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志二十
Treatise 20
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時憲一
Shixian Calendar 1
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明之大統術，本於元之授時。 成化以後，交食往往不驗。 萬曆末，徐光啟、李之藻等譯西人之書為新法，推交食、淩犯皆密合，然未及施用。 世祖定鼎以後，始絀明之舊曆，依新法推算，即承用二百六十餘年之時憲術也。 光啟等齗齗辨論，當時格而不行，乃為新朝改憲之資，豈非天意哉！ 聖祖邃於曆學，定用均輪法以齊七政，以康熙甲子為元。 雍正中，從監臣之請，推步改橢圓法，以雍正癸卯為元。 道光中，監臣以交食分秒不合，據實測之數損益原用數，以道光甲午為元。 自康熙至於道光，推步之術凡三改，而道光甲午元曆僅有恒星表。 至於推日月交食、步五星，均未及成書雲。 西人湯若望，與徐光啟共譯新法者也，以四十二事證西人之密、中術之疏，疇人子弟翕然信之。 宣城人梅文鼎研精天算，由授時以溯三統、四分以來諸家之術，又博考九執、回回術，而折衷於新法，皆洞其原本，究其異同，卒以績學受知聖祖，於是為推步之學者，始知中、西之學之一貫，不至眩晃於新法矣。 與湯若望同時入中國者為穆尼閣，傳其學於淄川薛鳳祚，而吳江人王錫闡自創新法，用以推日月食，不爽秒忽，兩家之學，皆不列於台官，然其精密，或為台臣所不及焉。 今為時憲誌，詳考其推步、七政、四餘、根理、法數著於篇，諸家論說有裨數理者，亦撮其大要載之。 明大統術、回回術，康熙初用之，以詳於明史，不具論。
Ming astronomical reckoning in the Datong tradition stemmed from the Yuan dynasty's Shoushi calendar. From the Chenghua era onward, predicted eclipses frequently did not accord with what was observed. In the late Wanli period, Xu Guangqi, Li Zhizao, and others translated Western works to establish the new methods; their predictions of eclipses and planetary conjunctions agreed closely with observation, but the system was never adopted before the dynasty fell. Once the Shunzhi Emperor had secured the throne, the court set aside the Ming calendar and reckoned time by the Western methods—the Shixian system that has governed Qing astronomy for more than two centuries and sixty years. Xu Guangqi and his colleagues had pressed their case insistently, yet the late Ming court blocked them; their work then supplied the foundation for the Qing calendar reform—as if Heaven had reserved it for the new dynasty. The Kangxi Emperor, himself a master of astronomy, adopted the uniform epicycle method to harmonize the motions of the seven luminaries, with the jiazi year of his reign as the new epoch. During the Yongzheng era, on the petition of the Astronomical Directorate, computation shifted to elliptical theory, with the guimao year of Yongzheng as the new epoch. In the Daoguang period, when eclipse predictions disagreed with observation down to the minute, the Directorate revised the constants on the basis of actual measurements, establishing the jiawu year of Daoguang as a new epoch. Between Kangxi and Daoguang the computational system was revised three times, but the Daoguang jiawu-epoch reform produced only a star table—the work on solar and lunar eclipses and the five planets remained unfinished. Treatises on eclipse prediction and planetary motion, however, were never brought to completion. Adam Schall, the Jesuit who with Xu Guangqi rendered the new astronomy into Chinese, set forth forty-two propositions demonstrating Western precision and the shortcomings of native methods; calendrical students accepted them wholesale. Mei Wending of Xuancheng mastered astronomy and mathematics, tracing from the Shoushi calendar back through the Santong, Sifen, and later traditions while also studying the Jiuzhi and Islamic systems and weighing them against the new methods. He grasped each system's foundations and compared their differences until the Kangxi Emperor took notice of his scholarship. Scholars of astronomical calculation then understood that Chinese and Western astronomy formed a single continuum, and were no longer swept away by novelty alone. Nicolas Longobardo arrived in China at the same time as Schall and passed his astronomy to Xue Fengzuo of Zichuan, while Wang Xichan of Wujiang devised independent methods that predicted eclipses to the exact second. Neither school held office in the Astronomical Directorate, yet their accuracy at times exceeded that of the official astronomers. This treatise on the Shixian calendar sets out in detail its methods of calculation, the seven luminaries, the four residuals, underlying theory, and computational constants, while also summarizing scholarly discussions that illuminate the mathematics involved. The Ming Datong and Islamic calendars, still referenced in the early Kangxi period, are covered at length in the History of Ming and are therefore not treated here.
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推步因革
Evolution of Astronomical Calculation
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順治元年六月，湯若望言：「臣於明崇禎二年來京，曾依西洋新法釐訂舊曆，今將新法所推本年八月初一日日食，京師及各省所見食限分秒，並起復方位，圖象進呈，乞屆期遣官測驗。」 從之。 七月，若望又推天象進呈。 是月禮部言：「欽天監改用新法，推注已成，請易新名，頒行天下。」 睿親王言：「宜名『時憲』，以稱朝廷憲天乂民至意。」 從之。 八月丙辰朔午時，日食二分四十八秒，大學士馮銓，同若望赴觀象台測驗覆奏，惟新法一一菂合，大統、回回二曆俱差時刻，敕：「舊法歲久自差，非官生推算之誤，新法既密合天行，監局宜學習勿怠玩。」 十月，頒順治二年時憲書。 若望又言：「敬授人時，全以節氣交宮與太陽出入晝刻為重。 今節氣之日時刻分與太陽出入晝夜時刻，俱照道里遠近推算，請刊入時憲書。」 從之。 十一月，以若望掌欽天監事。 若望等言：「臣等按新法推算月食時刻分秒，復定每年進呈書目，刪其復重，以免混淆。」 二年六月，若望等言：「舊法推算本年十二月己卯朔辰時日食三分強，回回科算見食一分弱。 依新法推之，止應食半分強，且在日出之前，地平上不見，請臨期遣官測驗。」 從之。 至期天陰雨，推驗事遂輟。 十一月，若望以明大學士徐光啟所譯崇禎曆書改名新法曆書進呈，上命發監局官生肄習，仍宣付史館，加若望太常寺卿銜。 十年，賜若望通玄教師，以獎其勤勞。
In the sixth month of Shunzhi 1, Schall reported: "I reached Beijing in Chongzhen 2 of the Ming and had already revised the old calendar by Western methods. I now submit charts of the solar eclipse predicted for the first day of the eighth month of this year—the magnitude in degrees, minutes, and seconds as it will appear in the capital and each province, with the bearings of first and last contact—and request that officials be dispatched to verify the prediction on the appointed day." The court approved. In the seventh month Schall submitted further astronomical predictions. That month the Ministry of Rites reported: "The Astronomical Directorate has adopted the new methods and finished its ephemerides; we ask that the calendar be given a new name and promulgated empire-wide." The Prince Regent Dorgon proposed: "Let it be called Shixian—'Season Constitution'—to reflect the court's purpose of aligning with Heaven and governing the people." The court approved. On the new moon of the eighth month, at noon, the eclipse reached two parts and forty-eight seconds of the sun's disk. Grand Secretary Feng Quan joined Schall at the Observatory to observe and report: only the new methods matched observation in every detail, while the Datong and Islamic calendars erred in timing. The throne replied: "The old systems have drifted with time—this is not the fault of the calculators. Since the new methods accord closely with the heavens, the Directorate must study them diligently and not treat them lightly." In the tenth month the Shixian almanac for Shunzhi 2 was promulgated. Schall added: "The calendar's chief duty is to regulate human time, and this rests above all on the solar terms' passage through the lodges and on the sun's times of rising, setting, and the length of day. The dates and times of the solar terms and the sun's rising, setting, and day-length should now be calculated for each place according to its distance from the capital; I ask that these tables be included in the Shixian almanac." The court approved. In the eleventh month Schall was appointed to direct the Astronomical Directorate. Schall and his colleagues reported: "We have computed lunar eclipse times to the second by the new methods, and have revised the annual list of works submitted to the throne, removing duplications to prevent confusion." In the sixth month of Shunzhi 2 they added: "By the old methods the solar eclipse on the new moon of the twelfth month, at the chen hour, should exceed three parts of the sun's disk; the Islamic section predicts a little more than one part visible. By the new methods the eclipse should amount to only a little over half a part, and it will occur before sunrise and therefore be invisible at the horizon; we again ask that officials verify this on the day." The court approved. On the appointed day clouds and rain prevented observation, and the verification was abandoned. In the eleventh month Schall submitted Xu Guangqi's Chongzhen Calendar Book under the new title Calendar of the New Methods. The emperor ordered it distributed for study within the Directorate and also deposited in the Historiography Office, and promoted Schall to Vice Minister of the Court of Imperial Sacrifices. In the tenth year of Shunzhi he received the title Master Who Penetrates the Mysterious in recognition of his service.
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若望之法，以天聰戊辰為元。 分周天為三百六十度。 太陽一日平行五十九分八秒十九微四十九纖三十六芒，最高一年行四十五秒，戊辰年平行距冬至五十三分三十五秒三十九微，最高距冬至五度五十九分五十九秒。 太陰一日平行一十三度一十分三十五秒一微，自行一十三度三分五十三秒五十六微，正交行三分一十秒，月孛行六分四十一秒，戊辰年平行距冬至六宮一度五十分五十四秒四十六微，自行距冬至六宮二十五度三十二分一十五秒三十四微，正交行距冬至一宮一十四秒，月孛行距冬至十一宮六度一十九分。 土星諸行應平行距冬至為十一宮十八度五十一分五十一秒，本年最高行距冬至為九宮八度五十七分五十九秒，平行距最高即引數，為二宮九度五十三分五十二秒，正交行距冬至為六宮七度九分八秒。 一平年平行為十二度十三分三十一秒，最高行一分二十秒十二微，以最高行減平行，得十二度十二分十五秒，乃一年之引數也。 一閏年平行為十二度十五分三十五秒，引數為十二度十四分十五秒。 正交行一年為四十二秒。 木星諸行應平行距冬至為八宮二十八度八分三十一秒，本天最高行為十一宮二十七度十一分十五秒，平行距最高即引數，為九宮初度五十七分十六秒，正交行為六宮二十四度四十一分五十二秒。 一平年距冬至平行為一宮零二十分三十二秒，最高行為五十七秒五十二微，兩數相減，得一宮零十九分三十四秒，乃一平年之引數也。 一閏年距冬至平行為一宮零二十五分三十一秒，引數為一宮零二十四分三十三秒。 正交行一年為一十四秒。 火星諸行應平行距冬至為五宮四度五十四分三十秒，本天最高在七宮二十九度三十分四十秒，平行距最高即引數，為九宮五度二十三分五十秒，正交行為三宮十七度二分二十九秒。 一平年距冬至平行為六宮十一度十七分一十秒，最高行一分十四秒，兩數相減，得六宮十一度十五分五十五秒。 一閏年距冬至平行為六宮十一度四十八分三十六秒，引數為六宮十一度四十七分二十一秒。 正交行一年為五十三秒。 金星諸行應平行距冬至與太陽同度，為初宮初度五十三分三十五秒三十九微，平行距最高即引數，為六宮零五十六分五十五秒，伏見行從極遠處起，為初宮九度十一分七秒，最高行在六宮零十六分六秒。 一平年距冬至為十一宮二十九度四十五分四十秒三十八微，自行引數為十一宮二十九度四十四分十七秒，伏見行為七宮十五度一分五十秒，最高行為一分二十一秒。 一閏年距冬至及自行加五十九分八秒，伏見行加三度六分二十四秒，乃一日之行也。 金星正交在最高前十六度，即五宮十四度十六分，其行極微，故未定其率，然於最高行無大差。 水星諸行應平行距冬至與太陽同度，平行距最高即引數，為二十九度二十分二秒，伏見行從極遠處起，為三宮二十九度五十四分一十六秒，最高在十一宮零五十二分四十二秒。 一平年距冬至亦與太陽同度，自行引數為十一宮二十九度四十三分五十一秒，伏見行滿三周外有一宮二十三度五十七分二十六秒。 一閏年引數為十二宮零四十二分五十九秒，伏見行全周外為一宮二十七度三分五十二秒，正交行或曰與最高同度難測，故不敢定雲。
Schall's system takes the wuchen year of the Later Jin (Tiancong) as its epoch. The celestial sphere is divided into 360 degrees. The sun's mean daily motion is 59′8″19‴49⁗36⁘; its apogee advances 45″ per year. For the wuchen epoch year, mean motion from the winter solstice is 53′35″39‴, and the apogee lies 5°59′59″ from the winter solstice. The moon's mean daily motion is 13°10′35″1‴; its true (anomalistic) motion 13°3′53″56‴; its node advances 3′10″ per day; the lunar apogee advances 6′41″ per day. For the wuchen epoch, mean longitude from the winter solstice is 6 lodges 1°50′54″46‴, true longitude 6 lodges 25°32′15″34‴, the node at 1 lodge 14″, and the lunar apogee at 11 lodges 6°19′. For Saturn at the epoch: mean longitude from the winter solstice is 11 lodges 18°51′51″; the apogee for that year lies 9 lodges 8°57′59″ from the winter solstice; the equation (mean minus apogee) is 2 lodges 9°53′52″; the node stands 6 lodges 7°9′8″ from the winter solstice. In a common year Saturn's mean motion is 12°13′31″ and its apogee advances 1′20″12‴; subtracting the latter from the former yields 12°12′15″ as the annual equation. In an intercalary year the mean motion is 12°15′35″ and the equation 12°14′15″. The node advances 42″ per year. For Jupiter: mean longitude from the winter solstice is 8 lodges 28°8′31″; the apogee lies at 11 lodges 27°11′15″; the equation is 9 lodges 0°57′16″; the node at 6 lodges 24°41′52″. In a common year Jupiter's mean motion from the winter solstice is 1 lodge 20′32″ and its apogee advances 57″52‴; the difference, 1 lodge 19′34″, is the annual equation for a common year. In an intercalary year the mean motion is 1 lodge 25′31″ and the equation 1 lodge 24′33″. The node advances 14″ per year. For Mars: mean longitude from the winter solstice is 5 lodges 4°54′30″; the apogee at 7 lodges 29°30′40″; the equation 9 lodges 5°23′50″; the node at 3 lodges 17°2′29″. In a common year Mars's mean motion from the winter solstice is 6 lodges 11°17′10″ and its apogee advances 1′14″; the difference is 6 lodges 11°15′55″. In an intercalary year the mean motion is 6 lodges 11°48′36″ and the equation 6 lodges 11°47′21″. The node advances 53″ per year. For Venus: mean longitude from the winter solstice coincides with the sun at 1 lodge 0°53′35″39‴; the equation is 6 lodges 0°56′55″; visibility motion begins at greatest elongation, 1 lodge 9°11′7″; the apogee lies at 6 lodges 0°16′6″. In a common year Venus's longitude from the winter solstice is 11 lodges 29°45′40″38‴; the anomalistic equation is 11 lodges 29°44′17″; visibility motion 7 lodges 15°1′50″; apogee motion 1′21″. In an intercalary year the daily increments are 59′8″ for mean and true longitude from the winter solstice, and 3°6′24″ for visibility motion. Venus's node lies 16° before the apogee, at 5 lodges 14°16′; its motion is so slight that no rate was assigned, though this scarcely affects apogee calculation. For Mercury: mean longitude from the winter solstice coincides with the sun; the equation is 29°20′2″; visibility motion at greatest elongation begins at 3 lodges 29°54′16″; the apogee at 11 lodges 0°52′42″. In a common year Mercury's longitude from the winter solstice again coincides with the sun; the anomalistic equation is 11 lodges 29°43′51″; visibility motion beyond three circuits is 1 lodge 23°57′26″. In an intercalary year the equation is 12 lodges 0°42′59″ and visibility motion beyond one circuit is 1 lodge 27°3′52″. The node's motion was left undetermined, some holding that when it coincides with the apogee it cannot be measured reliably.
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若望論新法大要凡四十二事：曰天地經緯，天有經緯，地亦有之，以二百五十里當天之一度，經緯皆然。 曰諸曜異天，諸曜高卑相距遠甚，駁舊曆認為同心之誤。 曰圓心不同，太陽本圈與地不同心，二心相距，古今不等。 曰蒙氣差，地有蒙氣，非先定蒙氣差不能密合。 曰測算異古法，測天以弧三角形，算以割圜八線表。 曰測算皆以黃道，測天用赤道儀，所得經度不合，新法就黃道經度，通以黃赤通率表，乃與天行密合。 曰改定諸應，從天聰二年戊辰前冬至後己卯日子正為始。 曰求真節氣，舊法平節氣，非真節氣，今改定。 曰盈縮真限，用授時消分為平歲，更以最高最卑差加減之，為定歲。 曰表測二分，舊法以圭表測冬至，非法之善者，今用春秋二分，較二至為密。 曰太陽出入及晨昏限，從京師起算，各處有加減。 曰晝夜不等，其差較一刻有奇，一緣黃道夏遲冬疾，一緣黃赤二道廣狹不同距，則率度不同分。 曰改定時刻，定晝夜為九十六刻。 曰置閏不同，舊法用平節氣置閏，非也，改用太陽所躔天度以定節氣。 曰太陰加減，朔望止一加減，餘日另有二三，均數多寡不等。 曰月行高卑遲疾，月行轉周之最高極遲，最卑極疾，五星準此。 曰朔後月見遲疾，一因自行度遲疾，一因黃道升降斜正，一因白道在緯南緯北。 曰交行加減，月在交上，以平求之必不合，因設一加減為交行均數。 曰月緯距度，舊法黃白二道相距五度，不知朔望外尚有損益，其至大之距，五度三分之一。 曰交食有無，月食以距黃道緯度較月與地景兩半徑並，日食則以距度較日月兩半徑並，距度為小則食，大則不食。 曰日月食限不同，月食則太陰與地景兩周相切，以其兩視半徑較白道距黃道度，又以距度推交周度定食限，日食必加入視差而後得距度。 曰日月食分異同，距度在月食為太陰心實距地景之心，在日食為日月兩心之距，但日食不據實距而據視距。 曰實食中食，以地心之直線上至黃道者為主，日月五星兩居此線之上，則實食也； 月與五星各居本輪之周，地心直線上至黃道，而兩本輪之心俱當線上，則中食也。 曰視食，日食有天上之實食，有人所見之視食，視食依人目與地面為準。 曰黃道九十度為東西差之中限，論天頂則高卑差為正下，南北差為斜下，而東西差獨中限之一線為正下，以外皆斜下。 論其道則南北差為股，東西差恒為勾，高卑差恒為弦。 至中限則股弦為一線，無勾矣。 曰三視差，以地半徑為一邊，以太陽太陰各距地之遠為一邊，以二曜高度為一邊，成三角形，用以得高卑差，又偏南而變緯度得南北差； 以黃道九十度限偏左偏右而變緯度，得東西差。 曰外三差，東西、南北、高卑之差，皆生於地徑，外三差不生於地徑而生於氣。 一，清蒙氣差； 二，清蒙徑差； 三，本輪徑差。 曰虧復不一，非二時折半之說，新法以視行推變時刻，則了然於虧復時刻不一之故。 曰交食異算，諸方各以地徑推算交食時刻及日食分。 曰日食變差，據法應食而實不見食，必此日此地之南北差變為東西差，此千百年偶遇一二次，非無有者。 曰推前驗後，新法諸表，上溯下沿，開卷了然，不費功力。 曰五星準日，舊法於合伏日數，時多時寡，徒以段目定之，不免有差，今改正。 曰伏見密合，舊法五星伏見惟用黃道距度，非也，今改正。 曰五星緯度，太陰本道斜交黃道，因生距度與陰、陽二曆，五星亦然，新法一一詳求，舊未能也。 曰金水伏見，金星或合太陽而不伏，水星離太陽而不見，用渾儀一測便知，非舊法所能。 曰五星測法，測五星須用恒星為準。 曰恒星東移，恒星以黃道極為極，各宿距星時近赤極，亦或時遠赤極，由黃赤二道各極不同，非距星有異行或易位。 曰定恒星大小有六等之別，前此未聞。 曰天漢釋疑，新法測以遠鏡，天漢乃無算小星攢聚而成。 曰四餘刪改，羅睺即白道之正交，計都即中交，月孛乃月行極高之點。 至紫[A12L]一餘，無數可定，明史附會，今俱改刪。 曰測器，新法增置者，曰象限儀、百遊儀、地平儀、弩儀、天環天球紀限儀、渾蓋簡平儀、黃赤全儀，而新製之遠鏡，尤為測星要器。 曰日晷，為地平晷、三晷、百遊晷、通光晷。 此外更有星晷、月晷，以備夜測之用。 若望所言，大抵據新法以詆舊術之疏，然新法之精蘊，亦盡於此矣。
Schall set forth forty-two principal points of the new astronomy. On terrestrial and celestial coordinates: as heaven has longitude and latitude, so does the earth; 250 li on earth correspond to one degree in the sky, for both coordinates alike. On the separate spheres of the luminaries: the planets stand at very different distances from the earth, refuting the old calendars' error of placing them on concentric spheres. On eccentric solar motion: the sun's orbit and the earth are not concentric, and the distance between their centers varies over time. On atmospheric refraction: the earth is wrapped in air, and predictions cannot match observation unless refraction is calculated in advance. On new methods of measurement and calculation: celestial observation employs spherical triangles, and computation uses the table of eight trigonometric lines derived from the circle. On basing all reckoning on the ecliptic: observations with the equatorial armillary yield longitudes that do not agree with theory until converted by the ecliptic-equatorial table; only then do predictions match the heavens. On revising all epoch constants: reckoning begins from the zi hour of the jimao day following the winter solstice before Tiancong 2 (wuchen). On true solar terms: the old system used mean terms rather than true terms; this has now been corrected. On true solar inequality: the mean tropical year follows the Shoushi reduction method, then apogee and perigee corrections yield the true year. On gnomon measurement at the equinoxes: measuring only the winter solstice, as the old method did, is inferior; using both equinoxes yields greater precision than relying on the solstices alone. On times of sunrise, sunset, and twilight: tables are computed from Beijing, with corrections for each locality. On unequal day and night: the difference amounts to somewhat more than one quarter-hour, partly because the sun moves more slowly in summer and faster in winter along the ecliptic, and partly because the ecliptic and equator differ in width and separation, altering the minutes per degree. On revising the time system: day and night are each divided into 96 quarters (ke). On intercalation: the old practice of inserting leap months by mean solar terms was incorrect; intercalation now follows the sun's actual longitude. On lunar corrections: at new and full moon only one equation applies, but on other days two or three additional corrections are needed, each of varying magnitude. On lunar anomaly: the moon moves slowest at apogee and fastest at perigee; the five planets obey the same principle. On the timing of first visibility after new moon: it depends on the moon's anomalistic motion, on the obliquity and inclination of the ecliptic, and on whether the lunar path lies north or south of the ecliptic. On node-motion corrections: when the moon is at a node, mean-motion calculation will not suffice, so an equation for node motion is introduced. On lunar latitude: the old system fixed the separation of the ecliptic and lunar path at five degrees, unaware that latitude varies beyond new and full moon; the maximum separation is five and one-third degrees. On whether an eclipse occurs: for a lunar eclipse, compare the moon's latitude with the sum of the semidiameters of the moon and earth's shadow; for a solar eclipse, compare the separation of sun and moon with the sum of their semidiameters. If the separation is less, an eclipse occurs; if greater, it does not. On differing eclipse limits: a lunar eclipse requires the moon's orbit and the earth's shadow to be tangent; compare their apparent semidiameters with the moon's latitude and derive the nodal arc to determine the limit. For solar eclipses, parallax must be included before the separation can be found. On eclipse magnitude: for lunar eclipses the relevant separation is the true distance between the centers of the moon and earth's shadow; for solar eclipses it is the apparent separation of the sun and moon, not their true distance. On true and central eclipse: when sun, moon, or planets all lie on the line from the earth's center to the ecliptic, a true eclipse occurs; a central eclipse occurs when the line from the earth's center meets the ecliptic and both orbital centers lie upon it. On apparent eclipse: every solar eclipse has both a true (geocentric) phase and an apparent phase as seen from the earth, the latter reckoned from the observer's eye and the local horizon. On the ninety-degree point of the ecliptic as the central limit for east-west parallax: at the zenith, altitude correction is vertical and north-south correction oblique; only along the central meridian is east-west correction vertical, elsewhere it is oblique. In geometric terms, north-south parallax is the base, east-west parallax the perpendicular, and altitude parallax the hypotenuse. At the central limit the base and hypotenuse coincide, and the perpendicular vanishes. On the three parallaxes: form a triangle with the earth's radius, the body's distance from earth, and its altitude to obtain altitude parallax; derive north-south parallax from its declination; and east-west parallax from its position east or west of the ecliptic's central meridian. On the outer three corrections: east-west, north-south, and altitude parallax all depend on the earth's size; three further corrections arise not from the earth's diameter but from the atmosphere. First, refraction in clear air; second, apparent enlargement of the disk through the atmosphere; third, correction for the orbital radius. On unequal times of first and last contact: the old notion of dividing the eclipse duration in half is wrong; computing by apparent motion shows why these times differ. On local eclipse calculation: each place computes eclipse times and magnitude according to its own longitude and latitude. On anomalous solar eclipses: when theory predicts an eclipse but none is seen, north-south parallax must have shifted into east-west parallax at that place and time—an event that may occur once or twice in a millennium. On forward projection and backward verification: the new tables permit instant lookup in either direction without laborious calculation. On planetary motion modeled on the sun: the old method's day-counts for conjunction and visibility varied inconsistently because they relied on tabular sections alone; this has been corrected. On precise prediction of planetary visibility: the old method relied only on ecliptic longitude, which was inadequate; this too has been corrected. On planetary latitude: the moon's orbit crosses the ecliptic obliquely, producing latitude and the yin-yang lunar tables; the five planets follow the same principle, which the new methods calculate in full detail—something the old system could not achieve. On the visibility of Venus and Mercury: Venus may conjoin the sun without disappearing from view, and Mercury may stand apart from the sun yet remain invisible—a fact revealed at once by the armillary sphere, beyond the reach of older methods. On measuring the five planets: fixed stars must serve as the reference points. On the eastward drift of fixed stars: they are measured from the ecliptic pole, so each lodge's determinant star appears sometimes nearer and sometimes farther from the equatorial pole, because the ecliptic and equatorial poles differ—not because the stars themselves change course or position. On stellar magnitudes ranked in six grades—a classification unknown before. On the Milky Way: observation through the telescope shows it to be an aggregation of innumerable faint stars. On revising the four residuals: Rahu is the ascending node of the lunar path, Ketu the descending node, and Yuebo the moon's apogee. As for the residual Ziqi, no parameters can be determined; the Ming History forced a connection with it—all such entries have now been removed. On instruments: the new methods added the quadrant, the universal instrument, the horizon circle, the cross-staff, the celestial ring and globe, the simplified armillary sphere, and the full ecliptic-equatorial sphere; above all, the newly made telescope became indispensable for stellar observation. On sundials: the horizon dial, triple dial, universal dial, and light-passing dial. Star and moon dials were also added for nighttime observation. Schall's exposition chiefly used the new methods to expose the shortcomings of older systems, yet the essential substance of the new astronomy is fully summarized here.
8
十四年四月，前回回科秋官正吳明炫言：「臣祖默河亦裏等十八姓，本西域人，自隋開皇己未抱其學重譯來朝，授為日官。 一千五十九年，專司星宿行度吉凶，每年推算太陰五星淩犯，天象占驗，日月交食，即以臣科白本進呈為定例。 順治三年，本監掌印湯若望令臣科凡日月交食及太陰五星淩犯、天象占驗俱不必奏進。 臣查若望所推七政，水星二、八月皆伏不見，今水星二月二十九日仍見東方，又八月二十四日夕見，關係象占，不敢不據實上聞。 乞復立回回科，以存絕學。」 奏下所司。 時新安衛官生楊光先叩閽進摘謬論，糾湯若望新法之謬，且言：「時憲書有『依西洋新法』五字尤不合。」 又進選擇議，糾若望選榮親王葬期用洪範五行，山向、日月俱犯忌殺。
In the fourth month of Shunzhi 14, Wu Mingxuan, former director of the Islamic astronomical section, reported: "My ancestor Mohe Yili and seventeen other clans came from the Western Regions. In Kaihuang jiwei of the Sui they brought their learning to court and were appointed calendrical officials. For 1,059 years our section has calculated the motions of the luminaries, planetary conjunctions, celestial omens, and eclipses, submitting annual reports as established precedent. In Shunzhi 3, Schall as director ordered that our section need no longer submit reports on eclipses, planetary conjunctions, or celestial omens. I find that by Schall's calculations Mercury should be invisible in the second and eighth months, yet on the twenty-ninth day of the second month it was still visible in the east, and on the evening of the twenty-fourth day of the eighth month it appeared again—matters bearing on celestial omens that I must report faithfully. I beg that the Islamic section be restored to preserve this endangered tradition." The memorial was referred to the appropriate offices. At this time Yang Guangxian, a student of the Directorate from Xin'an, presented his Extracted Errors impeaching Schall's new methods, and objected that the phrase "according to Western new methods" on the Shixian almanac was especially improper. He also submitted a Selection Discourse charging that Schall, in choosing the burial date for Prince Rong by the Hongfan Five Phases, had violated taboos in mountain orientation, sun, and moon.
9
康熙三年十二月，禮部議「時憲書麵『依西洋新法』五字擬改『奏準』二字」，從之。 四年，議政王等言：每日百刻，新法改為九十六刻； 二十八宿次序，湯若望將觜、參二宿改易前後； 又將四餘刪去紫炁，俱不合。 其選擇不用正五行，用洪範五行，以致山向、日月俱犯忌殺，事犯重大，將湯若望及科官等分別擬淩遲斬決。 敕湯若望從寬免死，時憲科李祖白等五人俱處斬。 於是復用大統舊術，以楊光先掌監務，光先抗疏屢辭，不允。 光先於推步之學本不深，唐熙七年，謂明年當閏十二月，尋知其誤，自行檢舉，而時憲書已頒行，乃諭天下停止閏月雲。 是年監副吳明烜言：「古法差謬，五官正暨回回科所進各不同，立加校正。」 下禮部議。 禮部覆奏：「五官正戈繼文等所算七政金水二星差錯太甚，主簿陳聿新所推七政未經測驗，亦有差錯，監副吳明烜所推七政與天象相近，理應頒行，仍令監臣同四科官，每日晝測晷景以定節氣，夜測月五星以定行度。」 從之。
In the twelfth month of Kangxi 3 the Ministry of Rites proposed replacing the five characters "according to Western new methods" on the almanac cover with the two characters "memorial approved," and the court agreed. In Kangxi 4 the Prince Regents reported: the old division of the day into one hundred quarters was changed to ninety-six by the new methods; the order of the twenty-eight lodges was altered when Schall reversed the positions of Zi and Shen; and Ziqi was removed from the four residuals—all improper changes. He had used the Hongfan Five Phases rather than the orthodox system in selecting the burial date, violating taboos in mountain orientation, sun, and moon—a grave offense for which Schall and the section officials were sentenced to execution. Schall was spared by imperial grace, but five Shixian-section officials including Li Zubai were executed. The old Datong system was restored and Yang Guangxian placed in charge of the Directorate; he repeatedly submitted memorials declining the post, but was not permitted to withdraw. Guangxian was never deeply versed in astronomical calculation. In Kangxi 7 he declared that the following year required an intercalary twelfth month, then discovered his error and reported it himself—but the almanac had already been issued, and the throne ordered the intercalary month cancelled throughout the empire. That year Deputy Director Wu Mingxuan reported: "The restored methods are in error, and the calculations submitted by the Five Offices and the Islamic section disagree; corrections must be made immediately." The matter was referred to the Ministry of Rites. The Ministry replied: "Director Ge Jiwen's calculations of Venus and Mercury are grossly in error; Registrar Chen Yuxin's figures have not been verified and are also wrong; Deputy Director Wu Mingxuan's results accord with observation and should be adopted. Directorate officials and the four sections should measure gnomon shadows by day to fix the solar terms and observe the moon and planets by night to determine their motions." The court approved.
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十一月，西人南懷仁言所頒各法不合天象，乃召南懷仁、利類思、安文思及監官馬祐、楊光先、吳明烜等至東華門，大學士李霨傳諭：「授時乃國家要政，爾等勿挾宿仇，以己為是，以彼為非。 是者當遵用，非者當更改，務期歸於至善。」 十二月，南懷仁劾吳明烜所造康熙八年七政時憲書糾謬，下王大臣、九卿、科道會議，議政王等言：「乞派大臣同南懷仁等測驗。」 乃遣圖海、李霨等二十人赴觀象台測驗。 八年二月，議政王等議覆：「圖海等赴觀象台測驗，南懷仁所言皆合，吳明烜所言皆謬，問監正馬祐等，亦言南懷仁所算實與天象合。 竊思百刻雖行之已久，但南懷仁九十六刻之法既合天象，自應頒用。 又南懷仁言羅睺、計都、月孛係推算所用，故載於七政之後，其紫炁星無用處，不應造入。 應自康熙九年為始，用九十六刻之曆」，時明烜言「臣祗知天文，不知曆法」，光先言「臣不知曆法，惟知曆理」。 光先語尤不遜，褫職。 三月，授南懷仁欽天監監副。 先是監官依古法推算，康熙八年十二月應置閏，南懷仁言雨水為正月中氣，是月二十九日值雨水，即為康熙九年正月，不應置閏，置閏當在明年二月。 監官多直懷仁，從其言，改閏九年二月，於是大統、回回兩法俱廢，專用西洋法，如順治之初。 八月，南懷仁劾楊光先誣陷湯若望叛逆，議政王等議「湯若望應復通微教師，照原品賜恤，楊光先應反坐」。 敕「免議」。
In the eleventh month Verbiest reported that the issued methods did not match the heavens. Verbiest, de Tournon, de Magalhaes, and Directorate officials including Ma You, Yang Guangxian, and Wu Mingxuan were summoned to the Donghua Gate, where Grand Secretary Li Ji conveyed the emperor's words: "Calendar reform is a vital state affair; do not let old grudges lead you to insist you alone are right. What is correct should be adopted; what is wrong should be revised, until the system reaches perfection." In the twelfth month Verbiest impeached errors in Wu Mingxuan's Shixian almanac for Kangxi 8. The case was referred to princes, ministers, the Nine Chief Ministers, and the censorate; the Prince Regents proposed that senior officials test the rival methods alongside Verbiest. Tu Hai, Li Ji, and twenty others were dispatched to the Observatory to conduct the tests. In the second month of Kangxi 8 the Prince Regents reported: "At the Observatory, Verbiest's predictions matched observation in every case and Wu Mingxuan's in none; Director Ma You and his colleagues confirmed that Verbiest's calculations accorded with the heavens. Although the hundred-quarter day had long been in use, Verbiest's ninety-six-quarter system matched the heavens and should be adopted. Verbiest also explained that Rahu, Ketu, and Yuebo are computational points listed after the seven luminaries, whereas the star Ziqi serves no purpose and should be omitted. The ninety-six-quarter calendar was to take effect from Kangxi 9. Mingxuan said, "I know astronomy but not calendar-making"; Guangxian said, "I know calendrical principle but not calendar-making." Guangxian's language was especially disrespectful, and he was dismissed. In the third month Verbiest was appointed deputy director of the Astronomical Directorate. Directorate officials using the old methods had placed an intercalary month in the twelfth month of Kangxi 8. Verbiest argued that rain-water, the mid-term of the first month, fell on the twenty-ninth day of that month, making it already the first month of Kangxi 9; intercalation belonged in the second month of the following year. Most officials supported Verbiest, intercalation was set for the second month of Kangxi 9, and both the Datong and Islamic systems were abandoned in favor of Western methods, as in early Shunzhi. In the eighth month Verbiest impeached Yang Guangxian for falsely accusing Schall of treason. The Prince Regents proposed restoring Schall's title and honors and imposing counter-penalties on Yang Guangxian. The emperor ordered the matter dropped.
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十三年二月，新造儀象誌告成，南懷仁加太常寺卿銜。 十四年二月，諭監副安泰從何君錫學古曆法。 十五年二月，欽天監奏五月朔日食，監副安泰依古法算，應食五分六十秒，南懷仁新法隻應食二十微三分秒之一。 至期登台測驗，酉正食甚，將及一分，戌初刻復圓，古法所推分數失之甚遠，而新法亦不甚合。 南懷仁曰：「此清蒙氣之所為，蒙氣能映小為大故也。」
In the second month of Kangxi 13 the new Instruments and Observations was completed, and Verbiest received the additional title of Vice Minister of the Court of Imperial Sacrifices. In the second month of Kangxi 14 the emperor ordered Deputy Director An Tai to study the ancient calendar under He Junxi. In the second month of Kangxi 15 the Directorate reported a solar eclipse on the new moon of the fifth month. By ancient methods An Tai predicted 5 parts 60 seconds; Verbiest's new method predicted only 20 microseconds and one-third of a second. On the day of the eclipse they observed from the terrace: greatest eclipse at you-zheng, nearly one part of the sun's disk, ending at the start of the xu quarter. The ancient prediction was far wrong, and the new method also fell somewhat short. Verbiest explained: "Clear atmospheric refraction magnifies small objects—that is why the eclipse appeared larger than calculated."
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十七年七月，欽天監進呈康熙永年表三十二卷。 二十二年十月，監臣推算盛京九十度表告成。 初，南懷仁奏：「各省北極高度不同，其交合之時刻食分俱不等，全憑各省之九十度表推算。 向來不知盛京北極高度，即用京師之九十度表，今測得盛京北極比京師高二度，請依其高度推算九十度表。」 從之。 至是，以盛京九十度表進呈，諭「永遠遵守」雲。
In the seventh month of Kangxi 17 the Directorate submitted the Kangxi Perpetual Tables in thirty-two volumes. In the tenth month of Kangxi 22 the ninety-degree table for Mukden was completed. Verbiest had earlier reported: "Each province's north polar altitude differs, so eclipse times and magnitudes differ and must be calculated from local ninety-degree tables. Previously Mukden's latitude was unknown and the capital's table was used; measurement now shows Mukden's north pole is two degrees higher than Beijing's—please compute a table for that latitude." The court approved. The Mukden table was then submitted with orders to observe it permanently.
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四十一年十月，大學士李光地以宣城貢生梅文鼎曆學疑問三卷進呈，上曰：「朕留心曆算多年，此事朕能決其是非。」 乃親加批點還之，事具梅文鼎傳。 文鼎論中、西二法之同異曰：「今之用新曆也，乃兼用其長，以補舊法之未備，非盡廢古法而從新法術也。 夫西曆之同乎中法者，不止一端。 其言五星之最高加減也，即中法之盈縮曆也，在太陰，則遲疾曆也。 其言五星之歲輪也，即中法之段目也。 其言恒星東行也，即中法之歲差也。 其言節氣之以日躔過宮也，即中法之定氣也。 其言各省真節氣不同也，即中法之裏差也。 但中法言盈縮遲疾，而西說以最高最卑明其故； 中法言段目，而西說以歲輪明其故； 中法言歲差，而西說以恒星東行明其故。 是則中曆所言者當然之運，而西曆所推者其所以然之理，此其可取者也。 若夫定氣裏差，中曆原有其法，但不以法曆耳，非古無而今始有也。 西曆始有者，則五星之緯度是也。 中曆之緯度，惟太陽、太陰有之，而五星則未有及之者。 今西曆之五星有交點、有緯行，亦如太陽太陰之詳明，是則中曆缺陷之大端，得西法以補其未備矣。 夫於中法之同，亦既有以明其所以然之故，而於中法之未備者，又有以補其缺，於是吾之積候者，得彼說而益信，而彼說之若難信者，亦因吾之積候而有以知其不誣，雖聖人復起，亦在所兼收而並取矣。」
In the tenth month of Kangxi 41 Grand Secretary Li Guangdi presented Mei Wending's three-volume Calendrical Questions; the emperor said, "I have studied astronomy for many years and can judge these matters myself." He annotated the work in his own hand and returned it, as recorded in Mei's biography. Mei wrote on the relation of Chinese and Western astronomy: "The new calendar today draws on both traditions to supplement what the old methods lacked—it does not simply discard native methods for Western ones. Western astronomy agrees with Chinese methods in more than one respect. Its treatment of planetary apogee correction corresponds to the Chinese expansion-contraction tables; for the moon, to the slow-swift tables. Its annual circles for the five planets correspond to the Chinese section tables. Its eastward drift of fixed stars is the Chinese precession. Its solar terms defined by the sun's lodge passage are the Chinese true solar terms. Its provincial variation in true solar terms is the Chinese li difference. Chinese methods record expansion, contraction, slow, and swift motion; Western theory explains these by apogee and perigee; Chinese methods use section tables; Western theory explains them through annual circles; Chinese methods record precession; Western theory explains it through the stars' eastward drift. Chinese astronomy describes what happens; Western astronomy explains why—therein lies its value. True solar terms and li difference were already known in Chinese astronomy but not always applied in official calendars—not inventions of the West. What Western astronomy first supplied is planetary latitude. Chinese methods calculated latitude only for the sun and moon, not yet for the five planets. Western methods now give the five planets nodes and latitude as fully as for the sun and moon, filling the greatest gap in Chinese astronomy. Where the traditions agree, Western theory clarifies the reasons; where Chinese methods were incomplete, Western methods supply the lack. Long observation confirms Western theory, and Western claims once doubted are vindicated by Chinese records. A sage would embrace both."
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五十年十月，上諭大學士等：「天文曆法，朕素留心，西法大端不誤，但分刻度數之間，積久不能無差。 今年夏至，欽天監奏午正三刻，朕細測日景，是午初三刻九分。 此時稍有舛錯，恐數十年後所差愈甚。 猶之錢糧，微塵秒忽，雖屬無幾，而總計之，便積少成多，此事實有證驗，非比書生論說可以虛詞塞責也。」 又諭禮部考取效力算法人員，臨軒親試，取顧琮等四十二人。 五十一年五月，駕幸避暑山莊，徵梅文鼎之孫梅瑴成詣行在。 先是命蘇州府教授陳厚耀，欽天監五官正何君錫之子何國柱、國宗，官學生明安圖，原任欽天監監副成德，皆扈從侍直，上親臨提命，許其問難如師弟子。 及徵瑴成至，奏對稱旨，遂與厚耀等同直內廷。 五十二年五月，修律呂、算法諸書，以誠親王允祉、皇十五子允烜、皇十六子允祿充承旨纂修，何國宗、梅瑴成充彙編，陳厚耀、魏廷珍、王蘭生、方苞等充分校。 所纂之書，每日進呈，上親加改正焉。
In the tenth month of Kangxi 50 the emperor told the grand secretaries: "I have long studied astronomy; Western methods are sound in principle, but minute errors accumulate over time. At this summer solstice the Directorate reported noon at the third quarter of wu-zheng; my own measurement of the sun's shadow showed the third quarter of wu-initial plus nine minutes. The error is already small, but in decades it will grow substantially. Like accounts in which tiny sums eventually mount up, this is verified by measurement, not empty scholarly debate." He also ordered the Ministry of Rites to examine candidates skilled in mathematics and test them at court; forty-two including Gu Cong were chosen. In the fifth month of Kangxi 51, while at the Summer Resort, the emperor summoned Mei Juecheng, grandson of Mei Wending. He had already summoned Chen Houyao, the He brothers, Ming Antu, and Cheng De to attend him, instructing them personally and permitting discussion as between teacher and pupils. When Juecheng arrived his answers pleased the emperor, and he joined Chen Houyao and the others in service within the palace. In the fifth month of Kangxi 52 works on pitch pipes and mathematics were compiled under Princes Yunzhi, Yunxuan, and Yunlu, with He Guozong and Mei Juecheng as collators and Chen Houyao, Wei Tingzhen, Wang Lansheng, and Fang Bao as proofreaders. Each day's compilation was presented to the emperor, who corrected it in his own hand.
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五十三年四月，諭誠親王允祉等：「古曆規模甚好，但其數目歲久不合，今修書宜依古曆規模，用今之數目算之。」 十月，又諭：「北極高度、黃赤距度最為緊要，著於澹寧居後逐日測量。」 乃製象限儀，儀徑五尺，範銅為之，晝測日度，夜測勾陳帝星。 又製中表、正表、倒表各二，俱高四尺，中表測日中心，正表、倒表測日上下邊之景。 惟六表所得日景尾數多參差不合。 海瑴成言：「表高景澹，尾數難真，自古患之。 昔郭守敬為銅表，端挾二龍，舉橫梁至四十尺，因其景虛澹，創為景符以取實影。 其制以銅葉博二寸，長加博之二，中穿一竅若針芥然，以方木為趺，一端設機軸，令可開闔。 稽其一端，使其針斜倚，北高南下，往來遷就於虛影之中。 竅達日光，僅如黍米，隱然見橫梁於其中。」 乃仿元史郭守敬製造景符六，如法用之，影尾數始毫末不爽。 測得申昜春園北極高三十九度五十九分三十秒，比京師觀象台高四分三十秒，黃赤大距二十三度二十九分，比舊測減二分雲。 十一月，誠親王允祉等言：「郭守敬造授時術，遣人二十七處分測，故能密合。 今除申昜春園及觀象台逐日測驗外，如福建、廣東、雲南、四川、陝西、河南、江南、浙江八省，於裏差尤為較著，請遣人逐日測量，得其真數，庶幾東西南北裏差及日天半徑，皆有實據。」 從之。
In the fourth month of Kangxi 53 the emperor instructed the compilers: "The ancient calendar's framework is excellent, but its constants have drifted; follow the ancient framework but use modern numbers." In the tenth month he ordered daily measurement of north polar altitude and ecliptic-equatorial obliquity behind the Danning Studio. A quadrant five feet in diameter was cast in bronze to measure the sun by day and the Pole Star and Imperial Star by night. Central, upright, and inverted gnomons six feet high were also made—the central measuring the sun's center, the upright and inverted its upper and lower limbs. But the tail figures from the six gnomons showed many discrepancies. Mei Juecheng observed: "When the gnomon is tall the shadow grows faint and precise tail readings are difficult—an ancient problem. Guo Shoujing had raised a bronze gnomon to forty feet with a crossbeam held by dragon figures; because the shadow was diffuse, he invented the shadow-marker to capture the true image. It was a copper plate two inches wide and twice as long, pierced by a pinhole; mounted on a pivoted wooden stand. One end was tilted northward so the hole could be adjusted within the diffuse shadow. Sunlight through the hole, no larger than a millet grain, projected the crossbeam's image." Six shadow-markers were made on this model, and the tail readings finally agreed to the finest detail. At Shenyang Spring Garden the north polar altitude was found to be 39°59′30″, 4′30″ higher than Beijing; ecliptic obliquity 23°29′, two minutes less than the previous value. In the eleventh month Prince Yunzhi reported: "Guo Shoujing's Shoushi calendar relied on measurements from twenty-seven locations—that is how he achieved such precision. Beyond daily measurements at Shenyang and the Observatory, the eight provinces Fujian, Guangdong, Yunnan, Sichuan, Shaanxi, Henan, Jiangnan, and Zhejiang—where li difference is especially pronounced—should receive daily observers so that local li differences and the sun's semidiameter may be established on solid evidence." The court approved.
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五十八年二月，以推算人不敷用，敕禮部錄送蒙養齋考試，取傅明安等二十八人，命在修書處行走。 六十年，御製算法書成，賜名數理精蘊。 諭：「此書賜梅文鼎一部，命悉心校對。」 遣其孫梅瑴成齎書賜之。 六十一年六月，曆書稿成，並律呂、算法，共為律曆淵源一百卷：一曰曆象考成上、下編，一曰律呂精義上、下編，續編，一曰數理精蘊上、下編。 雍正元年，頒曆象考成於欽天監，是為康熙甲子元法。 自雍正四年為始，造時憲書一遵曆象考成之法。 又議準其御製之書，無庸欽天監治理，其治曆法之西洋人授為監正。 八年六月，監正明安圖言：「日月行度，積久漸差，法須旋改，始能密合。 臣等遵御製曆象考成推算時憲，據監正戴進賢、監副徐懋德推測，覺有微差。 於本月初一日日食，臣等公同測驗，實測與推算分數不合，乞敕下戴進賢、徐懋德詳加校定修理。」 從之。 十年四月，修日躔、月離表成。
In the second month of Kangxi 58, with too few calculators available, the Ministry of Rites was ordered to assign twenty-eight candidates including Fu Ming'an who had passed the Mengyang Studio examination to the compilation office. In Kangxi 60 the emperor's work on mathematics was completed and titled Essence of Mathematical Principles. The emperor ordered a copy sent to Mei Wending for careful collation. Mei Juecheng was dispatched to deliver it. In the sixth month of Kangxi 61 the calendrical manuscripts were finished, together with works on pitch pipes and mathematics, as the hundred-volume Origins of Pitch and Calendar: Calendrical and Astronomical Investigations Completed in two parts, Essence of Pitch Pipes in three parts, and Essence of Mathematical Principles in two parts. In Yongzheng 1 Calendrical and Astronomical Investigations Completed was issued to the Directorate as the Kangxi jiazi-epoch system. From Yongzheng 4 onward the Shixian almanac followed that work exclusively. The emperor's own works were exempted from Directorate administration, and Western experts in astronomy were appointed directors. In the sixth month of Yongzheng 8 Director Ming Antu reported: "Solar and lunar motions drift over time; the methods must be periodically revised to maintain precision. We calculate the Shixian calendar by the imperial Calendrical and Astronomical Investigations Completed, but projections by Dai Jinxian and Xu Maode reveal slight discrepancies. At this month's solar eclipse our joint observations did not match the predictions; we ask that Dai Jinxian and Xu Maode undertake detailed revision." The court approved. In the fourth month of Yongzheng 10 the solar and lunar motion tables were completed.
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乾隆二年四月，協辦吏部尚書事顧琮言：「世宗皇帝允監臣言，請纂修日躔、月離二表，以推日月交合，並交宮過度，晦朔弦望，晝夜永短，以及淩犯，共三十九頁，續於曆象考成諸表之末。 查造此表者，監正西洋人戴進賢； 能用此表者，監副西洋人徐懋德與五官正明安圖。 擬令戴進賢為總裁，徐懋德、明安圖為副總裁，盡心考驗，增補圖說。 曆象考成內倘有酌改之處，亦令其悉心改正。」 敕：「即著顧琮專管。」 五月，琮復言：「乞命梅瑴成為總裁，何國宗協同總裁。」 從之。 十一月，命莊親王允祿為總理。
In the fourth month of Qianlong 2 Gu Cong reported that the Yongzheng Emperor had ordered solar and lunar tables for eclipses, lodge passage, phases, day-length, and planetary conjunctions—thirty-nine pages appended to Calendrical and Astronomical Investigations Completed. Dai Jinxian, the Western director, had compiled them; Xu Maode, the Western deputy director, and Ming Antu of the Five Offices could use them. Dai Jinxian was proposed as chief compiler with Xu Maode and Ming Antu as deputies to verify and supplement the diagrams. They were also to revise the parent work wherever needed." The emperor ordered Gu Cong to take charge. In the fifth month Gu proposed Mei Juecheng as chief compiler with He Guozong as co-chief. The court approved. In the eleventh month Prince Zhuang Yunlu was appointed general supervisor.
18
三年四月，莊親王允祿等言：「曆象考成一書，其數惟黃赤大距減少二分，餘皆仍新法算書西人第穀之舊。 康熙中西人有噶西尼、法蘭德等，發第穀未盡之義，其大端有三：其一謂太陽地半徑差，舊定為三分，今測祗有十秒； 其一謂清蒙氣差，舊定地平上為三十四分，高四十五度，祗有五秒，今測地平上止三十二分，高四十五度，尚有五十九秒； 其一謂日月五星之本天非平圓，皆為橢圓，兩端徑長，兩腰徑短。 以是三者，經緯度俱有微差。 戴進賢等習知其說，因未經徵驗，不敢遽以為是。 雍正八年六月朔日食，舊法推得九分二十二秒，今法推得八分十秒，驗諸實測，今法為近。 故奏準重修日躔、月離新表二差，以續於曆象考成之後。 臣等奉命增修表解圖說，以日躔新表推算，春分比前遲十三刻許，秋分比前早九刻許，冬夏至皆遲二刻許。 然以測午正日高，惟冬至比前高二分餘，夏至秋分僅差二三十秒。 蓋測量在地面，而推算則以地心，今所定地半徑差與蒙氣差皆與前不同，故推算每差數刻，而測量終不甚相遠也。 至其立法以本天為橢圓，雖推算較繁，而損益舊數以合天行，頗為新巧。 臣等闡明理數，著日躔九篇並表數，乞親加裁定，附曆象考成之後，顏曰御製後編。 凡前書已發明者，不復贅述。」 報聞。 七年，莊親王允祿等奏進日躔、月離、交宮共書十卷，是為雍正癸卯元法。
In the fourth month of Qianlong 3 Prince Zhuang reported: "In Calendrical and Astronomical Investigations Completed only the ecliptic obliquity differs, reduced by two minutes; the rest still follows Tycho Brahe. By Kangxi's reign Cassini and Flamsteed had advanced beyond Tycho on three points: solar parallax, formerly three minutes, is now measured at only ten seconds; atmospheric refraction, formerly thirty-four minutes at the horizon and five seconds at forty-five degrees, is now thirty-two minutes at the horizon and fifty-nine seconds at forty-five degrees; and the primary orbits of the sun, moon, and planets are elliptical, not circular. These three revisions produce slight changes in longitude and latitude. Dai Jinxian and his colleagues understood these theories but hesitated to adopt them without verification. The solar eclipse of Yongzheng 8, sixth month, sixth day: the old method predicted 9 parts 22 seconds, the new 8 parts 10 seconds; observation favored the new method. They therefore proposed revising the solar and lunar tables to continue Calendrical and Astronomical Investigations Completed. Ordered to expand the explanations, they found the new solar table placed spring equinox thirteen quarters later, autumn equinox nine quarters earlier, and both solstices two quarters later than before. Noon altitude measurements showed only winter solstice more than two minutes higher; summer solstice and autumn equinox differed by mere seconds. Because observation is from the surface but calculation from the earth's center, and parallax and refraction constants have changed, calculated times differ by quarters while measured altitudes scarcely change. Modeling orbits as ellipses complicates calculation but adjusts constants more ingeniously to match the heavens. They submitted nine treatises with tables as an Imperially Composed Sequel to Calendrical and Astronomical Investigations Completed. Points already covered in the earlier work were not repeated." The memorial was noted. In Qianlong 7 Prince Zhuang submitted ten volumes on solar motion, lunar motion, and lodge passage—the Yongzheng guimao-epoch system.
19
九年十月，監正戴進賢等言：「靈台儀象誌原載星辰約七十年差一度，為時已久，宜改定。 康熙十三年修誌之時，黃赤大距與今測不同，所列諸表，當逐一增修。 三垣二十八宿以及諸星，今昔多寡不同，亦應釐訂。」 敕莊親王、鄂爾泰、張照議奏。 十一月，議準仍以三人兼管。 是年更定羅睺、計都名目，又增入紫炁為四餘。 十七年，莊親王允祿等言儀象誌所載之星，多不順序，今依次改正，共成書三十卷，賜名儀象考成。 是月莊親王等復奏改正恒星經緯度表，並更定二十八宿值日觜參之前後。 敕大學士會同九卿議奏。 十二月，大學士傅恒等言：「請以乾隆十九年為始，時憲書之值宿，改觜前參後。」 從之。 既而欽天監又以推算土星有差減平行三十分，自乾隆以後至道光初，交食分秒漸與原推不合。
In the tenth month of Qianlong 9 Dai Jinxian reported that the seventy-year precession rate in the Observatory Treatise needed revision. The ecliptic obliquity at compilation in Kangxi 13 differed from current measurement; all tables required revision. The star catalogs for the Three Enclosures and twenty-eight lodges also needed updating. Prince Zhuang, Ortai, and Zhang Zhao were ordered to deliberate. In the eleventh month the three were confirmed as supervisors. That year Rahu and Ketu were renamed and Ziqi restored as a fourth residual. In Qianlong 17 Prince Zhuang reported that stars in the Instruments Treatise were reordered into thirty volumes titled Instruments Investigations Completed. That month they also revised the fixed-star table and lodge duty-day order, placing Zi before Shen. Grand secretaries and the Nine Chief Ministers were ordered to deliberate. In the twelfth month Fu Heng proposed adopting Zi-before-Shen lodge order from Qianlong 19. The court approved. Later Saturn's mean motion was reduced thirty minutes for discrepancies; from Qianlong to early Daoguang eclipse predictions gradually drifted.
20
道光十八年八月，管理欽天監事務工部尚書敬徵言：「自道光四年臣管理監務，查觀象台儀器，康熙十三年所製黃赤大距，皆為二十三度三十二分。 至乾隆九年重製璣衡撫辰儀，所測黃赤大距，則為二十三度二十九分，是原設諸儀已與天行不合，今又將百年，即撫辰儀亦有差失。 臣將撫辰儀更換軸心，諸儀亦量為安置。 另製小象限儀一，令官生晝測日行，夜測月星，每逢節氣交食，所測實數有與推算不合者，詳加考驗。 知由太陽緯度不合之數，測得黃赤大距較前稍小，其數僅二十三度二十七分。 由交節時刻之早晚，考知太陽行度有進退不齊之分。 夫太陽行度為推測之本，諸曜宗之。 而推日行，又以歲實、氣應兩心差曰本天最卑行度為據。 擬自道光十四年甲午為年根，按實測之數，將原用數稍為損益，推得日行交節時刻，似與實測之數較近。 至太陰行度，以交食為考驗之大端。 近年測過之月食，較原推早者多，遲者少。 故於月之平行、自行、交行內量為損益，按現擬之平行，仍用諸均之舊數，推得道光十四年後月食三次。 除十七年三月祗見初虧，九月天陰未測，僅測得道光十六年九月十五日月食，與新數所推相近，然僅食一次，尚未可憑，仍須隨時考驗。 現屆本年八月十五日月食，謹將新擬用數推算得時刻食分方位，比較原推早見分秒，另繕清單進呈。 至期臣等逐時測驗，再行據實具奏。」 報聞。
In the eighth month of Daoguang 18 Jing Zheng, overseeing the Directorate, reported that instruments from Kangxi 13 still showed an obliquity of 23°32′. The Jiyun armillary remade in Qianlong 9 measured 23°29′, proving the old instruments obsolete; after another century even the Jiyun instrument had drifted. Jing replaced the Jiyun armillary's axis and adjusted the other instruments. A small quadrant was added for daily solar and nightly lunar and stellar observation, with verification whenever predictions failed. Solar latitude discrepancies indicated an obliquity of only 23°27′. Nodal timing showed uneven solar motion. Solar motion is the foundation of all astronomical calculation. Solar motion depends on the tropical year and the perigee motion of the primary orbit. Taking Daoguang 14 jiawu as epoch and adjusting constants to observation, nodal times improved. Lunar motion is verified chiefly through eclipses. Recent lunar eclipses had mostly occurred earlier than predicted. Lunar mean, true, and node motions were adjusted accordingly, and three post-Daoguang 14 eclipses were projected. Only the eclipse of Daoguang 16, ninth month, fifteenth day, had been fully observed and matched the new constants; one case was insufficient, and further testing was required. For the upcoming eighth-month eclipse they submitted predictions by the new constants, earlier than the old method. They would observe and report when the day arrived." The memorial was noted.
21
二十二年六月，敬徵等又言：「每屆日月交食，按新擬用數推算，俱與實測相近。 至本年六月朔日食，新推較之實測，僅差數秒。 是新擬之數，於日行已無疑義，月行亦屬近合。 今擬先測恒星，以符運度，繼考日躔、月離，務合天行。 請以道光十四年甲午為元，按新數日行黃赤大距，修恒星、黃赤道經緯度表，即於測算時詳考五緯月行，俾恒星、五緯、日月交食等書，得以次第竣事。」 從之。 是年七月，以敬徵為修曆總裁，監正周餘慶、左監副高煜為副總裁。
In the sixth month of Daoguang 22 Jing Zheng reported that recent eclipses calculated by the new constants had matched observation. The current year's solar eclipse differed from observation by only seconds. Solar motion was now reliable; lunar motion was nearly so. They proposed measuring fixed stars first, then revising solar and lunar motion. Taking Daoguang 14 jiawu as epoch, they sought to revise star tables and complete works on the planets, moon, and eclipses." The court approved. In the seventh month Jing Zheng became chief compiler with Zhou Yuqing and Gao Yu as deputies.
22
二十五年七月，進呈黃道經緯度表、赤道經緯度表各十三卷，月五星相距表一卷，天漢界度表四卷，經星彙考、星首步天歌、恒星總紀各一卷，為儀象考成續編。 至日月交食、五星行度俱闕而未備雲。 時冬官正司廷棟撰淩犯視差新法，用弧三角布算，以限距地高及星距黃極以求黃經高弧三角，較舊法為簡捷。 乾隆以後，曆官能損益舊法，廷棟一人而已。 其不為曆官而知曆者，梅文鼎、薛鳳祚、王錫闡以下，江永、戴震、錢大昕、李善蘭為尤著。 其闡明中、西曆理，實遠出徐光啟、李之藻等之上焉。
In the seventh month of Daoguang 25 they submitted star tables and related works as the sequel to Instruments Investigations Completed. Works on eclipses and planetary motion remained incomplete. Si Tingdong of the Winter Offices devised a spherical-triangle method for encroachment parallax, simpler than the old approach. After Qianlong, only Si Tingdong among official astronomers could improve the methods. Outside official circles, Mei Wending, Xue Fengzuo, Wang Xichan, and later Jiang Yong, Dai Zhen, Qian Daxin, and Li Shanlan were especially distinguished. In explaining the principles of Chinese and Western astronomy, they far surpassed Xu Guangqi and Li Zhizao.