large remainder for quarters and full moons yields the definitive fractional day.
115
求朔弦望加時定度
Procedure: true longitudes at new moon, quarters, and full moon.
116
以章歲乘加時盈縮，差法除之，所得滿會數為盈縮大小分，以盈減縮加本日月所在，盈不足，以紀法進退度，為日月所在定度分。
Scale the double-hour surplus by 19, divide by the difference divisor, split the result into major and minor parts modulo 47, apply surplus-minus or shrinkage-plus to the base solar and lunar longitudes, borrowing 589 when needed—the outcome is the corrected degree and fraction.
117
推月行夜半入曆
Procedure: place midnight within the anomalistic table.
118
以周半乘朔小餘，如通數而一，以減入曆日餘。 余不足，加周法而減焉，卻一日。 卻得周日加其分，即得夜半入曆。
Multiply the lunation fraction by 127, divide by 31, and subtract from the anomalistic day remainder. If the subtraction underflows, add the week divisor and back up one day. After the retreat, add the weekly fractional part to reach midnight’s slot in the ephemeris.
119
求次日，轉一日，因日餘到二十七日，日余滿周日分去之，不直周日也。 其不滿直之，加周虛于余，餘皆次日入曆日餘也。
Advance one tabulated day; when the remainder crosses 27 days, strip multiples of the week fraction unless you land exactly on a week boundary. If you do not hit the boundary cleanly, add the week void to the residue—everything else becomes the next day’s anomalistic remainder.
120
求月夜半定度
Procedure: true lunar longitude at midnight.
121
以夜半入曆日餘，乘損益率，如周法得一，不盡為餘，以損益縮積，餘無所損，破全為法損之，為夜半盈縮也。 滿章歲為度，不盡為分。 通數乘分及余，余如周法從分，分滿紀法從度，以盈加縮減本夜半度及餘，為定度。
Multiply the midnight remainder by the rate, divide by the week divisor, fold it into the shrinkage integral—if nothing cancels, borrow a unit and subtract—to get midnight’s surplus or shrinkage. Divide by 19 for whole degrees; the tail is fractional parts. Multiply fractions by 31, carry into parts per the week rule, carry 589 into degrees, then add surplus or subtract shrinkage from the midnight longitude for the definitive position.
122
求變衰法
Procedure: interpolate the decay correction.
123
以入曆日餘乘列衰，如周法得一，不盡為余，即穀知其日變衰也。
Multiply the anomalistic remainder by the tabulated decay and divide by the week divisor; the quotient is how far the decay shifts on that day, reading the doubtful word in the text as meaning each successive step.
124
求次曆
Procedure: advance to the next anomalistic cycle.
125
以周虛乘列衰，如周法為常數，曆竟，輒以加變衰，滿列衰去之，轉為次曆變衰也。
Multiply the void 2666 by the listed decay and divide by the week divisor for a constant increment; at cycle end add it to the change-decay modulo the table to seed the next lunation.
126
求次日夜半定度
Procedure: true midnight longitude on the following day.
127
以變衰進加退減曆日轉分，分盈不足，章歲出入度也。 通數乘分及餘，而日轉加夜定度，為次日也。 竟曆不直周日，減餘三十八，乃以通數乘之，直周日者加餘八百三十七，又以少大分八百九十九，加次曆變衰，轉求如前。
Apply the decay as advance-plus or retreat-minus to the daily motion fraction, carrying 19 into degrees when the fraction overflows or underflows. Scale the fractional parts by 31, add the day’s motion to tonight’s fixed longitude, and you have tomorrow’s position. If the cycle ends off the week boundary, subtract 38 from the remainder and scale by 31; if it lands on the boundary, add 837, then add 899 to the minor fraction and fold in the next cycle’s change-decay, repeating the same steps.
128
求次日夜半盈縮
Procedure: surplus or shrinkage at the next midnight.
129
以變衰減加損益率，為變損益率，而以轉損益夜半盈縮。 曆竟損不足，反減為入次曆，減加餘如上數。
Combine the decay with the base increase-decrease rate to form the adjusted rate, then let the daily revolution update midnight’s surplus or shrinkage. If the cycle closes with a failed decrease, invert the subtraction to enter the next ephemeris, adjusting remainders as in the rules above.
130
求昏明月度
Procedure: lunar longitude at dusk and dawn.
131
以曆月行分乘所近節氣夜漏，二百而一為明分。 以減月行分為昏分。 分如章歲為度，以通數乘分，以加夜半定度，為昏明定度。 余分半法以上成，不滿廢之。
Multiply the tabulated lunar rate by the neighboring qi’s night water-clock reading and divide by 200 for the bright fraction. Subtract that from the lunar rate to obtain the dusk fraction. Convert parts to degrees via 19, scale with 31, and add to the midnight longitude for dusk and dawn positions. Round up fractional parts from half the divisor upward; discard what falls short.
132
求月行遲疾
Procedure: lunar anomaly in speed.
133
月經四表，出入三道，交錯分天，以月率除之，為曆之日。 周天乘朔望合，如會月而一，朔合分也。 通數乘合數，餘如會數而一，退分也。 以從月周，為日進分。 會數而一，為差率也。
As the moon crosses the four limits and weaves among three paths, partition the sky and divide by the monthly modulus to index the ephemeris day. Multiply the celestial circumference by the syzygy conjunction and divide by 11045 for the conjunction fraction. Multiply the conjunction count by 31 and reduce modulo 47 to obtain the retreat fraction. Add this to the lunar circuit to get the daily forward fraction. Divide by 47 to form the differential rate.
134
陰陽曆衰損益率兼數
Table heading: lunar and solar anomaly rates and their companion numbers.
135
一日一減益十七初
Day 1: single decrease, gain seventeen—opening entry.
136
二日限餘千二百九十微分四百五十七
Day 2 limit: remainder 1290, fine fraction 457.
137
一減益十六十七
One step of decrease, gain sixteen, cumulative seventeen.
138
此為前限
This marks the forward limit of the table.
139
三日三減益十五三十三
Day 3: triple decrease, gain fifteen, running total thirty-three.
140
四日四減益十二四十八
Day 4: quadruple decrease, gain twelve, total forty-eight.
141
五日四減益八六十
Day 5: four decreases, gain eight, total sixty.
142
六日三減益四六十八
Day 6: three decreases, gain four, total sixty-eight.
143
七日三減減不足，反損為加，謂益有一，當減三，為不足
Day 7: three decreases; if subtraction fails, invert loss into gain—meaning when increase reads one you owe three decreases, which is the shortfall case.
144
益一七十二
Entry: add one for a running total of seventy-two.
145
八日四加損二過極損之，謂月行半周，
Day 8: four additive losses of two; once the anomaly passes its peak, switch to subtraction—this is halfway through the lunar circuit.
146
度已過極，則當損之。
After the moon passes the apsidal line, corrections must subtract.
147
九日四加損六七十一
Day 9: four additive steps, loss six, balance seventy-one.
148
十日三加損十六十五
Day 10: three additive steps, loss sixteen, total fifteen.
149
十一日二加損十三五十五
Day 11: two additive steps, loss thirteen, total fifty-five.
150
十二日一加損十五四十二
Day 12: one additive step, loss fifteen, total forty-two.
151
十三日限餘三千九百一十二，微分一千七百五十二。
Day 13 boundary values: limit remainder 3912, fine fraction 1752.
152
此為後限
This marks the rear limit of the anomaly table.
153
一加曆初大，分日。 損十六二十七
Add one at the lunation’s opening and split the fractional day. Subtract sixteen, cumulative twenty-seven.
154
分日五千二百而三少加少者損十六大十一
Technical split-day instruction: with divisor 5202 and minor thirds, minor additions trigger a major loss of sixteen and remainder eleven.
155
少大法，四百七十三。
Minor-difference divisor: 473.
156
曆周，十萬七千五百六十五。
Anomalistic circumference for this table: 107,565.
157
差率，萬一千九百八十六。
Latitude-difference modulus: 11,986.
158
朔合分，萬八千三百二十八。
Syzygy conjunction numerator: 18,328.
159
微分，九百一十四。
Fine fraction constant: 914.
160
微分法，二千二百九。
Fine-fraction denominator: 2209.
161
推朔入陰陽曆
Procedure: place the new moon inside the solar-lunar latitude ephemeris.
162
以會月去上元積月，餘以朔合分及微分各乘之，微分滿其法從合分，合分滿周天去之，其餘不滿曆周者，為入陽曆； 滿去之，餘為入陰曆。 餘皆如月周得一日，算外，所求月合朔入曆，不盡為日餘。
Reduce epoch months modulo 11,045, scale by 18,328 and 914 with carries, reduce modulo the sky circle; a remainder under 107,565 means the moon is in the yang (north) column; if it exceeds that, strip the cycle and the residue places the moon in the yin (south) column. Divide the residue by the lunar circuit for whole days beyond the tally; what remains is the fractional day inside the table.
163
求次月
Procedure: advance one lunation.
164
加二日，日餘二千五百八十，微分九百一十四，如法成日，滿十三去之，除餘如分日。 陰陽曆竟互入端，入曆在前限餘前，後限餘後者月行中道也。
Add two days plus remainder 2580 and micro-part 914, normalize, drop multiples of thirteen, and reconcile with the split-day rule. When the solar-lunar cycle flips, if the index falls before the front limit or after the rear limit, the moon is taken to ride the ecliptic mean.
165
求朔望定數
Procedure: true syzygy correction.
166
各置入遲疾曆盈縮大小分，會數乘小分為微分，盈減縮加陰陽日余，日餘盈不足，進退日而定。 以定日餘乘損益率，如月周得一，以損益兼數，為加時定數。
Take the fast–slow surplus or deficit, turn the small part into micro-fractions with factor 47, add or subtract from the solar-lunar day fraction, and roll the day forward or back if it overflows. Multiply the corrected day fraction by the rate, divide by the lunar circuit, and apply the paired adjustment for the true syzygy instant.
167
推夜半入曆
Procedure: midnight index in the solar-lunar table.
168
以差率乘朔小餘，如微分法得一，以減入曆日餘，不足，加月周而減之，卻一日。 卻得分日加其分，以會數約微分為小分，即朔日夜半入曆。
Scale the lunation fraction by the difference rate, divide by 2209, subtract from the table remainder, borrowing the lunar circuit if needed, backing up one day when required. Restore the split-day fraction, reduce micro-parts with factor 47, and you have midnight of new moon inside the ephemeris.
169
求次日，加一日，日餘三十一，小分三十一，小分如會數從余，余滿月周去之，又加一日，曆竟下，日余滿分日去之，為入曆初也。 不滿分日者直之，加餘二千七百二，小分三十一，為入次曆。
For the next day add 1, 31, and 31 with carries through 47 and the lunar circuit; at cycle bottom clear the split-day overflow to re-enter the table head. If the split-day slot is short, add 2702 plus fractional 31 to step into the next lunation row.
170
求夜半定日
Procedure: true day count at midnight.
171
以通數乘入遲疾曆夜半盈縮及余，余滿周半為小分，以盈加縮減入陰陽日余，日餘盈不足，以月周進退日而定也。 以定日餘乘損益率，如月周得一，以損益兼數，為夜半定數也。
Multiply the fast–slow midnight correction by 31, carry 127 into a small fraction, adjust the solar-lunar day count, and borrow a lunar-circuit day if needed. Scale the midnight remainder by the rate, divide by the lunar circuit, and apply the paired number for the true midnight correction.
172
求昏明數
Procedure: dusk and dawn corrections.
173
以損益率乘所近節氣夜漏，二百而一為明，以減損益率為昏，而以損益夜半數為昏明定數。
Multiply the rate by the night run of the neighboring qi and divide by 200 for dawn; subtract from the rate for dusk; combine with the midnight adjustment for definitive twilight values.
174
求月去極度
Procedure: lunar distance from the celestial pole.
175
置加時若昏明定數，以十二除之為度，其餘三而一為少，不盡一為強，二少弱也，所得為月去黃道度也。 其陽曆以加日所在黃道曆去極度，陰曆以減之，則月去極度。 強正弱負，強弱相並，同名相從，異名相消。 其相減也，同名相消，異名相從，無對互之，二強進少而弱。
Convert the correction to degrees by dividing by twelve, express thirds as shao and mark strong or weak units—this yields the moon’s ecliptic latitude. For the yang column add the solar ecliptic polar distance; for the yin column subtract it—either way you reach the moon’s polar distance. Treat strong corrections as positive and weak as negative; combine like signs and cancel opposites. When subtracting mixed strong and weak parts, pair matching terms, let unlike terms accumulate, and convert double-strong into advancing shao with weakness.
176
上元己丑以來，至建安十一年丙戌，歲積七千三百七十八。
From epoch jichou to Jian’an 11 bingxu the accumulated year count is again 7378.
177
己丑戊寅丁卯丙辰乙巳甲午癸未
Stem-branch cycle line one: jichou, wuyin, dingmao, bingchen, yisi, jiawu, guiwei.
178
壬申辛酉庚戌己亥戊子丁丑丙寅
Stem-branch cycle line two: renshen, xinyou, gengxu, jihai, wuzi, dingchou, bingyin.
179
推五星
Procedure: computing the five planets.
180
五行：木，歲星； 火，熒惑； 土，填星； 金，太白； 水，辰星。 各以終日與天度相約，為周率、日率。 章歲乘周，為月法。 章月乘日，為月分。 分如法，為月數。 通數乘月法，日度法也。 斗分乘周率，為斗分。 日度法用紀法乘周率，故此同以分乘之。
Wood corresponds to Jupiter, the Year Star; fire to Mars, the Sparkling Deluder; earth to Saturn, the Filler Star; metal to Venus, the Great White; water to Mercury, the Morning Star. For each planet, ratio its sidereal period against the sky to derive a cycle rate and a day rate. Multiply the cycle rate by nineteen to form the lunation divisor. Multiply the day rate by 235 for the lunation numerator. Reduce the fraction by the standard divisor to get the lunation count. Multiply the lunation divisor by 31 to obtain the degree-measure divisor. Multiply the cycle rate by the dipper fraction 145 for the scaled dipper term. Because the degree divisor equals the era divisor times the cycle rate, the same fractional multiplication applies.
181
五星朔大餘、小餘。 以通法各乘月數，日法各除之，為大餘，不盡為小餘。 以六十去大餘。
Heading: planetary conjunction day fractions. Multiply each lunation count by the tong divisor and divide by the day denominator for the stem-branch remainder and fractional tail. Cast out multiples of sixty from the large remainder.
182
五星入月日、日餘。 各以通法乘月餘，以合月法乘朔小餘，並之，會數約之，所得各以日度法除之，則皆是。
Heading: day-in-month and remainder for planetary conjunctions. Combine scaled month remainders with the lunation fraction, reduce by 47, and divide by the degree divisor for the planetary ingress values.
183
五星度數、度餘。 減多為度餘分，以周天乘之，以日度法約之，所得為度，不盡為度余，過周天去之及斗分。
Heading: heliocentric arc and residue. Turn the dominant remainder into a celestial arc, scale by the sky circle, divide by the degree law, and strip full revolutions plus the dipper fraction.
184
紀月，七千二百八十五。
Era month count: 7285.
185
章閏，七。
Intercalary months per metonic cycle: 7.
186
章月，二百三十五。
Months per metonic cycle: 235.
187
歲中，十二。
Months per solar year: 12.
188
通法，四萬三千二十六。
Common divisor: 43,026.
189
日法，千四百五十七。
Day denominator: 1457.
190
會數，四十七。
Lunation factor: 47.
191
周天，二十一萬五千一百三十。
Celestial circle in parts: 215,130.
192
斗分，一百四十五。
Dipper fraction: 145.
193
木：周率，六千七百二十二。
Jupiter—cycle rate: 6722.
194
日率，七千三百四十一。
Jupiter—day rate: 7341.
195
合月數，十三。
Jupiter—lunations per cycle: 13.
196
月餘，六萬四千八百一。
Jupiter—lunation remainder: 64,801.
197
合月法，十二萬七千七百一十八。
Jupiter—lunation divisor: 127,718.
198
日度法，三百九十五萬九千二百五十八。
Jupiter—degree divisor: 3,959,258.
199
朔大餘，二十三。
Jupiter—conjunction large remainder: 23.
200
朔小餘，一千三百七。
Jupiter—conjunction small remainder: 1307.
201
入月日，十五。
Jupiter—day within month: 15.
202
日餘，三百四十八萬四千六百四十六。
Jupiter—day remainder: 3,484,646.
203
朔虛分，一百五十。
Jupiter—conjunction void: 150.
204
斗分，九十七萬四千六百九十。
Jupiter—dipper term: 974,690.
205
度數，三十三。
Jupiter—whole degrees: 33.
206
度餘，二百五十萬九千九百五十六。
Jupiter—degree remainder: 2,509,956.
207
火：周率，三千四百七。
Mars—cycle rate: 3407.
208
日率，七千二百七十一。
Mars—day rate: 7271.
209
合月數，二十六。
Mars—lunations per cycle: 26.
210
月餘，二萬五千六百二十七。
Mars—lunation remainder: 25,627.
211
合月法，六萬四千七百三十三。
Mars—lunation divisor: 64,733.
212
日度法，二百萬六千七百二十三。
Mars—degree divisor: 2,006,723.
213
朔大餘，四十七。
Mars—conjunction large remainder: 47.
214
朔小餘，一千一百五十七。
Mars—conjunction small remainder: 1157.
215
入月日，十二。
Mars—day within month: 12.
216
日余，九十七萬三千一十三。
Mars—day remainder: 970,013.
217
朔虛分，三百。
Mars—conjunction void: 300.
218
斗分，四十九萬四千一十五。
Mars—dipper term: 494,015.
219
度數，四十八。
Mars—whole degrees: 48.
220
度餘，一百九十九萬一千七百六。
Mars—degree remainder: 1,991,706.
221
土：周度，三千五百二十九。
Saturn—cycle rate (listed as week degree): 3529.
222
日率，三千六百五十三。
Saturn—day rate: 3653.
223
合月數，十二。
Saturn—lunations per cycle: 12.
224
月餘，五萬三千八百四十三。
Saturn—lunation remainder: 53,843.
225
合月法，六萬七千五十一。
Saturn—lunation divisor: 67,051.
226
日度法，二百七萬八千五百八十一。
Saturn—degree divisor: 2,078,581.
227
朔大餘，五十四。
Saturn—conjunction large remainder: 54.
228
朔小餘，五百三十四。
Saturn—conjunction small remainder: 534.
229
入月日，二十四。
Saturn—day within month: 24.
230
日余，十六萬六千二百七十二。
Saturn—day remainder: 166,272.
231
朔虛分，九百二十三。
Saturn—conjunction void: 923.
232
斗分，五十一萬一千七百五。
Saturn—dipper term: 511,705.
233
度數，十二。
Saturn—whole degrees: 12.
234
度餘，一百七十三萬三千一百四十八。
Saturn—degree remainder: 1,733,148.
235
金：周率，九千二十二。
Venus—cycle rate: 9022.
236
日率，七千二百一十三。
Venus—day rate: 7213.
237
合月數，九。
Venus—lunations per cycle: 9.
238
月餘，十五萬二千二百九十三。
Venus—lunation remainder: 152,293.
239
合月法，十七萬一千四百一十八。
Venus—lunation divisor: 171,418.
240
日度法，五百三十一萬三千九百五十八。
Venus—degree divisor: 5,313,958.
241
朔大餘，二十五。
Venus—conjunction large remainder: 25.
242
朔小餘，一千一百二十九。
Venus—conjunction small remainder: 1129.
243
入月日，二十七。
Venus—day within month: 27.
244
日余，五萬六千九百五十四。
Venus—day remainder: 56,954.
245
朔虛分，三百二十八。
Venus—conjunction void: 328.
246
斗分，一百三十萬八千一百九十。
Venus—dipper term: 1,308,190.
247
度數，二百九十二。
Venus—whole degrees: 292.
248
度余，五萬六千九百五十四。
Venus—degree remainder: 56,954.
249
水：周率，一萬一千五百六十一。
Mercury—cycle rate: 11,561.
250
日率，一千八百三十四。
Mercury—day rate: 1834.
251
合月數，一。
Mercury—lunations per cycle: 1.
252
月餘，二十一萬一千三百三十一。
Mercury—lunation remainder: 211,331.
253
合月法，二十一萬九千六百五十九。
Mercury—lunation divisor: 219,659.
254
日度法，六百八十萬九千四百二十九。
Mercury—degree divisor: 6,809,429.
255
朔大餘，二十九。
Mercury—conjunction large remainder: 29.
256
朔小餘，七百七十三。
Mercury—conjunction small remainder: 773.
257
入月日，二十八。
Mercury—day within month: 28.
258
日餘，六百四十一萬九百六十七。
Mercury—day remainder: 6,410,967.
259
朔虛分，六百八十四。
Mercury—conjunction void: 684.
260
斗分，一百六十七萬六千三百四十五。
Mercury—dipper term: 1,676,345.
261
度數，五十七。
Mercury—whole degrees: 57.
262
度餘，六百四十一萬九百六十七。
Mercury—degree remainder: 6,410,967.
263
推五星
Procedure: computing planetary conjunctions.
264
置上元盡所求年，以周率乘之，滿日率得一，名積合，不盡為合餘。 以周率除之，得一，星合往年。 二，合前往年。 無所得，合其年。 合餘減周率為度分。 金、水積合，奇為晨，耦為夕。
From the epoch to the target year, multiply by the cycle rate and divide by the day rate; the quotient is completed conjunction cycles and the residue is the fractional conjunction. Divide once by the cycle rate: a quotient of one means the conjunction fell in the preceding year. A quotient of two places the event in the year before that. No whole quotient means the conjunction falls in the current year. Subtract the cycle rate from the conjunction remainder to isolate the degree fraction. For Venus and Mercury, an odd cycle count signals a dawn apparition, an even one a dusk apparition.
265
推星合月
Procedure: month of planetary conjunction.
266
以月數、月餘各乘積合，滿合月法從月，不盡為月餘。 以紀月去積月，余為入紀月。 副以章閏乘之，滿章月得一閏，以減入紀月，餘以歲中去之，命以天正算外，合月也。 其在閏交際，以朔禦之。
Multiply the conjunction count by the lunation numerator and divisor, carrying overflows to whole months. Reduce the month total modulo 7285 for the position inside the era cycle. Apply the 7/235 leap rule, adjust the era-month index, divide by 12, and count from the standard first month to name the conjunction month. If the result straddles an intercalary boundary, anchor it with the new-moon rule.
267
推入月日
Procedure: day within month at conjunction.
268
以通法乘月餘，合月法乘朔小餘，並以會數約之，所得滿日度法得一，則星合入月日也。 不滿為日餘，命以朔算外。
Combine the scaled month remainder with the lunation fraction, reduce by 47, divide by the degree divisor for the conjunction’s calendar day. The leftover is the fractional day; count beyond the new moon for the precise instant.
269
推星合度
Procedure: ecliptic longitude at conjunction.
270
以周天乘度分，滿日度法得一度，不盡為餘，命度以牛前五起。
Multiply the degree fraction by the celestial circle, divide by the degree divisor, and count mansions from five degrees before Ox.
271
右求星合。
Above: steps to find a planetary conjunction.
272
求後合月
Procedure: next conjunction month.
273
以月數加月數，以月餘加月餘，滿合月法得一月，不滿歲中，即合其年，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。
Add another cycle’s month count and remainder, carrying into whole months; if under twelve months the event stays in the same year, otherwise strip years and account for leap months. A second overflow pushes the conjunction two years ahead. For Venus and Mercury, adding a dawn apparition yields the next dusk one, and vice versa.
274
求後合朔日
Procedure: new moon of the next conjunction month.
275
以朔大小餘，加合月大小餘，上成月者，又加大餘二十九，小餘七百七十三，小餘滿日法從大餘，命如前。
Sum the lunation fractions, and if a month rolls over add 29 and 773 with carries, assigning stem-branches as earlier.
276
求後入月日術
Procedure: day-in-month after the next conjunction.
277
以入月日、日餘，加合入月日及余，余滿日度法得一日，其前合朔小餘滿其虛分者，減一日。。 後小餘滿七百七十三以上者，去二十九日，不滿，去三十日，其餘則後合，入月日也。
Add the previous ingress day and fraction; carry by the degree divisor; if the earlier lunation fraction exceeded the void limit, borrow one day. If the new fractional part is at least 773, subtract 29 days; otherwise subtract 30—what remains is the next ingress date.
278
求後度
Procedure: longitude after the next conjunction.
279
以度加度，度余加度余，滿日度法得一度。
Add the degree increments and their fractional parts, carrying the divisor into whole degrees.
280
伏三十二日。 三百四十八萬四千六百四十六分。
Jupiter: 32 days of invisibility. Fractional part: 3,484,646.
281
見三百六十六日。
Visible apparition: 366 days.
282
伏行五度。 二百五十萬九千九百五十六分。
Arc during invisibility: five degrees. Fractional arc: 2,509,956.
283
見行四十度。 除逆退十二度，定行二十八度。
Arc while visible: forty degrees. Subtract twelve degrees of retrograde motion; net direct travel is twenty-eight degrees.
284
火：伏百四十三日。 九十七萬三千一十三分。
Mars: 143 days invisible. Fraction: 970,013.
285
見六百三十六日。
Visible span: 636 days.
286
伏行一百一十度。 四十七萬八千九百九十八分。
Arc hidden: 110 degrees. Fraction: 478,998.
287
見行三百二十度。 除逆十七度，定行三百三度。
Arc visible: 320 degrees. Subtract 17° of retrograde; net advance 303°.
288
土：伏三十三日。 十六萬六千二百七十二分。
Saturn: 33 days invisible. Fraction: 166,272.
289
見三百四十五日。
Visible span: 345 days.
290
伏行三度。 一百七十三萬三千一百四十八分。
Hidden arc: three degrees. Fraction: 1,733,148.
291
見行十五度。 除逆六度，定行九度。
Visible arc: fifteen degrees. Subtract six degrees retrograde; net nine degrees direct.
292
金：晨伏東方八十二日。 十一萬三千九百八分。
Venus: 82 days as morning star lost in the east. Fraction: 113,908.
293
見西方。 二百四十六日。 除逆六度，定行二百四十六度。
Then appears in the west. Duration: 246 days. Subtract six degrees of retrograde; net travel 246 degrees.
294
晨伏行百度。 十一萬三千九百八分。
While lost in the dawn twilight the planet advances one hundred degrees. Carry fraction 113,908.
295
見東方。 日度加西。 伏十日，退八度。
It then becomes visible in the east. Add the western elongation to the solar longitude. It disappears for ten days, slipping back eight degrees.
296
水：晨伏三十三日。 六百一萬二千五百五分。
Mercury: thirty-three morning days in conjunction. Attached fraction: 6,012,505.
297
見西方。 三十二日。 除逆一度，定行三十二度。
Evening apparition in the west. Visible span: thirty-two days. Subtract one degree of retrograde; net motion thirty-two degrees forward.
298
伏行六十五度。 六百一萬二千五百五分。
During invisibility it creeps sixty-five degrees. Same fractional adjunct: 6,012,505.
299
見東方。 日度如西，伏十八日，退十四度
Reappears in the east as morning star. Mirror the western case: eighteen days invisible, fourteen degrees retrograde.
300
五星曆步術
Ephemeris algorithm for the five planets’ stations.
301
以法伏日度及餘，加星合日度余，余滿日度法得一，從全命之如前，得星見日及度也。 以星行分母乘見度，餘如日度法得一，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，遞則減之，伏不盡度，經鬥除分，以行母為率，分有損益，前後相禦。 凡言如盈約滿，皆求實之除也； 去及除之，取盡之除也。
Add the tabulated invisibility arc to the conjunction longitude, carry the degree divisor, and assign mansions as before to get the date and degree of first visibility. Multiply the visible arc by the motion denominator, divide by the degree divisor, rounding up halves. Accumulate daily motion with its own denominator, converting between retrograde and direct bases by cross-multiplying denominators. A station continues prior motion; at a loop subtract; incomplete arcs crossing Dipper drop fractions; rates follow each phase’s denominator, with corrections chaining across segments. Phrases such as ‘as if full’ mean divide to obtain the true quotient. ‘Cast out’ and ‘divide’ mean reduce until the remainder is cleared.
302
木：晨與日合，伏，順，十六日百七十四萬二千三百二十三分，行星二度三百二十三萬四千六百七分，而晨見東方，在日後。 順，疾，日行五十八分之十一，五十八日行十一度。 更順，遲，日行九分，五十八日行九度。 留，不行二十五日而旋。 逆，日行七分之一，八十四日退十二度。 復留，二十五日而順，日行五十八分之九，五十八日行九度。 順，疾，日行十一分，五十八日行十一度，在日前，夕伏西方。 十六日百七十四萬二千三百二十三分，行星二度三百二十三萬四千六百七分，而與日合。 凡一終，三百九十八日三百四十八萬四千六百四十六分，行星四十三度二百五十萬九千九百五十六分。
Jupiter: after inferior conjunction it runs direct for sixteen days plus fraction, covering just over two degrees, then emerges east of the sun at dawn. First direct phase: 11° every 58 days (11/58 of a degree per day). Slower direct motion: 9/58 degree per day for fifty-eight days. It stands still twenty-five days, then reverses. Retrograde at 1/7° per day for eighty-four days—twelve degrees backward. Second station, then direct again at 9/58° per day for fifty-eight days. Speeds up to 11/58° per day, passes west of the sun, and sets in the west. After the same sixteen-day interval and arc it reaches superior conjunction. One complete synodic period: 398 days plus fraction and 43° net travel.
303
火：晨與日合，伏，順，七十一日百四十八萬九千八百六十八分，行星五十五度百二十四萬二千八百六十分半，而晨見東方，在日後。 順，日行二十三分之十四，百八十四日行一百一十二度。 更順，遲，日行二十三分之十二，九十二日行四十八度。 留，不行十一日。 旋，逆，日行六十二分之十七，六十二日退十七度。 復留，十一日而順，日行十二分，九十二日行四十八度。 復順，疾，日行十四分，百八十四日行百一十二度，在日前，夕伏西方。 七十一日百四十八萬九千八百六十八分，行星五十五度百二十四萬二千八百六十分半，而與日合。 凡一終，七百七十九日九十七萬三千一十三分，行星四百一十四度四十七萬八千九百九十八分。
Mars emerges from conjunction after seventy-one days, having advanced 55°, and appears eastern in the dawn. Direct motion 14/23° per day for 184 days—112°. Slower stretch: 12/23° per day for ninety-two days. Eleven-day stationary point. Retrogrades 17/62° per day for sixty-two days. Second station, then resumes 12° per ninety-two days (rate 12/92 simplified). Accelerates to 14°/184d, stands west of the sun, and vanishes at dusk. Returns to conjunction after the symmetric interval and arc. Full Martian synodic cycle totals 779 days and 414° of motion.
304
土：晨與日合，伏，順，十六日百一十二萬二千四百二十六分半，行星一度百九十九萬五千八百六十四分半，而晨見東方，在日後。 順，日行三十五分之三，八十七日半行七度半。 留，不行三十四日。 旋，逆，日行十七分之一，百二日退六度。 復三十四日而順，日行三分，八十七日行七度半，在日前，夕伏西方。 十六日百一十二萬二千四百二十六分半，行星一度百九十萬五千八百六十四分半，而與日合也。 凡一終，三百七十八日十六萬六千二百七十二分，行星十二度百七十三萬三千一百四十八分。
Saturn leaves conjunction with sixteen days of direct drift amounting to barely one degree, then shows in the east. Creeps 3/35° per day for eighty-seven and a half days—seven and a half degrees. Thirty-four-day halt. Retrogrades 1/17° per day for 102 days—six degrees. Resumes direct motion at 3/35° per day, passes west of the sun, and sets. Returns to conjunction after the stated interval (text gives 1,905,864.5 for the degree fraction). Saturn’s synodic period spans 378 days and twelve degrees of net motion.
305
金：晨與日合，伏，逆，五日退四度，而晨見東方，在日後。 逆，日行五分度之三，十日退六度。 留，不行八日。 旋，順，遲，日行四十六分之三十三，四十六日行三十三度而順。 疾，日行一度九十一分之十五，九十一日行一百六度。 更順，益疾，日行一度九十一分之二十二，九十一日行百一十三度，在日後，晨伏東方。 順，四十一日五萬六千九百五十四分，行星五十度五萬六千九百五十四分，而與日合。 一合，二百九十二日五萬六千九百五十四分，行星亦如之。
Venus: brief retrograde right after inferior conjunction, four degrees in five days, then visible east. Continues retrograde at three-fifths of a degree per day for ten days. Eight-day station. Resumes direct motion at 33/46° per day for forty-six days. Accelerates to 1+15/91 degrees daily for ninety-one days. Still faster—22/91 added per day—until greatest western elongation and morning disappearance. Forty-one-day sprint of fifty degrees brings superior conjunction. A single synodic arc spans 292 days and the same fractional travel.
306
金：夕與日合，伏，順，四十一日五萬六千九百五十四分，行星五十度五萬九千九百五十四分，而夕見西方，在日前。 順，疾，日行一度九十一分之二十二，九十一日行百一十三度。 更順，減疾，日行一度十五分，九十一日行百六度而順。 遲，日行四十六分之三十三，四十六日行三十三度。 留，不行八日。 旋，逆，日行五分之三，十日退六度，在日前，夕伏西方，逆，疾，五日退四度，而與日合。 凡再合一終，五百八十四日十一萬三千九百八分，行星亦如之。
Evening apparition begins after forty-one days and fifty degrees (fraction 59,954 in text). Descends from superior conjunction at peak speed. Slows to 1+15/91° per day for ninety-one days. Approaches inferior conjunction at 33/46° per day. Eight-day pause. Retrogrades toward inferior conjunction, vanishes west, then completes the inner loop in five days. Two successive inferior-superior cycles total 584 days plus the listed fraction.
307
水：晨與日合，伏，逆，九日退七度，而晨見東方，在日後。 更逆，疾，一日退一度。 留，不行二日。 旋，順，遲，日行九分之八，九日行八度而順。 疾，日行一度四分之一，二十日行二十五度，在日後。 晨伏東方，順，十六日六百四十一萬九百六十七分，行星三十二度六百四十一萬九百六十七分，而與日合，一合，五十七日六百四十一萬九百六十七分，行星亦如之。
Mercury’s dawn apparition follows nine days and seven degrees of retrograde. Accelerating retreat: one degree per day. Two-day station. Resumes at 8/9° per day for nine days. Speeds to one and a quarter degrees daily for twenty-five degrees total, still trailing the sun. Morning invisibility lasts sixteen days with matching degree fraction; full synodic span fifty-seven days with the same remainder.
308
水：夕與日合，伏，順，十六日六百四十一萬九百六十七分行星三十二度六百四十一萬九百六十七分，而夕見西方，在日前。 順，疾，日行一度四分之一，二十日行二十五度而順。 遲，日行九分之八，九日行八度。 留，不行二日。 旋，逆，一日退一度，在日前，夕伏西方。 逆，遲，九日退七度，與日合。 凡再合一終，一百一十五日六百一萬二千五百五分，行星亦如之。
Evening apparition: identical sixteen-day ingress, then visibility west of the sun. Evening arc accelerates at 1.25°/day for twenty days. Slows to eight ninths of a degree daily. Two-day halt. Drops back one degree per day and vanishes west of the sun. Completes inferior conjunction after nine days and seven degrees retrograde. Two Mercury loops span 115 days plus the tabulated fraction.