1
夫明時治曆，自黃帝、堯、舜與三代之盛王，莫不重之，其文備見於傳記矣。 雖去古既遠，其法不詳，然原其要，不過隨時考驗，以合於天而已。 漢劉歆作《三統曆》，始立積年日法，以為推步之準。 後世因之，歷唐而宋，其更元改法者，凡數十家，豈故相為乖異哉？ 蓋天有不齊之運，而曆為一定之法，所以既久而不能不差，既差則不可不改也。
Clarifying the seasons and regulating the calendar was esteemed by rulers from the Yellow Emperor through Yao and Shun to the greatest kings of the Three Dynasties; the full account survives in the historical records. Though we are far removed from antiquity and the methods are no longer fully known, the essential principle was always the same: observe and test as times require, until the calendar matches the heavens. In the Han, Liu Xin wrote the Triple Concordance Calendar and was the first to set out accumulated years and day-tables as the basis for calendrical calculation. Later dynasties followed suit; from Tang through Song, dozens of schools revised the epoch and changed the methods—not because they wished to contradict one another. Heaven's motions are not perfectly uniform, yet a calendar must be a fixed system; over time errors inevitably accumulate, and once they appear the calendar must be revised.
2
元初承用金《大明曆》，庚辰歲，太祖西征，五月望，月蝕不效； 二月、五月朔，微月見於西南。 中書令耶律楚材以《大明曆》後天，乃損節氣之分，減周天之秒，去交終之率，治月轉之余，課兩曜之後先，調五行之出沒，以正《大明曆》之失。 且以中元庚午歲，國兵南伐，而天下略定，推上元庚午歲天正十一月壬戌朔，子正冬至，日月合璧，五星聯珠，同會虛宿六度，以應太祖受命之符。 又以西域、中原地裏殊遠，創為裏差以增損之，雖東西萬里，不復差忒。 遂題其名曰《西征庚午元曆》，表上之，然不果頒用。
In the early Yuan the court still used the Jin dynasty's Great Enlightenment Calendar; in a gengchen year, during the Grand Progenitor's western campaign, the lunar eclipse predicted for the fifteenth of the fifth month failed to occur as forecast; On new-moon days in the second and fifth months, a thin crescent appeared in the southwest when it should not have. Grand Councilor Yelü Chucai judged the Great Enlightenment Calendar to run behind the heavens; he trimmed the solar-term divisions, reduced seconds in the tropical year, revised nodal-cycle rates, refined lunar-motion remainders, recalculated the sun and moon's relative positions, and adjusted planetary risings and settings to correct the calendar's errors. He also took the middle origin in a gengwu year—when imperial forces marched south and the realm was nearly pacified—and derived an upper origin in gengwu: on the winter solstice at midnight, new moon on renxu day in the eleventh month, with sun and moon united and the five planets aligned at six degrees of Xu, in token of the Grand Progenitor's Mandate. Because the Western Regions and the Central Plain differ greatly in latitude, he invented a latitude correction so that observations thousands of li apart would no longer disagree. He titled the work the Western Campaign Gengwu Origin Calendar and submitted it to the throne, but it was never actually put into use.
3
至元四年，西域劄馬魯丁撰進《萬年曆》，世祖稍頒行之。 十三年，平宋，遂詔前中書左丞許衡、太子贊善王恂、都水少監郭守敬改治新曆。 衡等以為金雖改曆，止以宋《紀元曆》微加增益，實未嘗測驗於天，乃與南北日官陳鼎臣、鄧元麟、毛鵬翼、劉巨淵、王素、嶽鉉、高敬等參考累代曆法，復測候日月星辰消息運行之變，參別同異，酌取中數，以為曆本。 十七年冬至，曆成，詔賜名曰《授時曆》。 十八年，頒行天下。 二十年，詔太子諭德李謙為《曆議》，發明新曆順天求合之微，考證前代人為附會之失，誠可以貽之永久，自古及今，其推驗之精，蓋未有出於此者也。 今衡、恂、守敬等所撰《曆經》及謙《曆議》故存，皆可考據，是用具著於篇。 惟《萬年曆》不復傳，而《庚午元曆》雖未嘗頒用，其為書猶在，因附著於後，使來者有考焉。 作《曆志》。
In Zhiyuan 4, Jamal al-Din from the Western Regions presented the Myriad Years Calendar, and Kublai gradually put it into circulation. In year 13, after the conquest of Song, the court ordered Xu Heng, Wang Xun, and Guo Shoujing to draft a new calendar. Xu Heng and his colleagues argued that the Jin revision had merely tweaked the Song Jiyuan Calendar without genuine astronomical testing; with calendar officials from north and south they reviewed earlier systems, re-observed the motions of sun, moon, and planets, compared results, and chose balanced values as the new calendar's basis. The calendar was finished at the winter solstice of year 17, and the court named it the Season-Granting Calendar. In year 18 it was issued empire-wide. In year 20 Li Qian, instructor to the crown prince, was ordered to write a Discourse on the Calendar explaining how the new system aligns with the heavens and exposing earlier calendars' forced adjustments—a work fit to endure; in precision of calculation and verification, none had surpassed it. The Calendar Classic by Xu Heng, Wang Xun, and Guo Shoujing, and Li Qian's Discourse, still survive and can be verified; they are therefore recorded here in detail. The Myriad Years Calendar no longer survives, but the Gengwu Origin Calendar, though never adopted, is still extant and is appended here for future reference. Thus the Treatise on the Calendar is composed.
4
授時曆議上驗氣
Discourse on the Season-Granting Calendar, Part I: Verifying the Solar Terms
5
天道運行，如環無端，治曆者必就陰消陽息之際，以為立法之始。 陰陽消息之機，何從而見之？ 惟候其日晷進退，則其機將無所遁。 候之之法，不過植表測景，以究其氣至之始。 智作能述，前代諸人為法略備，茍能精思密索，心與理會，則前人述作之外，未必無所增益。
Heaven's course runs in an endless cycle; calendar-makers must fix the moment when yin declines and yang pauses as the starting point for their system. How is one to detect the turning of yin and yang? Only by watching the gnomon shadow lengthen and shorten can that turning be captured with certainty. The method is simply to set up a gnomon, measure the shadow, and determine when each solar term arrives. Earlier masters had largely perfected the methods; yet with careful thought and close observation, aligning mind with principle, further improvements beyond their work remain possible.
6
舊法擇地平衍，設水準繩墨，植表其中，以度其中晷。 然表短促，尺寸之下所為分秒太、半、少之數，未易分別。 表長，則分寸稍長，所不便者，景虛而淡，難得實景。 前人欲就虛景之中考求真實，或設望筒，或置小表，或以木為規，皆取表端日光下徹圭面。 今以銅為表，高三十六尺，端挾以二龍，舉一橫梁，下至圭面，共四十尺，是為八尺之表五。 圭表刻為尺寸，舊寸一，今申而為五，厘毫差易分。 別創為景符，以取實景。 其制以銅葉，博二寸，長加博之二，中穿一竅，若針芥然，以方跂為趺，一端設為機軸，令可開闔，榰其一端，使其勢斜倚，北高南下，往來遷就於虛景之中，竅達日光，僅如米許，隱然見橫梁於其中。 舊法以表端測晷，所得者日體上邊之景，今以橫梁取之，實得中景，不容有毫末之差。
The old practice was to choose level ground, establish a true vertical with level and plumb line, set the gnomon at the center, and measure the noon shadow. A short gnomon made it hard to read fine gradations of fen and miao below the inch mark. A longer gnomon eased reading the scale, but the shadow grew faint and diffuse, making a precise edge difficult to capture. To extract a true reading from a blurred shadow, earlier observers used sighting tubes, auxiliary gnomons, or wooden templates—all channeling sunlight from the gnomon tip to the measuring plane. The new gnomon is cast in bronze, thirty-six feet tall, with two dragons at the top holding a crossbeam; from crossbeam to measuring surface is forty feet—five times the classical eight-foot standard. The measuring table is marked in cun and chi; where the old scale used one division per inch, the new scale uses five, making fine readings of li and hao far easier. They also invented a shadow template to capture a sharp, true shadow image. It was a copper plate two inches wide and four inches long, pierced with a pinhole at the center, mounted on a square base with a hinged pivot so one end could be tilted north-high, south-low; slid within the blurred shadow until a pinhole beam the size of a rice grain fell through and the crossbeam appeared faintly inside. The old method measured from the gnomon tip and captured the upper limb of the sun; the crossbeam method yields the sun's center, eliminating even the slightest error.
7
地中八尺表景，冬至長一丈三尺有奇，夏至尺有五寸。 今京師長表，冬至之景七丈九尺八寸有奇，在八尺表則一丈五尺九寸六分； 夏至之景一丈一尺七寸有奇，在八尺表則二尺三寸四分。 雖晷景長短所在不同，而其景長為冬至，景短為夏至，則一也。 惟是氣至時刻考求不易，蓋至日氣正，則一歲氣節從而正矣。 劉宋祖沖之嘗取至前後二十三四日間晷景，折取其中，定為冬至，且以日差比課，推定時刻。 宋皇祐間，周琮則取立冬、立春二日之景，以為去至既遠，日差頗多，易為推考。 《紀元》以後諸曆，為法加詳，大抵不出沖之之法。 新曆積日累月，實測中晷，自遠日以及近日，取前後日率相埒者，參考同異，初非偏取一二日之景，以取數多者為定，實減《大明曆》一十九刻二十分。 仍以累歲實測中晷日差分寸，定擬二至時刻於後。
At the geographical center, an eight-foot gnomon casts a shadow of somewhat more than thirteen feet at winter solstice and one foot five inches at summer solstice. At the capital the long gnomon gives a winter-solstice shadow of nearly seventy-nine chi eight cun; scaled to an eight-foot gnomon that is fifteen chi nine cun six fen; the summer-solstice shadow is a little over eleven chi seven cun; on an eight-foot gnomon, two chi three cun four fen. Shadow lengths differ by latitude, but everywhere the longest shadow marks winter solstice and the shortest marks summer solstice. Pinpointing the exact moment of solstice is difficult; yet if the solstice is fixed correctly, the entire year's solar terms fall into place. Zu Chongzhi of Liu Song measured noon shadows for roughly twenty-three or twenty-four days on either side of solstice, took the midpoint as winter solstice, and used daily shadow differences to compute the exact moment. Under Song Emperor Renzong, Zhou Cong used shadows on Beginning of Winter and Beginning of Spring, arguing that being far from solstice the daily change was larger and easier to calculate with. Post-Jiyuan calendars refined the procedures but largely followed Zu Chongzhi's approach. The new calendar relied on months of actual noon-shadow measurements, comparing days with similar daily rates before and after solstice rather than cherry-picking one or two readings; it shortened the Great Enlightenment Calendar's solstice by nineteen ke twenty fen. Using years of measured daily shadow differences, the solstice moments are determined as follows.
8
推至元十四年丁丑歲冬至
Calculating the winter solstice of Zhiyuan 14 (dingchou year)
9
其年十一月十四日己亥，景長七丈九尺四寸八分五厘五毫； 至二十一日丙午，景長七丈九尺五寸四分一厘； 二十二日丁未，景長七丈九尺四寸五分五厘。 以己亥、丁未二日之景相校，余三分五毫為晷差，進二位； 以丙午、丁未二日之景相校，余八分六厘為法； 除之，得三十五刻； 用減相距日八百刻，余七百六十五刻； 折取其中，加半日刻，共為四百三十二刻半； 百約為日，得四日； 余以十二乘之，百約為時，得三時，滿五十又作一時，共得四時； 余以十二收之，得三刻； 命初起距日己亥算外，得癸卯日辰初三刻為丁丑歲冬至。 此取至前後四日景。
On the fourteenth of the eleventh month (jihai day) that year, the shadow measured seven zhang nine chi four cun eight fen five li five hao; by the twenty-first (bingwu day) it was seven zhang nine chi five cun four fen one li; On the twenty-second (dingwei day) it was seven zhang nine chi four cun five fen five li. Comparing the jihai and dingwei shadows leaves a gnomon difference of three fen five hao, carried forward two decimal places; comparing bingwu and dingwei shadows gives a divisor of eight fen six li; dividing yields thirty-five ke; subtracting from eight hundred ke between the days leaves seven hundred sixty-five ke; halving and adding half a day gives four hundred thirty-two and a half ke; converting by hundreds gives four days; multiplying the remainder by twelve and converting gives three double-hours; rounding up at fifty adds another, for four double-hours total; the remainder divided by twelve gives three ke; counting forward from jihai day yields guimao day at the third ke of chen hour as the dingchou winter solstice. This calculation uses shadows from four days on either side of solstice.
10
十一月初九日甲午，景七丈八尺六寸三分五厘五毫； 至二十六日辛亥，景七丈八尺七寸九分三厘五毫； 二十七日壬子，景七丈八尺五寸五分。 以甲午、壬子景相減，復以辛亥、壬子景相減，準前法求之，亦得癸卯日辰初三刻。 至二十八日癸丑，景七丈八尺三寸四厘五毫，用壬子、癸丑二日之景與甲午景，準前法求之，亦合。 此取至前後八九日景。
On the ninth of the eleventh month (jiawu day), shadow seven zhang eight chi six cun three fen five li five hao; by the twenty-sixth (xinhai day), shadow seven zhang eight chi seven cun nine fen three li five hao; On the twenty-seventh (renzi day), shadow seven zhang eight chi five cun five fen. Subtracting jiawu from renzi and xinhai from renzi shadows and applying the same method again yields guimao day, third ke of chen hour. On the twenty-eighth (guichou day) the shadow was seven zhang eight chi three cun four li five hao; using renzi, guichou, and jiawu shadows with the same method also agrees. This uses shadows from eight or nine days on either side of solstice.
11
十一月丙戌朔，景七丈五尺九寸八分六厘五毫； 二日丁亥，景七丈六尺三寸七分七厘； 至十二月初六日庚申，景七丈五尺八寸五分一厘。 準前法求之，亦在辰初三刻。 此取至前後一十七日景。
On new moon of the eleventh month (bingxu day), shadow seven zhang five chi nine cun eight fen six li five hao; on the second (dinghai day), shadow seven zhang six chi three cun seven fen seven li; by the sixth of the twelfth month (gengshen day), shadow seven zhang five chi eight cun five fen one li. Applying the same method again gives the third ke of chen hour. This uses shadows from seventeen days on either side of solstice.
12
十月二十一日丙子，景七丈九寸七分一厘； 至十二月十六日庚午，景七丈七寸六分； 十七日辛未，景七丈一寸五分六厘五毫。 準前法求之，亦得辰初三刻。 此取至前後二十七日景。
On the twenty-first of the tenth month (bingzi day), shadow seven zhang nine cun seven fen one li; by the sixteenth of the twelfth month (gengwu day), shadow seven zhang seven cun six fen; On the seventeenth (xinwei day), shadow seven zhang one chi five fen six li five hao. The same method again yields the third ke of chen hour. This uses shadows from twenty-seven days on either side of solstice.
13
六月初五日癸亥，景一丈三尺八分； 距十五年五月癸未朔，景一丈三尺三分八厘五毫； 初二日甲申，景一丈二尺九寸二分五毫。 準前法求之，亦合。 此取至前後一百六十日景。
On the fifth of the sixth month (guihai day), shadow one zhang three chi eight fen; at the fifteenth year's fifth-month new moon (guimao day), shadow one zhang three chi three fen eight li five hao; On the second (jiashen day), shadow one zhang two chi nine cun two fen five hao. The same method again agrees. This uses shadows from one hundred sixty days on either side of solstice.
14
推十五年戊寅歲夏至
Calculating the summer solstice of year 15 (wuyin year)
15
五月十九日辛丑，景一丈一尺七寸七分七厘五毫； 距二十八日庚戌，景一丈一尺七寸八分； 二十九日辛亥，景一丈一尺八寸五厘五毫。 用辛丑、庚戌二日之景相減，余二厘五毫，進二位為實； 復用庚戌、辛亥景相減，余二分五厘五毫為法； 除之，得九刻，用減相距日九百刻，余八百九十一刻； 半之，加半日刻，百約，得四日； 余以十二乘之，百約，得十一時； 余以十二收為刻，得三刻； 命初起距日辛丑算外，得乙巳日亥正三刻夏至。 此取至前後四日景。
On the nineteenth of the fifth month (xinchou day), shadow one zhang one chi seven cun seven fen seven li five hao; by the twenty-eighth (gengxu day), shadow one zhang one chi seven cun eight fen; On the twenty-ninth (xinhai day), shadow one zhang one chi eight cun five li five hao. Subtracting xinchou and gengxu shadows leaves two li five hao as dividend, carried two places; subtracting gengxu and xinhai shadows gives a divisor of two fen five li five hao; division yields nine ke; subtracting from nine hundred ke between days leaves eight hundred ninety-one ke; halving, adding half a day, and converting by hundreds gives four days; multiplying the remainder by twelve and converting gives eleven double-hours; the remainder divided by twelve gives three ke; counting forward from xinchou day yields yisi day at the third ke of midnight (hai hour) as summer solstice. This calculation uses shadows from four days on either side of solstice.
16
十四年十二月十五日己巳，景七丈一尺三寸四分三厘； 距十五年十一月初二日辛巳，景七丈七寸五分九厘五毫； 初三日壬午，景七丈一尺四寸六厘。 用己巳、壬午景相減，以辛巳、壬午景相減除之，亦合。 此用至前後一百五十六日景。
On the fifteenth of the twelfth month of year 14 (jisi day), shadow seven zhang one chi three cun four fen three li; at the second of the eleventh month of year 15 (xinsi day), shadow seven zhang seven cun five fen nine li five hao; On the third (renwu day), shadow seven zhang one chi four cun six li. Subtracting jisi and renwu shadows and dividing by xinsi-minus-renwu also agrees. This uses shadows from one hundred fifty-six days on either side of solstice.
17
十四年十二月十二日丙寅，景七丈二尺九寸七分二厘五毫； 十三日丁卯，景七丈二尺四寸五分四厘五毫； 十四日戊辰，景七丈一尺九寸九厘； 距十五年十一月初四日癸未，景七丈一尺九寸五分七厘五毫； 初五日甲申，景七丈二尺五寸五厘； 初六日乙酉，景七丈三尺三分三厘五毫。 前後互取，所得時刻皆合。 此取至前後一百五十八九日景。
On the twelfth of the twelfth month of year 14 (bingyin day), shadow seven zhang two chi nine cun seven fen two li five hao; on the thirteenth (dingmao day), shadow seven zhang two chi four cun five fen four li five hao; on the fourteenth (wuchen day), shadow seven zhang one chi nine cun nine li; at the fourth of the eleventh month of year 15 (guimao day), shadow seven zhang one chi nine cun five fen seven li five hao; on the fifth (jiashen day), shadow seven zhang two chi five cun five li; on the sixth (yiyou day), shadow seven zhang three chi three fen three li five hao. Using shadows from either side reciprocally, all resulting times agree. This uses shadows from one hundred fifty-eight or nine days on either side of solstice.
18
十四年十二月初七日辛酉，景七丈五尺四寸一分七厘； 初八日壬戌，景七丈四尺九寸五分九厘五毫； 初九日癸亥，景七丈四尺四寸八分六厘； 距十五年十一月初九日戊子，景七丈四尺五寸二分五毫； 初十日己丑，景七丈五尺三厘五毫； 十一日庚寅，景七丈五尺四寸四分九厘五毫。 以壬戌、己丑景相減為實，以辛酉、壬戌景相減為法，除之； 或以壬戌、癸亥景相減，或以戊子、己丑景相減，若己丑、庚寅景相減，推前法求之，皆合。 此取至前後一百六十三四日景。
On the seventh of the twelfth month of year 14 (xinyou day), shadow seven zhang five chi four cun one fen seven li; on the eighth (renxu day), shadow seven zhang four chi nine cun five fen nine li five hao; on the ninth (guihai day), shadow seven zhang four chi four cun eight fen six li; at the ninth of the eleventh month of year 15 (wuzi day), shadow seven zhang four chi five cun two fen five hao; on the tenth (jichou day), shadow seven zhang five chi three li five hao; on the eleventh (gengyin day), shadow seven zhang five chi four cun four fen nine li five hao. Subtracting renxu from jichou shadows gives the dividend, xinyou-minus-renxu the divisor; dividing; Whether subtracting renxu and guihai, wuzi and jichou, or jichou and gengyin shadows and applying the previous method, all agree. This uses shadows from one hundred sixty-three or four days on either side of solstice.
19
推十五年戊寅歲冬至
Calculating the winter solstice of year 15 (wuyin year)
20
其年十一月十九日戊戌，景七丈八尺三寸一分八厘五毫； 距閏十一月初九日戊午，景七丈八尺三寸六分三厘五毫； 初十日己未，景七丈八尺八分二厘五毫。 用戊戌、戊午二日景相減，余四分五厘為晷差，進二位，以戊午、己未景相減，余二寸八分一厘為法，除之，得一十六刻，加相距日二千刻，半之，加半日刻，百約，得十日； 余以十二乘之，百約為時，滿五十又進一時，共得七時； 余以十二收為刻； 命初起距日己亥算外，得戊申日未初三刻為戊寅歲冬至。 此取至前後十日景。
On the nineteenth of the eleventh month that year (wuxu day), shadow seven zhang eight chi three cun one fen eight li five hao; at the ninth of the intercalary eleventh month (wuwu day), shadow seven zhang eight chi three cun six fen three li five hao; on the tenth (jiwei day), shadow seven zhang eight chi eight fen two li five hao. Subtracting wuxu and wuwu shadows leaves a gnomon difference of four fen five li, carried two places; subtracting wuwu and jiwei shadows gives a divisor of two cun eight fen one li; dividing yields sixteen ke; adding two thousand ke between days, halving, adding half a day, and converting by hundreds gives ten days; multiplying the remainder by twelve and converting gives seven double-hours, rounding up at fifty; the remainder divided by twelve gives ke; counting forward from jihai day yields wushen day at the third ke of wei hour as the wuyin winter solstice. This calculation uses shadows from ten days on either side of solstice.
21
十一月十二日辛卯，景七丈五尺八寸八分一厘五毫； 十三日壬辰，景七丈六尺三寸一厘五毫； 閏十一月十五日甲子，景七丈六尺三寸六分六厘五毫； 十六日乙丑，景七丈五尺九寸五分三厘； 十七日丙寅，景七丈五尺五寸四厘五毫。 用壬辰、甲子景相減為實，以辛卯、壬辰景相減為法，除之，亦得戊申日未初三刻。 或用甲子、乙丑景相減，推之，亦合。 若用辛卯、乙丑景相減為實，用乙丑、丙寅景相減，除之，並同。 此取至前後十六七日景。
On the twelfth of the eleventh month (xinmao day), shadow seven zhang five chi eight cun eight fen one li five hao; on the thirteenth (renchen day), shadow seven zhang six chi three cun one li five hao; on the fifteenth of the intercalary eleventh month (jiazi day), shadow seven zhang six chi three cun six fen six li five hao; on the sixteenth (yichou day), shadow seven zhang five chi nine cun five fen three li; on the seventeenth (bingyin day), shadow seven zhang five chi five cun four li five hao. Subtracting renchen and jiazi shadows as dividend and xinmao-minus-renchen as divisor yields wushen day at the third ke of wei hour. Using jiazi-minus-yichou shadows and applying the method also agrees. Using xinmao-minus-yichou as dividend and yichou-minus-bingyin as divisor gives the same result. This uses shadows from sixteen or seventeen days on either side of solstice.
22
十一月初八日丁亥，景七丈四尺三分七厘五毫； 閏十一月二十日己巳，景七丈四尺一寸二分； 二十一日庚午，景七丈三尺六寸一分四厘五毫。 用丁亥、己巳景相減為實，以己巳、庚午景相減，除之，亦同。 此取至前後二十一日景。
On the eighth of the eleventh month (dinghai day), shadow seven zhang four chi three fen seven li five hao; on the twentieth of the intercalary eleventh month (jisi day), shadow seven zhang four chi one cun two fen; on the twenty-first (gengwu day), shadow seven zhang three chi six cun one fen four li five hao. Subtracting dinghai and jisi shadows as dividend and jisi-minus-gengwu as divisor gives the same result. This uses shadows from twenty-one days on either side of solstice.
23
六月二十六日戊寅，景一丈四尺四寸五分二厘五毫； 二十七日己卯，景一丈四尺六寸三分八厘； 至十六年四月二日戊寅，景一丈四尺四寸八分一厘。 以二戊寅景相減，用後戊寅、己卯景相減，推之，亦同。 此取至前後一百五十日景。
On the twenty-sixth of the sixth month (wuyin day), shadow one zhang four chi four cun five fen two li five hao; on the twenty-seventh (jimao day), shadow one zhang four chi six cun three fen eight li; by the second of the fourth month of year 16 (wuyin day), shadow one zhang four chi four cun eight fen one li. Subtracting the two wuyin shadows and using the later wuyin-minus-jimao and applying the method gives the same result. This uses shadows from one hundred fifty days on either side of solstice.
24
五月二十八日庚戌，景一丈一尺七寸八分； 至十六年四月二十九日乙巳，景一丈一尺八寸六分三厘； 三十日丙午，景一丈一尺七寸八分三厘。 用庚戌、丙午景相減，以乙巳、丙午景相減，推之，亦同。 此取至前後百七十八日景。
On the twenty-eighth of the fifth month (gengxu day), shadow one zhang one chi seven cun eight fen; by the twenty-ninth of the fourth month of year 16 (yisi day), shadow one zhang one chi eight cun six fen three li; on the thirtieth (bingwu day), shadow one zhang one chi seven cun eight fen three li. Subtracting gengxu and bingwu shadows and yisi-minus-bingwu and applying the method gives the same result. This uses shadows from one hundred seventy-eight days on either side of solstice.
25
推十六年己卯歲夏至
Calculating the summer solstice of year 16 (jimao year)
26
四月十九日乙未，景一丈二尺三寸六分九厘五毫； 二十日丙申，景一丈二尺二寸九分三厘五毫； 至五月十九日乙丑，景一丈二尺二寸六分四厘。 以丙申、乙丑景相減，余二分九厘五毫為晷差，進二位； 以乙未、丙申景相減，得七分六厘為法； 除之，得三十八刻； 加相距日二千九百刻，半之，加半日刻，百約，得十五日； 余以十二乘之，百約，得二時； 余以十二收之，得二刻； 命初起距日丙申算外，得辛亥日寅正二刻為夏至。 此取至前後十五日景。
On the nineteenth of the fourth month (yiwei day), shadow one zhang two chi three cun six fen nine li five hao; on the twentieth (bingshen day), shadow one zhang two chi two cun nine fen three li five hao; by the nineteenth of the fifth month (yichou day), shadow one zhang two chi two cun six fen four li. Subtracting bingshen and yichou shadows leaves a gnomon difference of two fen nine li five hao, carried two places; subtracting yiwei and bingshen shadows gives a divisor of seven fen six li; dividing yields thirty-eight ke; adding two thousand nine hundred ke between days, halving, adding half a day, and converting by hundreds gives fifteen days; multiplying the remainder by twelve and converting gives two double-hours; the remainder divided by twelve gives two ke; counting forward from bingshen day yields xinhai day at the second ke of midnight (yin hour) as summer solstice. This calculation uses shadows from fifteen days on either side of solstice.
27
三月二十一日戊辰，景一丈六尺三寸九分五毫； 六月十六日壬辰，景一丈六尺九分九厘五毫； 十七日癸巳，景一丈六尺三寸一分一厘。 用戊辰、癸巳景相減，以壬辰、癸巳景相減，準前法推之，亦合。 此取至前後四十二日景。
On the twenty-first of the third month (wuchen day), shadow one zhang six chi three cun nine fen five hao; on the sixteenth of the sixth month (renchen day), shadow one zhang six chi nine fen nine li five hao; on the seventeenth (guisi day), shadow one zhang six chi three cun one fen one li. Subtracting wuchen and guisi shadows and renchen-minus-guisi and applying the same method also agrees. This uses shadows from forty-two days on either side of solstice.
28
三月初二日己酉，景二丈一尺三寸五厘； 至七月初七日壬子，景二丈一尺一寸九分五厘五毫； 初八日癸丑，景二丈一尺四寸八分六厘五毫。 用己酉、壬子景相減，以壬子、癸丑景相減，如前法推之，亦合。 此取至前後六十一二日景。
On the second of the third month (jiyou day), shadow two zhang one chi three cun five li; by the seventh of the seventh month (renzi day), shadow two zhang one chi one cun nine fen five li five hao; on the eighth (guichou day), shadow two zhang one chi four cun eight fen six li five hao. Subtracting jiyou and renzi shadows and renzi-minus-guichou and applying the same method also agrees. This uses shadows from sixty-one or two days on either side of solstice.
29
三月戊申朔，景二丈一尺六寸一分一厘； 至七月初八日癸丑，景二丈一尺四寸八分六厘五毫； 初九日甲寅，景二丈一尺九寸一分五厘五毫。 用戊申、癸丑景相減，以癸丑、甲寅景相減，準前法推之，亦同。 此取至前後六十二三日景。
On new moon of the third month (wushen day), shadow two zhang one chi six cun one fen one li; by the eighth of the seventh month (guichou day), shadow two zhang one chi four cun eight fen six li five hao; on the ninth (jiayin day), shadow two zhang one chi nine cun one fen five li five hao. Subtracting wushen and guichou shadows and guichou-minus-jiayin and applying the same method gives the same result. This uses shadows from sixty-two or three days on either side of solstice.
30
二月十八日乙未，景二丈六尺三分四厘五毫； 至七月二十一日丙寅，景二丈五尺八寸九分九厘； 二十二日丁卯，景二丈六尺二寸五分九厘。 用乙未、丙寅景相減，以丙寅、丁卯景相減，如前法推之，亦同。 此取至前後七十五六日景。
On the eighteenth of the second month (yiwei day), shadow two zhang six chi three fen four li five hao; by the twenty-first of the seventh month (bingyin day), shadow two zhang five chi eight cun nine fen nine li; on the twenty-second (dingmao day), shadow two zhang six chi two cun five fen nine li. Subtracting yiwei and bingyin shadows and bingyin-minus-dingmao and applying the same method gives the same result. This uses shadows from seventy-five or six days on either side of solstice.
31
二月三日庚辰，景三丈二尺一寸九分五厘五毫； 至八月初五日庚辰，景三丈一尺五寸九分六厘五毫； 初六日辛巳，景三丈二尺二分六厘五毫。 用前庚辰與辛巳景相減，以後庚辰、辛巳景相減，如前推之，亦同。 此取至前後九十日景。
On the third of the second month (gengchen day), shadow three zhang two chi one cun nine fen five li five hao; by the fifth of the eighth month (gengchen day), shadow three zhang one chi five cun nine fen six li five hao; on the sixth (xinsi day), shadow three zhang two chi two fen six li five hao. Subtracting the earlier gengchen-minus-xinsi and later gengchen-minus-xinsi shadows and applying the same method gives the same result. This uses shadows from ninety days on either side of solstice.
32
正月十九日丁卯，景三丈八尺五寸一厘五毫； 至八月十八日癸巳，景三丈七尺八寸二分三厘； 十九日甲午，景三丈八尺三寸一分五毫。 用丁卯、甲午景相減，以癸巳、甲午景相校，如前推之，亦同。 此取至前後一百三四日景。
On the nineteenth of the first month (dingmao day), shadow three zhang eight chi five cun one li five hao; by the eighteenth of the eighth month (guisi day), shadow three zhang seven chi eight cun two fen three li; on the nineteenth (jiawu day), shadow three zhang eight chi three cun one fen five hao. Subtracting dingmao and jiawu shadows and comparing guisi and jiawu shadows and applying the same method gives the same result. This uses shadows from one hundred thirty or forty days on either side of solstice.
33
推十六年己卯歲冬至
Calculating the winter solstice of year 16 (jimao year)
34
十月二十四日戊戌，景七丈六尺七寸四分； 至十一月二十五日己巳，景七丈六尺五寸八分； 二十六日庚午，景七丈六尺一寸四分二厘五毫。 用戊戌、己巳景相減，余一寸六分為晷差，進二位； 以己巳、庚午景相減，余四寸三分七厘五毫為法； 除之，得三十六刻； 以相減距日三千一百刻，余三千六十四刻； 半之，加五十刻，百約，得一十五日； 余以十二乘之，百約為時，滿五十，又進一時，共得十時； 余以十二收之為刻，得二刻； 命初起距日戊戌算外，得癸丑日戌初二刻冬至。 此取至前後十五六日景。
On the twenty-fourth of the tenth month (wuxu day), shadow seven zhang six chi seven cun four fen; by the twenty-fifth of the eleventh month (jisi day), shadow seven zhang six chi five cun eight fen; on the twenty-sixth (gengwu day), shadow seven zhang six chi one cun four fen two li five hao. Subtracting wuxu and jisi shadows leaves a gnomon difference of one cun six fen, carried two places; subtracting jisi and gengwu shadows gives a divisor of four cun three fen seven li five hao; dividing yields thirty-six ke; subtracting from three thousand one hundred ke between the measured days leaves three thousand sixty-four ke; halving, adding fifty ke, and converting by hundreds gives fifteen days; multiplying the remainder by twelve and converting gives ten double-hours, rounding up at fifty; the remainder divided by twelve gives two ke; counting forward from wuxu day yields guichou day at the second ke of xu hour as winter solstice. This calculation uses shadows from fifteen or sixteen days on either side of solstice.
35
十月十八日壬辰，景七丈四尺五分二厘五毫； 十九日癸巳，景七丈四尺五寸四分五厘； 二十日甲午，景七丈五尺二分五厘； 至十一月二十八日壬申，景七丈五尺三寸二分； 二十九日癸酉，景七丈四尺八寸五分二厘五毫； 十二月甲戌朔，景七丈四尺三寸六分五厘； 初二日乙亥，景七丈三尺八寸七分一厘五毫。 用甲午、癸酉景相減，癸巳、甲午景相減，如前推之，亦同。 若以壬申、癸酉景相減為法，推之亦同。 此取至前後十八九日景。
On the eighteenth of the tenth month (renchen day), shadow seven zhang four chi five fen two li five hao; on the nineteenth (guisi day), shadow seven zhang four chi five cun four fen five li; on the twentieth (jiawu day), shadow seven zhang five chi two fen five li; by the twenty-eighth of the eleventh month (renshen day), shadow seven zhang five chi three cun two fen; on the twenty-ninth (guiyou day), shadow seven zhang four chi eight cun five fen two li five hao; On new moon of the twelfth month (jiaxu day), shadow seven zhang four chi three cun six fen five li; on the second (yihai day), shadow seven zhang three chi eight cun seven fen one li five hao. Subtracting jiawu and guiyou shadows and guisi-minus-jiawu and applying the same method gives the same result. Using renshen-minus-guiyou shadows as the divisor and applying the same method also agrees. This uses shadows from eighteen or nineteen days on either side of solstice.
36
若用癸巳與甲戌景相減，以壬辰、癸巳景相減，推之，或癸巳、甲午景相減，推之，或用甲戌、癸酉景相減，推之，或甲戌、乙亥景相減，推之，或以壬辰、乙亥景相減，用壬辰、癸巳景相減，推之並同。 此取至前後二十日景。
Whether one subtracts guisi-minus-jiaxu, renchen-minus-guisi, guisi-minus-jiawu, jiaxu-minus-guiyou, jiaxu-minus-yihai, or renchen-minus-yihai paired with renchen-minus-guisi and applies the same method, all agree. This uses shadows from twenty days on either side of solstice.
37
十月十六日庚寅，景七丈三尺一分五厘； 十二月初三日丙子，景七丈三尺三寸二分； 初四日丁丑，景七丈二尺八寸四分二厘五毫。 用庚寅、丁丑景相減，以丙子、丁丑景相減，推之亦同。 此取至前後二十三日景。
On the sixteenth of the tenth month (gengyin day), shadow seven zhang three chi one fen five li; on the third of the twelfth month (bingzi day), shadow seven zhang three chi three cun two fen; on the fourth (dingchou day), shadow seven zhang two chi eight cun four fen two li five hao. Subtracting gengyin and dingchou shadows and bingzi-minus-dingchou and applying the same method gives the same result. This uses shadows from twenty-three days on either side of solstice.
38
十月十四日戊子，景七丈一尺九寸二分二厘五毫； 十五日己丑，景七丈二尺四寸六分九厘； 十二月初五日戊寅，景七丈二尺二寸七分二厘五毫。 用己丑、戊寅景相減，以戊子、己丑景相減，推之，或用己丑、庚寅相減，推之亦同。 此取至前後二十四日景。
On the fourteenth of the tenth month (wuzi day), shadow seven zhang one chi nine cun two fen two li five hao; on the fifteenth (jichou day), shadow seven zhang two chi four cun six fen nine li; on the fifth of the twelfth month (wuyin day), shadow seven zhang two chi two cun seven fen two li five hao. Subtracting jichou and wuyin shadows and wuzi-minus-jichou, or jichou-minus-gengyin, and applying the same method gives the same result. This uses shadows from twenty-four days on either side of solstice.
39
十月初七日辛巳，景六丈七尺七寸四分五厘； 初八日壬午，景六丈八尺三寸七分二厘五毫； 初九日癸未，景六丈八尺九寸七分七厘五毫； 十二月十二日乙酉，景六丈八尺一寸四分五厘。 用壬午、乙酉景相減，以辛巳、壬午相減，推之，壬午、癸未景相減，推之亦同。 此取至前後三十一二日景。
On the seventh of the tenth month (xinsi day), shadow six zhang seven chi seven cun four fen five li; on the eighth (renwu day), shadow six zhang eight chi three cun seven fen two li five hao; on the ninth (guiwei day), shadow six zhang eight chi nine cun seven fen seven li five hao; on the twelfth of the twelfth month (yiyou day), shadow six zhang eight chi one cun four fen five li. Subtracting renwu and yiyou shadows and xinsi-minus-renwu, or renwu-minus-guiwei, and applying the same method gives the same result. This uses shadows from thirty-one or two days on either side of solstice.
40
十月乙亥朔，景六丈三尺八寸七分； 十二月十八日辛卯，景六丈四尺二寸九分七厘五毫； 十九日壬辰，景六丈三尺六寸二分五厘。 用乙亥、壬辰景相減，以辛卯、壬辰景相減，推之亦同。 此取至前後三十八日景。
On new moon of the tenth month (yihai day), shadow six zhang three chi eight cun seven fen; on the eighteenth of the twelfth month (xinmao day), shadow six zhang four chi two cun nine fen seven li five hao; on the nineteenth (renchen day), shadow six zhang three chi six cun two fen five li. Subtracting yihai and renchen shadows and xinmao-minus-renchen and applying the same method gives the same result. This uses shadows from thirty-eight days on either side of solstice.
41
九月二十二日丙寅，景五丈七尺八寸二分五厘； 十二月二十八日辛丑，景五丈七尺五寸八分； 二十九日壬寅，景五丈六尺九寸一分五厘。 用丙寅、辛丑景相減，以辛丑、壬寅景相減，推之亦同。 此取至前後四十七八日景。
On the twenty-second of the ninth month (bingyin day), shadow five zhang seven chi eight cun two fen five li; on the twenty-eighth of the twelfth month (xinchou day), shadow five zhang seven chi five cun eight fen; on the twenty-ninth (renyin day), shadow five zhang six chi nine cun one fen five li. Subtracting bingyin and xinchou shadows and xinchou-minus-renyin and applying the same method gives the same result. This uses shadows from forty-seven or eight days on either side of solstice.
42
九月二十日甲子，景五丈六尺四寸九分二厘五毫； 至十二月二十九日壬寅，景五丈六尺九寸一分五厘； 至十七年正月癸卯朔，景五丈六尺二寸五分。 用甲子、癸卯相減，壬寅、癸卯景相減，推之亦同。 此取至前後五十日景。
On the twentieth of the ninth month (jiazi day), shadow five zhang six chi four cun nine fen two li five hao; by the twenty-ninth of the twelfth month (renyin day), shadow five zhang six chi nine cun one fen five li; by new moon of the first month of year 17 (guimao day), shadow five zhang six chi two cun five fen. Subtracting jiazi-minus-guimao and renyin-minus-guimao shadows and applying the same method gives the same result. This uses shadows from fifty days on either side of solstice.
43
右以累年推測到冬夏二至時刻為準，定擬至元十八年辛巳歲前冬至，當在己未日夜半後六刻，即丑初一刻。 歲餘歲差
On the basis of solstice moments computed from years of observation, the winter solstice before Zhiyuan 18 (xinsi year) is fixed at jiwei day, six ke after midnight—the first ke of chou hour. Year Remainder and Precession
44
周天之度，周歲之日，皆三百六十有五。 全策之外，又有奇分，大率皆四分之一。 自今歲冬至距來歲冬至，曆三百六十五日，而日行一周，凡四周，曆千四百六十，則餘一日，析而四之，則四分之一也。 然天之分常有餘，歲之分常不足，其數有不能齊者，惟其所差至微，前人初未覺知。 迨漢末劉洪，始覺冬至後天，謂歲周余分太強，乃作《乾象曆》，減歲餘分二千五百為二千四百六十二。 至晉虞喜，宋何承天、祖沖之，謂歲當有差，因立歲差之法。 其法損歲餘，益天周，使歲餘浸弱，天周浸強，強弱相減，因得日躔歲退之差。 歲餘、天周，二者實相為用，歲差由斯而立，日躔由斯而得，一或損益失當，詎能與天葉哉？
A full circuit of the heavens and a full tropical year both number three hundred sixty-five. Beyond the whole number there is always a fractional remainder, generally one quarter. From one winter solstice to the next is three hundred sixty-five days; the sun completes one circuit per year, and in four years—one thousand four hundred sixty days—one day remains; divided by four, that is one quarter. Heaven's fraction tends to run slightly high and the year's fraction slightly low; the two cannot always be made to match—but the difference is minuscule, and earlier astronomers did not notice it at first. Not until Liu Hong in late Han did anyone notice that winter solstice ran behind the heavens; judging the year's surplus fraction too large, he wrote the Supernal Image Calendar and cut the year-remainder fraction from 2500 to 2462. By Jin's Yu Xi and Song's He Chengtian and Zu Chongzhi, who argued that the year must drift against the stars, the method of precession was established. The method trimmed the year remainder and enlarged the celestial circuit, gradually weakening the former and strengthening the latter; subtracting the two yields the rate at which the sun's position retreats year by year. Year remainder and celestial circuit work in tandem: precession and solar lodge position both depend on them. If either is adjusted wrongly, how can the calendar match the heavens?
45
今自劉宋大明壬寅以來，凡測景驗氣得冬至時刻真數者有六，取相距積日時刻，以相距之年除之，各得其時所用歲餘。 復自大明壬寅距至元戊寅積日時刻，以相距之年除之，得每歲三百六十五日二十四分二十五秒，比《大明曆》減去一十一秒，定為方今所用歲餘。 余七十五秒，用益所謂四分之一，共為三百六十五度二十五分七十五秒，定為天周。 余分強弱相減，余一分五十秒，用除全度，得六十六年有奇，日卻一度，以六十六年除全度，適得一分五十秒，定為歲差。
Since Liu Song's Great Enlightenment renyin year, six winter-solstice moments have been fixed by shadow measurement and qi testing; dividing the elapsed days and ke between each pair by the intervening years yields the year remainder used in each era. From Great Enlightenment renyin to Zhiyuan wuyin, dividing total elapsed days and ke by the years between gives 365 days 24 fen 25 miao per year—11 miao less than the Great Enlightenment Calendar—fixed as the present year remainder. Adding 75 miao to the nominal quarter-day gives 365 degrees 25 fen 75 miao, fixed as the celestial circuit. Subtracting the surplus and deficit fractions leaves 1 fen 50 miao; dividing the full circle by this gives just over sixty-six years per degree of solar retreat; dividing the full circle by sixty-six years yields exactly 1 fen 50 miao, fixed as the precession rate.
46
復以《堯典》中星考之，其時冬至日在女、虛之交。 及考之前史，漢元和二年，冬至日在鬥二十一度； 晉太元九年，退在鬥十七度； 宋元嘉十年，在鬥十四度末； 梁大同十年，在鬥十二度； 隋開皇十八年，猶在鬥十二度； 唐開元十二年，在鬥九度半； 今退在箕十度。 取其距今之年、距今之度較之，多者七十餘年，少者不下五十年，輒差一度。 宋慶元間，改《統天曆》，取大衍歲差率八十二年及開元所距之差五十五年，折取其中，得六十七年，為日卻行一度之差。 施之今日，質諸天道，實為密近。
Checking against the culminating stars recorded in the Canon of Yao, the winter-solstice sun then stood at the border of the Nu and Xu lodges. Earlier histories record that in Han Yuanhe 2 the winter-solstice sun stood at 21 degrees of the Dipper lodge; by Jin Taiyuan 9 it had retreated to 17 degrees of Dipper; by Song Yuanjia 10 it stood near the end of 14 degrees of Dipper; by Liang Datong 10 it stood at 12 degrees of Dipper; by Sui Kaihuang 18 it still stood at 12 degrees of Dipper; by Tang Kaiyuan 12 it stood at 9½ degrees of Dipper; Today it has retreated to 10 degrees of the Winnowing Basket lodge. Comparing elapsed years with degrees of retreat, intervals of fifty to seventy-plus years consistently show one degree of difference. When Song revised the Total Heaven Calendar in the Qingyuan era, it split the difference between the Dayan rate of 82 years and the 55-year interval since Kaiyuan, yielding 67 years per degree of solar retreat. Applied to the present and tested against observation, it proves remarkably close.
47
然古今曆法，合於今必不能通於古，密於古必不能驗於今。 今《授時曆》，以之考古，則增歲餘而損歲差； 以之推來，則增歲差而損歲餘； 上推春秋以來冬至，往往皆合； 下求方來，可以永久而無弊； 非止密於今日而已。 仍以《大衍》等六曆，考驗春秋以來冬至疏密，凡四十九事，具列如後。
Yet no calendar can fit both past and present: one accurate for today will fail antiquity, and one precise for antiquity will miss today. The Season-Granting Calendar, applied to antiquity, increases the year remainder and reduces the precession rate; projected forward, it increases precession and reduces the year remainder; projecting backward, winter solstices since the Spring and Autumn era mostly agree; projected forward, it can serve indefinitely without error; it is precise not only for the present but across time. Using the Dayan and five other calendars, forty-nine winter-solstice comparisons since the Spring and Autumn era are tabulated below.
48
○冬至刻
Winter Solstice Moments (ke)
49
《大衍》《宣明》《紀元》《統天》《大明》《授時》獻公十五年戊寅歲，正月甲寅朔旦冬至。
Dayan, Xuanming, Jiyuan, Tongtian, Daming, and Shoushi calendars — Duke Xian year 15 (wuyin): winter solstice at dawn on new moon, jiayin day, first month.
50
丙辰 〈（22）〉 乙卯 〈（88）〉 丁巳 〈（33）〉 乙卯 （2） 丁巳 〈（35）〉 甲寅 〈（99）〉 僖公五年丙寅歲，正月辛亥朔旦冬至。
The day bingchen. (22 ke) The day yimao. (88 ke) The day dingsi. (33 ke) The day yimao. (2 ke) The day dingsi. (35 ke) The day jiayin. (99 ke) Duke Xi year 5 (bingyin): winter solstice at dawn on new moon, xinhai day, first month.
51
辛亥 〈（94）〉 辛亥 〈（66）〉 壬子 〈（74）〉 辛亥 〈（27）〉 壬子 〈（89）〉 辛亥 〈（14）〉 昭公二十年己卯歲，正月己丑朔旦冬至。
The day xinhai. (94 ke) The day xinhai. (66 ke discrepancy) The day renzi. (74 ke discrepancy) The day xinhai. (27 ke discrepancy) The day renzi. (89 ke discrepancy) The day xinhai. (14 ke discrepancy) Duke Zhao year 20 (jimao): winter solstice at dawn on new moon, jichou day, first month.
52
己丑 〈（45）〉 己丑 〈（20）〉 庚寅 〈（25）〉 戊子 〈（92）〉 庚寅 〈（29）〉 戊子 〈（83）〉 宋元嘉十二年乙亥歲，十一月十五日戊辰景長。
The day jichou. (45 ke discrepancy) The day jichou. (20 ke discrepancy) The day gengyin. (25 ke discrepancy) The day wuzi. (92 ke discrepancy) The day gengyin. (29 ke discrepancy) The day wuzi. (83 ke discrepancy) Song Yuanjia 12 (yihai): on the fifteenth of the eleventh month (wuchen day), winter solstice by shadow length.
53
戊辰 〈（35）〉 戊辰 〈（32）〉 戊辰 〈（39）〉 戊辰 〈（51）〉 戊辰 〈（41）〉 戊辰 〈（47）〉 元嘉十三年丙子歲，十一月二十六日甲戌景長。
The day wuchen. (35 ke discrepancy) The day wuchen. (32 ke discrepancy) The day wuchen. (39 ke discrepancy) The day wuchen. (51 ke discrepancy) The day wuchen. (41 ke discrepancy) The day wuchen. (47 ke discrepancy) Yuanjia 13 (bingzi): on the twenty-sixth of the eleventh month (jiaxu day), winter solstice by shadow length.
54
癸酉 〈（59）〉 癸酉 〈（57）〉 癸酉 〈（63）〉 癸酉 〈（75）〉 癸酉 〈（65）〉 癸酉 〈（71）〉 元嘉十五年戊寅歲，十一月十八日甲申景長。
The day guiyou. (59 ke discrepancy) The day guiyou. (57 ke discrepancy) The day guiyou. (63 ke discrepancy) The day guiyou. (75 ke discrepancy) The day guiyou. (65 ke discrepancy) The day guiyou. (71 ke discrepancy) Yuanjia 15 (wuyin): on the eighteenth of the eleventh month (jiashen day), winter solstice by shadow length.
55
甲申 （8） 甲申 （6） 甲申 〈（12）〉 甲申 〈（24）〉 甲申 〈（14）〉 甲申 〈（19）〉 元嘉十六年己卯歲，十月二十九日己丑景長。
The day jiashen. (8 ke) The day jiashen. (6 ke) The day jiashen. (12 ke discrepancy) The day jiashen. (24 ke discrepancy) The day jiashen. (14 ke discrepancy) The day jiashen. (19 ke discrepancy) Yuanjia 16 (jimao): on the twenty-ninth of the tenth month (jichou day), winter solstice by shadow length.
56
己丑 〈（33）〉 己丑 （3） 己丑 〈（37）〉 己丑 〈（48）〉 己丑 〈（37）〉 己丑 〈（44）〉 元嘉十七年庚辰歲，十一月初十日甲午景長。
The day jichou. (33 ke discrepancy) The day jichou. (3 ke) The day jichou. (37 ke discrepancy) The day jichou. (48 ke discrepancy) The day jichou. (37 ke discrepancy) The day jichou. (44 ke discrepancy) Yuanjia 17 (gengchen): on the tenth of the eleventh month (jiawu day), winter solstice by shadow length.
57
甲午 〈（57）〉 甲午 〈（55）〉 甲午 〈（61）〉 甲午 〈（72）〉 甲午 〈（63）〉 甲午 〈（68）〉 元嘉十八年辛巳歲，十一月二十一日己亥景長。
The day jiawu. (57 ke discrepancy) The day jiawu. (55 ke discrepancy) The day jiawu. (61 ke discrepancy) The day jiawu. (72 ke discrepancy) The day jiawu. (63 ke discrepancy) The day jiawu. (68 ke discrepancy) Yuanjia 18 (xinsi): on the twenty-first of the eleventh month (jihai day), winter solstice by shadow length.
58
己亥 〈（82）〉 己亥 〈（79）〉 己亥 〈（85）〉 己亥 〈（97）〉 己亥 〈（87）〉 己亥 〈（93）〉 元嘉十九年壬午歲，十一月初三日乙巳景長。
The day jihai. (82 ke discrepancy) The day jihai. (79 ke discrepancy) The day jihai. (85 ke discrepancy) The day jihai. (97 ke discrepancy) The day jihai. (87 ke discrepancy) The day jihai. (93 ke discrepancy) Yuanjia 19 (renwu): on the third of the eleventh month (yisi day), winter solstice by shadow length.
59
乙巳 （6） 乙巳 （4） 乙巳 〈（10）〉 乙巳 〈（21）〉 乙巳 〈（11）〉 乙巳 〈（17）〉 大明五年辛丑歲，十一月乙酉冬至。
The day yisi. (6 ke) The day yisi. (4 ke) The day yisi. (10 ke discrepancy) The day yisi. (21 ke discrepancy) The day yisi. (11 ke discrepancy) The day yisi. (17 ke discrepancy) Daming 5 (xinchou): winter solstice on yiyou day, eleventh month.
60
甲申 〈（70）〉 甲申 〈（68）〉 甲申 〈（73）〉 甲申 〈（89）〉 甲申 〈（74）〉 甲申 〈（79）〉 陳天嘉六年乙酉歲，十一月庚寅景長。
The day jiashen. (70 ke discrepancy) The day jiashen. (68 ke discrepancy) The day jiashen. (73 ke discrepancy) The day jiashen. (89 ke discrepancy) The day jiashen. (74 ke discrepancy) The day jiashen. (79 ke discrepancy) Chen Tiancai 6 (yiyou): eleventh month (gengyin day), winter solstice by shadow length.
61
庚寅 〈（12）〉 庚寅 〈（13）〉 庚寅 （5） 庚寅 〈（24）〉 庚寅 〈（80）〉 庚寅 〈（17）〉 光大二年戊子歲，十一月乙巳景長。
The day gengyin. (12 ke discrepancy) The day gengyin. (13 ke discrepancy) The day gengyin. (5 ke) The day gengyin. (24 ke discrepancy) The day gengyin. (80 ke discrepancy) The day gengyin. (17 ke discrepancy) Guangda 2 (wuzi): eleventh month (yisi day), winter solstice by shadow length.
62
乙巳 （8） 乙巳 〈（86）〉 乙巳 〈（79）〉 乙巳 〈（97）〉 乙巳 〈（81）〉 乙巳 〈（90）〉 太建四年壬辰歲，十一月二十九日丁卯景長。
The day yisi. (8 ke) The day yisi. (86 ke discrepancy) The day yisi. (79 ke discrepancy) The day yisi. (97 ke discrepancy) The day yisi. (81 ke discrepancy) The day yisi. (90 ke discrepancy) Taijian 4 (renchen): on the twenty-ninth of the eleventh month (dingmao day), winter solstice by shadow length.
63
丙寅 〈（83）〉 丙寅 〈（78）〉 丙寅 〈（77）〉 丙寅 〈（95）〉 丙寅 〈（98）〉 丙寅 〈（87）〉 太建六年甲午歲，十一月二十日丁丑景長。
The day bingyin. (83 ke discrepancy) The day bingyin. (78 ke discrepancy) The day bingyin. (77 ke discrepancy) The day bingyin. (95 ke discrepancy) The day bingyin. (98 ke discrepancy) The day bingyin. (87 ke discrepancy) Taijian 6 (jiawu): on the twentieth of the eleventh month (dingchou day), winter solstice by shadow length.
64
丁丑 〈（32）〉 丁丑 〈（33）〉 丁丑 〈（25）〉 丁丑 〈（43）〉 丁丑 〈（27）〉 丁丑 〈（36）〉 太建九年丁酉歲，十一月二十三日壬辰景長。
The day dingchou. (32 ke discrepancy) The day dingchou. (33 ke discrepancy) The day dingchou. (25 ke discrepancy) The day dingchou. (43 ke discrepancy) The day dingchou. (27 ke discrepancy) The day dingchou. (36 ke discrepancy) Taijian 9 (dingyou): on the twenty-third of the eleventh month (renchen day), winter solstice by shadow length.
65
癸巳 （4） 癸巳 （6） 壬辰 〈（99）〉 癸巳 〈（16）〉 癸巳 （空） 癸巳 （8） 太建十年戊戌歲，十一月五日戊戌景長。
The day guisi. (4 ke) The day guisi. (6 ke) The day renchen. (99 ke discrepancy) The day guisi. (16 ke discrepancy) The day guisi. (blank cell) The day guisi. (8 ke) Taijian 10 (wuxu): on the fifth of the eleventh month (wuxu day), winter solstice by shadow length.
66
戊戌 〈（30）〉 戊戌 〈（30）〉 戊戌 〈（23）〉 戊戌 〈（40）〉 戊戌 〈（24）〉 戊戌 〈（33）〉 隋開皇四年甲辰歲，十一月十一日己巳景長。
The day wuxu. (30 ke discrepancy) The day wuxu. (30 ke discrepancy) The day wuxu. (23 ke discrepancy) The day wuxu. (40 ke discrepancy) The day wuxu. (24 ke discrepancy) The day wuxu. (33 ke discrepancy) Sui Kaihuang 4 (jiachen): on the eleventh of the eleventh month (jisi day), winter solstice by shadow length.
67
己巳 〈（77）〉 己巳 〈（78）〉 己巳 〈（69）〉 己巳 〈（86）〉 己巳 〈（71）〉 己巳 〈（86）〉 開皇五年乙巳歲，十一月二十二日乙亥景長。
The day jisi. (77 ke discrepancy) The day jisi. (78 ke discrepancy) The day jisi. (69 ke discrepancy) The day jisi. (86 ke discrepancy) The day jisi. (71 ke discrepancy) The day jisi. (86 ke discrepancy) Kaihuang 5 (yisi): on the twenty-second of the eleventh month (yihai day), winter solstice by shadow length.
68
乙亥 （1） 乙亥 （2） 甲戌 〈（92）〉 乙亥 〈（11）〉 甲戌 〈（55）〉 乙亥 〈（10）〉 開皇六年丙午歲，十一月三日庚辰景長。
The day yihai. (1 ke) The day yihai. (2 ke) The day jiaxu. (92 ke discrepancy) The day yihai. (11 ke discrepancy) The day jiaxu. (55 ke discrepancy) The day yihai. (10 ke discrepancy) Kaihuang 6 (bingwu): on the third of the eleventh month (gengchen day), winter solstice by shadow length.
69
庚辰 〈（25）〉 庚辰 〈（26）〉 庚辰 〈（18）〉 庚辰 〈（34）〉 庚辰 〈（19）〉 庚辰 〈（34）〉 開皇七年丁未歲，十一月十四日乙酉景長。
The day gengchen. (25 ke discrepancy) The day gengchen. (26 ke discrepancy) The day gengchen. (18 ke discrepancy) The day gengchen. (34 ke discrepancy) The day gengchen. (19 ke discrepancy) The day gengchen. (34 ke discrepancy) Kaihuang 7 (dingwei): on the fourteenth of the eleventh month (yiyou day), winter solstice by shadow length.
70
乙酉 〈（50）〉 乙酉 〈（51）〉 乙酉 〈（42）〉 乙酉 〈（59）〉 乙酉 〈（44）〉 乙酉 〈（59）〉 開皇十一年辛亥歲，十一月二十八日丙午景長。
The day yiyou. (50 ke discrepancy) The day yiyou. (51 ke discrepancy) The day yiyou. (42 ke discrepancy) The day yiyou. (59 ke discrepancy) The day yiyou. (44 ke discrepancy) The day yiyou. (59 ke discrepancy) Kaihuang 11 (xinhai): on the twenty-eighth of the eleventh month (bingwu day), winter solstice by shadow length.
71
丙午 〈（48）〉 丙午 〈（49）〉 丙午 〈（43）〉 丙午 〈（57）〉 丙午 〈（41）〉 丙午 〈（56）〉 開皇十四年甲寅歲，十一月辛酉朔旦冬至。
The day bingwu. (48 ke discrepancy) The day bingwu. (49 ke discrepancy) The day bingwu. (43 ke discrepancy) The day bingwu. (57 ke discrepancy) The day bingwu. (41 ke discrepancy) The day bingwu. (56 ke discrepancy) Kaihuang 14 (jiayin): winter solstice at dawn on new moon, xinyou day, eleventh month.
72
壬戌 〈（21）〉 壬戌 〈（22）〉 壬戌 〈（13）〉 壬戌 〈（30）〉 壬戌 〈（14）〉 壬戌 〈（29）〉 唐貞觀十八年甲辰歲，十一月乙酉景長。
The day renxu. (21 ke discrepancy) The day renxu. (22 ke discrepancy) The day renxu. (13 ke discrepancy) The day renxu. (30 ke discrepancy) The day renxu. (14 ke discrepancy) The day renxu. (29 ke discrepancy) Tang Zhenguan 18 (jiachen): eleventh month (yiyou day), winter solstice by shadow length.
73
甲申 〈（43）〉 甲申 〈（45）〉 甲申 〈（31）〉 甲申 〈（50）〉 甲申 〈（32）〉 甲申 〈（44）〉 貞觀二十三年己酉歲，十一月辛亥景長。
The day jiashen. (43 ke discrepancy) The day jiashen. (45 ke discrepancy) The day jiashen. (31 ke discrepancy) The day jiashen. (50 ke discrepancy) The day jiashen. (32 ke discrepancy) The day jiashen. (44 ke discrepancy) Zhenguan 23 (jiyou): eleventh month (xinhai day), winter solstice by shadow length.
74
庚戌 〈（65）〉 庚戌 〈（68）〉 庚戌 〈（53）〉 庚戌 〈（72）〉 庚戌 〈（54）〉 庚戌 〈（66）〉 龍朔二年壬戌歲，十一月四日己未至戊午景長。
The day gengxu. (65 ke discrepancy) The day gengxu. (68 ke discrepancy) The day gengxu. (53 ke discrepancy) The day gengxu. (72 ke discrepancy) The day gengxu. (54 ke discrepancy) The day gengxu. (66 ke discrepancy) Longshuo 2 (renxu): from the fourth of the eleventh month (jiwei through wuwu), winter solstice by shadow length.
75
戊午 〈（83）〉 戊午 〈（86）〉 戊午 〈（69）〉 戊午 〈（88）〉 戊午 〈（71）〉 戊午 〈（82）〉 儀鳳元年丙子歲，十一月壬申景長。
The day wuwu. (83 ke discrepancy) The day wuwu. (86 ke discrepancy) The day wuwu. (69 ke discrepancy) The day wuwu. (88 ke discrepancy) The day wuwu. (71 ke discrepancy) The day wuwu. (82 ke discrepancy) Yifeng 1 (bingzi): eleventh month (renshen day), winter solstice by shadow length.
76
壬申 〈（25）〉 壬申 〈（28）〉 壬申 〈（10）〉 壬申 〈（28）〉 壬申 〈（12）〉 壬申 〈（22）〉 永淳元年壬午歲，十一月癸卯景長。
The day renshen. (25 ke discrepancy) The day renshen. (28 ke discrepancy) The day renshen. (10 ke discrepancy) The day renshen. (28 ke discrepancy) The day renshen. (12 ke discrepancy) The day renshen. (22 ke discrepancy) Yongchun 1 (renwu): eleventh month (guimao day), winter solstice by shadow length.
77
癸卯 〈（72）〉 癸卯 〈（75）〉 癸卯 〈（57）〉 癸卯 〈（76）〉 癸卯 〈（58）〉 癸卯 〈（68）〉 開元十年壬戌歲，十一月癸酉景長。
The day guimao. (72 ke discrepancy) The day guimao. (75 ke discrepancy) The day guimao. (57 ke discrepancy) The day guimao. (76 ke discrepancy) The day guimao. (58 ke discrepancy) The day guimao. (68 ke discrepancy) Kaiyuan 10 (renxu): eleventh month (guiyou day), winter solstice by shadow length.
78
癸酉 〈（49）〉 癸酉 〈（54）〉 癸酉 〈（31）〉 癸酉 〈（50）〉 癸酉 〈（32）〉 癸酉 〈（46）〉 開元十一年癸亥歲，十一月戊寅景長。
The day guiyou. (49 ke discrepancy) The day guiyou. (54 ke discrepancy) The day guiyou. (31 ke discrepancy) The day guiyou. (50 ke discrepancy) The day guiyou. (32 ke discrepancy) The day guiyou. (46 ke discrepancy) Kaiyuan 11 (guihai): eleventh month (wuyin day), winter solstice by shadow length.
79
戊寅 〈（74）〉 戊寅 〈（77）〉 戊寅 〈（55）〉 戊寅 〈（74）〉 戊寅 〈（56）〉 戊寅 〈（70）〉 開元十二年甲子歲，十一月癸未冬至。
The day wuyin. (74 ke discrepancy) The day wuyin. (77 ke discrepancy) The day wuyin. (55 ke discrepancy) The day wuyin. (74 ke discrepancy) The day wuyin. (56 ke discrepancy) The day wuyin. (70 ke discrepancy) Kaiyuan 12 (jiazi): winter solstice on guiwei day, eleventh month.
80
癸未 〈（98）〉 甲申 （3） 癸未 〈（80）〉 癸未 〈（99）〉 癸未 〈（81）〉 癸未 〈（95）〉 宋景德四年丁未歲，十一月戊辰日南至。
The day guiwei. (98 ke discrepancy) The day jiashen. (3 ke) The day guiwei. (80 ke discrepancy) The day guiwei. (99 ke discrepancy) The day guiwei. (81 ke discrepancy) The day guiwei. (95 ke discrepancy) Song Jingde 4 (dingwei): winter solstice on wuchen day, eleventh month.
81
戊辰 〈（15）〉 戊辰 〈（26）〉 丁卯 〈（74）〉 丁卯 〈（82）〉 丁卯 〈（74）〉 丁卯 〈（80）〉 皇祐二年庚寅歲，十一月三十日癸丑景長。
The day wuchen. (15 ke discrepancy) The day wuchen. (26 ke discrepancy) The day dingmao. (74 ke discrepancy) The day dingmao. (82 ke discrepancy) The day dingmao. (74 ke discrepancy) The day dingmao. (80 ke discrepancy) Huangyou 2 (gengyin): on the thirtieth of the eleventh month (guichou day), winter solstice by shadow length.
82
癸丑 〈（65）〉 癸丑 〈（79）〉 癸丑 〈（22）〉 癸丑 〈（25）〉 癸丑 〈（22）〉 癸丑 〈（23）〉 元豐六年癸亥歲，十一月丙午景長。
The day guichou. (65 ke discrepancy) The day guichou. (79 ke discrepancy) The day guichou. (22 ke discrepancy) The day guichou. (25 ke discrepancy) The day guichou. (22 ke discrepancy) The day guichou. (23 ke discrepancy) Yuanfeng 6 (guihai): eleventh month (bingwu day), winter solstice by shadow length.
83
丙午 〈（73）〉 丙午 〈（85）〉 丙午 〈（26）〉 丙午 〈（27）〉 丙午 〈（26）〉 丙午 〈（26）〉 元豐七年甲子歲，十一月辛亥景長。
The day bingwu. (73 ke discrepancy) The day bingwu. (85 ke discrepancy) The day bingwu. (26 ke discrepancy) The day bingwu. (27 ke discrepancy) The day bingwu. (26 ke discrepancy) The day bingwu. (26 ke discrepancy) Yuanfeng 7 (jiazi): eleventh month (xinhai day), winter solstice by shadow length.
84
辛亥 〈（97）〉 壬子 〈（10）〉 辛亥 〈（50）〉 辛亥 〈（51）〉 辛亥 〈（50）〉 辛亥 〈（51）〉 元祐三年戊辰歲，十一月壬申景長。
The day xinhai. (97 ke discrepancy) The day renzi. (10 ke discrepancy) The day xinhai. (50 ke discrepancy) The day xinhai. (51 ke discrepancy) The day xinhai. (50 ke discrepancy) The day xinhai. (51 ke discrepancy) Yuanyou 3 (wuchen): eleventh month (renshen day), winter solstice by shadow length.
85
壬申 〈（94）〉 癸酉 （8） 壬申 〈（48）〉 壬申 〈（48）〉 壬申 〈（48）〉 壬申 〈（48）〉 元祐四年己巳歲，十一月丁丑景長。
The day renshen. (94 ke discrepancy) The day guiyou. (8 ke) The day renshen. (48 ke discrepancy) The day renshen. (48 ke discrepancy) The day renshen. (48 ke discrepancy) The day renshen. (48 ke discrepancy) Yuanyou 4 (jisi): eleventh month (dingchou day), winter solstice by shadow length.
86
戊寅 〈（19）〉 戊寅 〈（32）〉 丁丑 〈（72）〉 丁丑 〈（72）〉 丁丑 〈（72）〉 丁丑 〈（72）〉 元祐五年庚午歲，十一月壬午冬至。
The day wuyin. (19 ke discrepancy) The day wuyin. (32 ke discrepancy) The day dingchou. (72 ke discrepancy) The day dingchou. (72 ke discrepancy) The day dingchou. (72 ke discrepancy) The day dingchou. (72 ke discrepancy) Yuanyou 5 (gengwu): winter solstice on renwu day, eleventh month.
87
癸未 〈（44）〉 癸未 〈（56）〉 壬午 〈（96）〉 壬午 〈（97）〉 壬午 〈（96）〉 壬午 〈（96）〉 元祐七年壬申歲，十一月癸巳冬至。
The day guiwei. (44 ke discrepancy) The day guiwei. (56 ke discrepancy) The day renwu. (96 ke discrepancy) The day renwu. (97 ke discrepancy) The day renwu. (96 ke discrepancy) The day renwu. (96 ke discrepancy) Yuanyou 7 (renshen): winter solstice on guisi day, eleventh month.
88
癸巳 〈（92）〉 甲午 （5） 癸巳 〈（45）〉 癸巳 〈（45）〉 癸巳 〈（45）〉 癸巳 〈（45）〉 元符元年戊寅歲，十一月甲子冬至。
The day guisi. (92 ke discrepancy) The day jiawu. (5 ke) The day guisi. (45 ke discrepancy) The day guisi. (45 ke discrepancy) The day guisi. (45 ke discrepancy) The day guisi. (45 ke discrepancy) Yuanfu 1 (wuyin): winter solstice on jiazi day, eleventh month.
89
乙丑 〈（39）〉 乙丑 〈（52）〉 甲子 〈（91）〉 甲子 〈（91）〉 甲子 〈（91）〉 甲子 〈（91）〉 崇寧三年甲申歲，十一月丙申冬至。
The day yichou. (39 ke discrepancy) The day yichou. (52 ke discrepancy) The day jiazi. (91 ke discrepancy) The day jiazi. (91 ke discrepancy) The day jiazi. (91 ke discrepancy) The day jiazi. (91 ke discrepancy) Chongning 3 (jiashen): winter solstice on bingshen day, eleventh month.
90
丙申 〈（86）〉 丙申 〈（99）〉 丙申 〈（37）〉 丙申 〈（36）〉 丙申 〈（37）〉 丙申 〈（37）〉 紹熙二年辛亥歲，十一月壬申冬至。
The day bingshen. (86 ke discrepancy) The day bingshen. (99 ke discrepancy) The day bingshen. (37 ke discrepancy) The day bingshen. (36 ke discrepancy) The day bingshen. (37 ke discrepancy) The day bingshen. (37 ke discrepancy) Shaoxi 2 (xinhai): winter solstice on renshen day, eleventh month.
91
癸酉 〈（12）〉 癸酉 〈（27）〉 壬申 〈（57）〉 壬申 〈（47）〉 壬申 〈（57）〉 壬申 〈（46）〉 慶元三年丁巳歲，十一月癸卯日南至。
The day guiyou. (12 ke discrepancy) The day guiyou. (27 ke discrepancy) The day renshen. (57 ke discrepancy) The day renshen. (47 ke discrepancy) The day renshen. (57 ke discrepancy) The day renshen. (46 ke discrepancy) Qingyuan 3 (dingsi): winter solstice on guimao day, eleventh month.
92
甲辰 〈（59）〉 甲辰 〈（74）〉 甲辰 （3） 癸卯 〈（92）〉 甲辰 （3） 癸卯 〈（92）〉 嘉泰三年癸亥歲，十一月甲戌日南至。
The day jiachen. (59 ke discrepancy) The day jiachen. (74 ke discrepancy) The day jiachen. (3 ke) The day guimao. (92 ke discrepancy) The day jiachen. (3 ke) The day guimao. (92 ke discrepancy) Jiatai 3 (guihai): winter solstice on jiaxu day, eleventh month.
93
丙子 （5） 丙子 〈（21）〉 乙亥 〈（49）〉 乙亥 〈（37）〉 乙亥 〈（49）〉 乙亥 〈（37）〉 嘉定五年壬申歲，十一月壬戌日南至。
The day bingzi. (5 ke) The day bingzi. (21 ke discrepancy) The day yihai. (49 ke discrepancy) The day yihai. (37 ke discrepancy) The day yihai. (49 ke discrepancy) The day yihai. (37 ke discrepancy) Jiading 5 (renshen): winter solstice on renxu day, eleventh month.
94
癸亥 〈（25）〉 癸亥 〈（41）〉 壬戌 〈（69）〉 壬戌 〈（56）〉 壬戌 〈（68）〉 壬戌 〈（56）〉 紹定三年庚寅歲，十一月丙申日南至。
The day guihai. (25 ke discrepancy) The day guihai. (41 ke discrepancy) The day renxu. (69 ke discrepancy) The day renxu. (56 ke discrepancy) The day renxu. (68 ke discrepancy) The day renxu. (56 ke discrepancy) Shaoding 3 (gengyin): winter solstice on bingshen day, eleventh month.
95
丁酉 〈（65）〉 丁酉 〈（83）〉 丁酉 （7） 丙申 〈（63）〉 丁酉 （7） 丙申 〈（92）〉 淳祐十年庚戌歲，十一月辛巳日南至。
The day dingyou. (65 ke discrepancy) The day dingyou. (83 ke discrepancy) The day dingyou. (7 ke) The day bingshen. (63 ke discrepancy) The day dingyou. (7 ke) The day bingshen. (92 ke discrepancy) Chunyou 10 (gengxu): winter solstice on xinsi day, eleventh month.
96
壬午 〈（94）〉 壬午 〈（71）〉 辛巳 〈（96）〉 辛巳 〈（77）〉 辛巳 〈（94）〉 辛巳 〈（78）〉 本朝至元十七年庚辰歲，十一月己未夜半後六刻冬至。
The day renwu. (94 ke discrepancy) The day renwu. (71 ke discrepancy) The day xinsi. (96 ke discrepancy) The day xinsi. (77 ke discrepancy) The day xinsi. (94 ke discrepancy) The day xinsi. (78 ke discrepancy) Zhiyuan 17 (gengchen): winter solstice at jiwei day, eleventh month, six ke after midnight.
97
己未 〈（87）〉 庚申 （5） 己未 〈（25）〉 己未 （4） 己未 〈（24）〉 己未 （6）
The day jiwei. (87 ke discrepancy) The day gengshen. (5 ke) The day jiwei. (25 ke discrepancy) The day jiwei. (4 ke) The day jiwei. (24 ke discrepancy) The day jiwei. (6 ke)
98
右自春秋獻公以來，凡二千一百六十餘年，用《大衍》、《宣明》、《紀元》、《統天》、《大明》、《授時》六曆推算冬至，凡四十九事。 《大衍曆》合者三十二，不合者十七； 《宣明曆》合者二十六，不合者二十三； 《紀元曆》合者三十五，不合者十四； 《統天曆》合者三十八，不合者十一； 《大明曆》合者三十四，不合者十五； 《授時曆》合者三十九，不合者十事。
From Duke Xian through 2,160+ years, six calendars were tested against forty-nine recorded winter solstices. Dayan Calendar: 32 matches, 17 mismatches; Xuanming Calendar: 26 matches, 23 mismatches; Jiyuan Calendar: 35 matches, 14 mismatches; Tongtian Calendar: 38 matches, 11 mismatches; Daming Calendar: 34 matches, 15 mismatches; Shoushi Calendar: 39 matches, 10 mismatches.
99
今按獻公十五年戊寅歲正月甲寅朔旦冬至，《授時曆》得甲寅，《統天曆》得乙卯，後天一日； 至僖公五年丙寅歲正月辛亥朔旦冬至，《授時》、《統天》皆得辛亥，與天合； 下至昭公二十年己卯歲正月己丑朔旦冬至，《授時》、《統天》皆得戊子，並先一日，若曲變其法以從之，則獻公、僖公皆不合矣。 以此知《春秋》所書昭公冬至，乃日度失行之驗。 一也。 《大衍曆》考古冬至，謂劉宋元嘉十三年丙子歲十一月甲戌日南至，《大衍》與《皇極》、《麟德》三曆皆得癸酉，各先一日，乃日度失行，非三曆之差。 今以《授時曆》考之，亦得癸酉。 二也。 大明五年辛丑歲十一月乙酉冬至，諸曆皆得甲申，殆亦日度之差。 三也。 陳太建四年壬辰歲十一月丁卯景長，《大衍》、《授時》皆得丙寅，是先一日； 太建九年丁酉歲十一月壬辰景長，《大衍》、《授時》皆得癸巳，是後一日； 一失之先，一失之後，若合於壬辰，則差於丁酉，合於丁酉，則差於壬辰，亦日度失行之驗。 五也。 開皇十一年辛亥歲十一月丙午景長，《大衍》、《統天》、《授時》皆得丙午，與天合； 至開皇十四年甲寅歲十一月辛酉冬至，而《大衍》、《統天》、《授時》皆得壬戌，若合於辛亥，則失於甲寅，合於甲寅，則失於辛亥，其開皇十四年甲寅歲冬至，亦日度失行。 六也。 唐貞觀十八年甲辰歲十一月乙酉景長，諸曆得甲申，貞觀二十三年己酉歲十一月辛亥景長，諸曆皆得庚戌，《大衍曆議》以永淳、開元冬至推之，知前二冬至乃史官依時曆以書，必非候景所得，所以不合，今以《授時曆》考之亦然。 八也。 自前宋以來，測景驗氣者凡十七事，其景德丁未歲戊辰日南至，《統天》、《授時》皆得丁卯，是先一日； 嘉泰癸亥歲甲戌日南至，《統天》、《授時》皆得乙亥，是後一日； 一失之先，一失之後，若曲變其數以從景德，則其餘十六事多後天，從嘉泰，則其餘十六事多先天，亦日度失行之驗。 十也。
Duke Xian year 15: Shoushi gives jiayin; Tongtian gives yimao, one day late; Duke Xi year 5: both Shoushi and Tongtian give xinhai, matching the heavens; Duke Zhao year 20: both calendars give wuzi, one day early; forcing a fit would break the earlier matches. This shows the Spring and Autumn record for Duke Zhao reflects solar motion error, not calendar fault. First case. Yuanjia 13: Dayan, Huangji, and Linde all give guiyou, one day early—solar motion error, not calendar error. Shoushi Calendar also gives guiyou. Second case. Daming 5: all calendars give jiashen; likely solar motion error again. Third case. Taijian 4: Dayan and Shoushi both give bingyin, one day early; Taijian 9: Dayan and Shoushi both give guisi, one day late; One early, one late; matching one year breaks the other—solar motion error. Fifth case. Kaihuang 11: all three calendars give bingwu, matching the heavens; Kaihuang 14: all give renxu; the recorded solstice again reflects solar motion error. Sixth case. Zhenguan 18 and 23 shadow records were historiographers' calendar entries, not gnomon observations—Shoushi confirms Dayan's judgment. Eighth case. Song verification: Jingde winter solstice—Tongtian and Shoushi give dingmao, one day early; Jiatai winter solstice—both give yihai, one day late; Fitting one case breaks the other fifteen—solar motion error. Tenth case.
100
前十事皆《授時曆》所不合，以此理推之，非不合矣，蓋類其同則知其中，辨其異則知其變。 今於冬至略其日度失行及史官依時曆書之者凡十事，則《授時曆》三十九事皆中，《統天曆》與今曆不合者僅有獻公一事，《大衍曆》推獻公冬至後天二日，《大明》後天三日，《授時曆》與天合。 下推至元庚辰冬至，《大衍》後天八十一刻，《大明》後天一十九刻，《統天曆》先天一刻，《授時曆》與天合。 以前代諸曆校之，《授時》為密，庶幾千歲之日至，可坐而致云。
The ten mismatches reflect solar error; grouping agreements reveals the pattern. Excluding ten error cases, Shoushi matches all 39; Tongtian misses only Duke Xian; Dayan and Daming run late. Forward to Zhiyuan gengchen: Shoushi alone matches; Dayan 81 ke late, Daming 19 ke late. Among all calendars, Shoushi is most precise—the solstice of ages hence may be calculated at desk.
101
古今曆參校疏密
Comparing Calendar Precision Across Ages
102
《授時曆》與古曆相校，疏密自見，蓋上能合於數百載之前，則下可行之永久，此前人定說。 古稱善治曆者，若宋何承天，隋劉焯，唐傅仁均、僧一行之流，最為傑出。 今以其曆與至元庚辰冬至氣應相校，未有不舛戾者，而以新曆上推往古，無不吻合，則其疏密從可知已。
If a calendar fits centuries back, it will hold for ages forward—an established principle. He Chengtian, Liu Chuo, Fu Renjun, and Yixing were the greatest calendar-makers. Their calendars miss the Zhiyuan solstice; Shoushi matches every ancient case—its superiority is clear.
103
宋文帝元嘉十九年壬午歲十一月乙巳日十一刻冬至，距本朝至元十七年庚辰歲，計八百三十八年。 其年十一月，氣應己未六刻冬至，《元嘉曆》推之，得辛酉，後《授時》二日，《授時》上考元嘉壬午歲冬至，得乙巳，與元嘉合。
Yuanjia 19 to Zhiyuan 17: 838 years. Yuankai misses by two days; Shoushi back-calculation matches Yuankai exactly.
104
隋大業三年丁卯歲十一月庚午日五十二刻冬至，距至元十七年庚辰歲，計六百七十三年。 《皇極曆》推之，得庚申冬至，後《授時》一日； 《授時》上考大業丁卯歲冬至，得庚午，與《皇極》合。
Daye 3 to Zhiyuan 17: 673 years. Huangji gives gengshen, one day after Shoushi; Shoushi back-calculation matches Huangji.
105
唐武德元年戊寅歲十一月戊辰日六十四刻冬至，距至元十七年庚辰歲，計六百六十二年。 《戊寅曆》推之，得庚申冬至，後《授時》一日； 《授時曆》上考武德戊寅歲，得戊辰冬至，與《戊寅曆》合。
Wude 1 to Zhiyuan 17: 662 years. Wuyin Calendar gives gengshen, one day after Shoushi; Shoushi back-calculation matches Wuyin Calendar.
106
開元十五年丁卯歲十一月己亥日七十二刻冬至，距至元十七年庚辰歲，計五百五十三年。 《大衍曆》推之，得己未冬至，後《授時》八十一刻； 《授時曆》上考開元丁卯歲，得己亥冬至，與《大衍曆》合，先四刻。
Kaiyuan 15 to Zhiyuan 17: 553 years. Dayan gives jiwei, 81 ke after Shoushi; Shoushi back-calculation matches Dayan, four ke earlier.
107
長慶元年辛丑歲十一月壬子日七十六刻冬至，距至元十七年庚辰歲，計四百五十九年。 《宣明曆》推之，得庚申冬至，後《授時》一日； 《授時曆》上考長慶辛丑歲，得壬子冬至，與《宣明曆》合。
Changqing 1 to Zhiyuan 17: 459 years. Xuanming gives gengshen, one day after Shoushi; Shoushi back-calculation matches Xuanming.
108
宋太平興國五年庚辰歲十一月丙午日六十三刻冬至，距至元十七年庚辰歲，計三百年。 《乾元曆》推之，得庚申冬至，後《授時》一日； 《授時曆》上考太平興國庚辰歲，得丙午冬至，與《乾元》合。
Taiping Xingguo 5 to Zhiyuan 17: 300 years. Qianyuan gives gengshen, one day after Shoushi; Shoushi back-calculation matches Qianyuan.
109
咸平三年庚子歲十一月辛卯日五十三刻冬至，距至元十七年庚辰歲，計二百八十年。 《儀天曆》推之，得庚申冬至，後《授時》一日； 《授時》上考咸平庚子歲，得辛卯冬至，與《儀天》合。
Xianping 3 to Zhiyuan 17: 280 years. Yitian gives gengshen, one day after Shoushi; Shoushi back-calculation matches Yitian.
110
崇寧四年乙酉歲十一月辛丑日六十二刻冬至，距至元十七年庚辰歲，計一百七十五年。 《紀元曆》推之，得己未日冬至，後《授時》十九刻； 《授時曆》上考崇寧乙酉歲，得辛丑日冬至，與《紀元曆》合，先二刻。
Chongning 4 to Zhiyuan 17: 175 years. Jiyuan gives jiwei, 19 ke after Shoushi; Shoushi back-calculation matches Jiyuan, two ke earlier.
111
金大定十九年己亥歲十一月己巳日六十四刻冬至，距至元十七年庚辰歲，計一百一年。 《大明曆》推之，得己未冬至，後《授時》一十九刻； 《授時曆》上考大定己亥歲，己巳冬至，與《大明曆》合，先九刻。 〈（《大明》冬至蓋測驗未密故也。）〉
Dading 19 to Zhiyuan 17: 101 years. Daming gives jiwei, 19 ke after Shoushi; Shoushi back-calculation matches Daming, nine ke earlier. (Daming error likely from imprecise observation.)
112
慶元四年戊午歲十一月己酉日一十七刻冬至，距至元十七年庚辰歲，計八十二年。 《統天曆》推之，得己未冬至，先《授時》一刻； 《授時曆》上考慶元戊午歲，得己酉日冬至，與《統天曆》合。
Qingyuan 4 to Zhiyuan 17: 82 years. Tongtian gives jiwei, one ke before Shoushi; Shoushi back-calculation matches Tongtian.
113
周天列宿度
Circuit of Heaven and Lodge Degrees
114
列宿著於天，為舍二十有八，為度三百六十五有奇。 非日躔無以校其度，非列舍無以紀其度，周天之度，因二者以得之。 天體渾圓，當二極南北之中，絡以赤道，日月五星之行，常出入於比。 天左旋，日月五星溯而右轉，昔人曆象日月星辰，謂此也。 然列舍相距度數，歷代所測不同，非微有動移，則前人所測或有未密。 古用窺管，今新制渾儀，測用二線，所測度數分秒與前代不同者，今列於左。
Twenty-eight lodges span 365-odd degrees across the sky. Solar motion and lodge positions together define the celestial circuit. The heavens are spherical, girdled by the equator; sun, moon, and planets cross it regularly. Heaven spins westward; the luminaries creep eastward against it. Lodge intervals differ across dynasties—either precession or measurement error. New armillary measurements differ from earlier tube observations; results are tabulated below.
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〈（表略）〉 日躔
(table omitted) Solar Lodge Motion
116
日之麗天，縣象最著，大明一生，列宿俱熄。 古人欲測躔度所在，必以昏旦夜半中星衡考其所距，從考其所當； 然昏旦夜半時刻未易得真，時刻一差，則所距、所當，不容無舛。 晉姜岌首以月食沖檢，知日度所在； 《紀元曆》復以太白誌其相距遠近，於昏後明前驗定星度，因得日躔。 今用至元丁丑四月癸酉望月食既，推求得冬至日躔赤道箕宿十度，黃道九度有奇。 仍自其年正月至己卯歲終，三年之間，日測太陰所離宿次及歲星、太白相距度，定驗參考，共得一百三十四事，皆躔箕宿，適與月食所沖允合。 以金趙知微所修《大明曆法》推之，冬至猶躔鬥初度三十六分六十四秒，比新測實差七十六分六十四秒。
Sunrise outshines every star—the brightest sign in the sky. Ancient method: use culminating stars at dusk, dawn, and midnight to locate solar lodge position; A single timing error corrupts the whole calculation. Jiang Ji first used lunar eclipse opposition to fix solar position; Jiyuan Calendar used Venus distances at twilight to determine solar lodge. From the Zhiyuan dingchou eclipse: winter solstice at Ji 10° equatorial, 9°+ ecliptic. Three years of observations—134 cases—all confirm Ji, matching the eclipse. Daming Calendar misses by 76 fen 64 miao versus new observation.
117
日行盈縮
Solar Apogee and Perigee Motion
118
日月之行，有冬有夏，言日月行度，冬夏各不同也。 人徒知日行一度，一歲一周天，曾不知盈縮損益，四序有不同者。 北齊張子信積候合蝕加時，覺日行有入氣差，然損益未得其正。 趙道嚴復準晷景長短，定日行進退，更造盈縮以求虧食。 至劉焯立躔度，與四序升降，雖損益不同，後代祖述用之。
Solar and lunar daily motion varies between winter and summer. Most know one degree per day—but not seasonal speed variation. Zhang Zixin noticed seasonal solar variation but could not quantify it. Zhao Daoyan used shadow lengths to model solar speed for eclipses. Liu Chuo's seasonal model became the standard for later calendars.
119
夫陰陽往來，馴積而變，冬至日行一度強，出赤道二十四度弱，自此日軌漸北，積八十八日九十一分，當春分前三日，交在赤道，實行九十一度三十一分而適平。 自後其盈日損，復行九十三日七十一分，當夏至之日，入赤道內二十四度弱，實行九十一度三十一分，日行一度弱，向之盈分盡損而無余。 自此日軌漸南，積九十三日七十一分，當秋分後三日，交在赤道，實行九十一度三十一分而復平。 自後其縮日損，行八十八日九十一分，出赤道外二十四度弱，實行九十一度三十一分，復當冬至，向之縮分盡損而無余。 盈縮均有損益，初為益，末為損。 自冬至以及春分，春分以及夏至，日躔自北陸轉而西，西而南，於盈為益，益極而損，損至於無余而縮。 自夏至以及秋分，秋分以及冬至，日躔自南陸轉而東，東而北，於縮為益，益極而損，損至於無余而復盈。 盈初縮末，俱八十八日九十一分而行一象； 縮初盈末，俱九十三日七十一分而行一象； 盈縮極差，皆二度四十分。 由實測晷景而得，仍以算術推考，與所測允合。
From winter solstice the sun accelerates northward, reaching uniform speed three days before equinox. Surplus fades until summer solstice, when the sun slows to one-odd degrees daily. The sun decelerates southward, regaining uniform speed three days after equinox. Deficit fades until winter solstice completes the cycle. Surplus and deficit each rise then fall. Winter to summer: surplus peaks at solstice then yields to deficit. Summer to winter: deficit peaks at solstice then yields to surplus. Surplus start and deficit end: 88 days 91 parts per quadrant; Deficit start and surplus end: 93 days 71 parts per quadrant; Maximum speed variation: 2°40' each way. Derived from gnomon observation and confirmed by calculation.
120
月行遲疾
Lunar Anomalistic Motion
121
古曆謂月平行十三度十九分度之七。 漢耿壽昌以為日月行至牽牛、東井，日過度，月行十五度，至婁、角，始平行，赤道使然。 賈逵以為今合朔、弦、望、月食加時，所以不中者，蓋不知月行遲疾意。 李梵、蘇統皆以月行當有遲疾，不必在牽牛、東井、婁、角之間，乃由行道有遠近出入所生。 劉洪作《乾象曆》，精思二十餘年，始悟其理，列為差率，以囿進退損益之數。 後之作曆者，咸因之。 至唐一行，考九道委蛇曲折之數，得月行疾徐之理。
Ancient mean lunar motion: 13° 7/19 per day. Geng Shouchang attributed speed variation to equatorial latitude effects. Jia Kui blamed timing errors on ignoring lunar anomalistic motion. Li Fan and Su Tong traced variation to orbital distance, not fixed lodges. Liu Hong spent twenty years deriving lunar anomaly tables. All later calendars followed his method. Yixing refined the theory through the nine lunar paths.
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先儒謂月與五星，皆近日而疾，遠日而遲。 曆家立法，以入轉一周之日，為遲疾二曆，各立初末二限，初為益，末為損。 在疾初遲末，其行度率過於平行； 遲初疾末，率不及於平行。 自入轉初日行十四度半強，從是漸殺，曆七日，適及平行度，謂之疾初限，其積度比平行余五度四十二分。 自是其疾日損，又曆七日，行十二度微強，向之益者盡損而無余，謂之疾末限。 自是復行遲度，又曆七日，適及平行度，謂之遲初限，其積度比平行不及五度四十二分。 自此其遲日損，行度漸增，又曆七日，復行十四度半強，向之益者亦損而無余，謂之遲末限。 入轉一周，實二十七日五十五刻四十六分，遲疾極差皆五度四十二分。 舊曆日為一限，皆用二十八限。 今定驗得轉分進退時各不同，今分日為十二，共三百三十六限，半之為半周限，析而四之為象限。
Classical view: faster near the sun, slower when distant. Anomalistic month split into slow and fast tables with initial and final limits. Fast start and slow end: motion exceeds mean speed; Slow start and fast end: motion falls below mean speed. Fast initial limit: 7 days from 14°+ to mean, +5°42' accumulated. Fast final limit: 7 more days to 12°+, surplus exhausted. Slow initial limit: 7 days to mean, −5°42' accumulated. Slow final limit: 7 more days back to 14°+, deficit exhausted. Anomalistic month: 27d 55k 46p; max variation ±5°42'. Old method: 28 daily limits. New method: 336 limits (day split into twelve parts).
123
白道交周
Lunar Nodal Cycle
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當二極南北之中，橫絡天體以紀宿度者，赤道也。 出入赤道，為日行之軌者，黃道也。 所謂白道，與黃道交貫，月行之所由也。 古人隨方立名，分為八行，與黃道而九，究而言之，其實一也。 惟其隨交遷徙，變動不居，故強以方色名之。
The equator girdles the heavens at the celestial midpoint. The ecliptic is the sun's path across the equator. The white path (lunar orbit) crosses the ecliptic. Eight directional paths plus the ecliptic are one continuous orbit. Nodal drift forced directional color-naming on a single path.
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月道出入日道，兩相交值，當朔則日為月所掩，當望則月為日所沖，故皆有食。 然涉交有遠近，食分有深淺，皆可以數推之。 所謂交周者，月道出入日道一周之日也。 日道距赤道之遠，為度二十有四。 月道出入日道，不逾六度； 其距赤道也，遠不過三十度，近不下十八度。 出黃道外為陽，入黃道內為陰，陰陽一周，分為四象。 月當黃道為正交，出黃道外六度為半交，復當黃道為中交，入黃道內六度為半交，是為四象。 象別七日，各行九十一度，四象周曆，是謂一交之終，以日計之，得二十七日二十一刻二十二分二十四秒。 每一交，退天一度二百分度之九十三，凡二百四十九交，退天一周有奇，終而復始。 正交在春正，半交出黃道外六度，在赤道內十八度。 正交在秋正，半交出黃道外六度，在赤道外三十度。 中交在春正，半交入黃道內六度，在赤道內三十度。 中交在秋正，半交入黃道內六度，在赤道外十八度。 月道與赤道正交，距春秋二正黃赤道正交宿度，東西不及十四度三分度之二。 夏至在陰曆內，冬至在陽曆外，月道與赤道所差者多； 夏至在陽曆外，冬至在陰曆內，月道與赤道所差者少。 蓋白道二交，有斜有直，陰陽二曆，有內有外，直者密而狹，斜者疏而闊，其差亦從而異。 今立象置法求之，差數多者不過三度五十分，少者不下一度三十分，是為月道與赤道多少之差。
Crossing paths cause solar eclipses at new moon and lunar at full. Eclipse depth depends on nodal distance—all calculable. Nodal cycle: days for the moon to recross the ecliptic. Ecliptic inclination: 24° from the equator. Lunar inclination: no more than 6° from the ecliptic; from the equator: 18°–30°. Outside ecliptic = yang, inside = yin; one cycle = four quadrants. Major, half, mid, and half crossings define four nodal quadrants. Nodal cycle: 4 × 7 days × 91° = 27d 21k 22f 24m. 249 nodal cycles = one sidereal regression of the nodes. Spring major crossing: half-point at ecliptic +6°, equator −18°. Autumn major crossing: half-point at ecliptic +6°, equator +30°. Spring mid-crossing: half-point at ecliptic −6°, equator −30°. Autumn mid-crossing: half-point at ecliptic −6°, equator +18°. Nodal colure within 14⅔° of equinox lodges. Yin-year solstice placement increases lunar-equatorial spread; Yang-year placement reduces the spread. Oblique nodes yield wider latitude variation than direct ones. Lunar declination range: 1°30' to 3°50'.
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晝夜刻
Day-Night Ke Divisions
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日出為晝，日入為夜，晝夜一周，共為百刻。 以十二辰分之，每辰得八刻三分刻之一。 無間南北，所在皆同。 晝短則夜長，夜短則晝長，此自然之理也。 春秋二分，日當赤道出入，晝夜正等，各五十刻。 自春分以及夏至，日入赤道內，去極浸近，夜短而晝長。 自秋分以及冬至，日出赤道外，去極浸遠，晝短而夜長。 以地中揆之，長不過六十刻，短不過四十刻。 地中以南，夏至去日出入之所為遠，其長有不及六十刻者； 冬至去日出入之所為近，其短有不止四十刻者。 地中以北，夏至去日出入之所為近，其長有不止六十刻者； 冬至去日出入之所為遠，其短有不及四十刻者。 今京師冬至日出辰初二刻，日入申正二刻，故晝刻三十八，夜刻六十二； 夏至日出寅正二刻，日入戌初二刻，故晝刻六十二，夜刻三十八。 蓋地有南北，極有高下，日出入有早晏，所以不同耳。 今《授時曆》晝夜刻，一以京師為正，其各所實測北極高下，具見《天文志》。
One day-night cycle = 100 ke. Each double-hour = 8⅓ ke. Uniform everywhere—or so the old view held. Day and night lengths are inversely related. Equinoxes: 50 ke day, 50 ke night. Spring to summer: days lengthen as the sun moves north. Autumn to winter: nights lengthen as the sun moves south. At the terrestrial midpoint: 40–60 ke daylight. South of center: summer days may not reach 60 ke; winter nights may exceed 40 ke. North of center: summer days may exceed 60 ke; winter nights may fall short of 40 ke. Capital winter solstice: 38 ke day, 62 ke night; Capital summer solstice: 62 ke day, 38 ke night. Latitude and polar height account for the variation. Shoushi uses the capital standard; local polar heights are in the Astronomy Treatise.