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皇祐渾儀
The Huangyou Armillary Sphere
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堯敕羲和制橫簫以考察星度，其機衡用玉，欲其燥濕不變，運動有常，堅久而不能廢也。 至於後世，鑄銅為圓儀，以法天體。 自洛下閎造《太初曆》，用渾儀，及東漢孝和帝時，太史惟有赤道儀，歲時測候，頗有進退。 帝以問典星待詔姚崇等，皆曰：「星圖有規法，日月實從黃道，今無其器，是以失之。」 至永元十五年，賈逵始設黃道儀。 桓帝延熹七年，張衡更制之，以四分為度。 其後，陸績、王蕃、孔挺、斛蘭、梁令瓚、李淳風並嘗製作。 五代亂亡，遺法蕩然矣。 真宗祥符初，韓顯符作渾儀，但遊儀雙環夾望筒旋轉，而黃、赤道相固不動。 皇祐初，又命日官舒易簡、于淵、周琮等參用淳風、令瓚之制，改鑄黃道渾儀，又為漏刻、圭表，詔翰林學士錢明逸詳其法，內侍麥允言總其工。 既成，置渾儀于翰林天文院之候台，漏刻於文德殿之鐘鼓樓，圭表于司天監。 帝為制《渾儀總要》十卷，論前代得失，已而留中不出。 今具黃道遊儀之法，著於此焉。
Yao charged Xihe with fashioning transverse sight-tubes to measure the stars, and had the mechanisms and pivots made of jade so they would not warp with damp or dryness, would move evenly, and would endure firm and unchanging for long use. In later ages, bronze was cast into a spherical instrument to represent the celestial sphere. From Luoxia Hong's compilation of the Taichu Calendar with the armillary sphere down to the Eastern Han Emperor He, the Grand Astrologer's office possessed only an equatorial instrument, and its seasonal observations drifted noticeably. The emperor asked the awaited-edict astrologers Yao Chong and others, and they all replied: "The star chart follows fixed rules, and the sun and moon truly travel the ecliptic; because we lack the proper instrument today, our readings go astray." In the fifteenth year of Yongyuan (103 CE), Jia Kui first installed an ecliptic instrument. In the seventh year of Yanxi (164 CE) under Emperor Huan, Zhang Heng redesigned it, using quarter-degrees as the unit. Later, Lu Ji, Wang Fan, Kong Ting, Hu Lan, Liang Lingzan, and Li Chunfeng each produced instruments of their own. The turmoil and collapse of the Five Dynasties swept away the transmitted methods entirely. Early in the Xiangfu era under Emperor Zhenzong, Han Xianfu built an armillary sphere in which only the pivoting rings gripping the sight-tube could turn, while the ecliptic and equatorial rings remained fixed. Early in the Huangyou era, the court again ordered the solar officials Shu Yijian, Yu Yuan, and Zhou Cong to draw on the designs of Li Chunfeng and Liang Lingzan, recast an ecliptic armillary sphere, and also build clepsydra graduations and a gnomon table; Hanlin Academician Qian Mingyi was directed to work out the methods, and Inner Attendant Mai Yunyan to supervise construction. When they were finished, the armillary sphere was installed on the observation terrace of the Hanlin Astronomical Bureau, the clepsydra in the bell-and-drum tower of Wende Hall, and the gnomon table at the Directorate of Astronomy. The emperor drafted Essentials of the Armillary Sphere in ten scrolls, reviewing what earlier ages had done well or poorly, but afterward kept the work in the palace archives and never published it. The methods of the ecliptic pivoting instrument are set forth in full below.
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第一重，名六合儀。
The first layer is called the Six Harmonies Instrument.
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陽經雙環：週邊二丈三尺二寸八分，直徑七尺七寸六分，闊六寸，厚六分。 南北並立，兩面各列周天三百六十五度少強，北極出地三十五度少強。
Yang meridian double ring: circumference 2 zhang 3 chi 2 cun 8 fen, diameter 7 chi 7 cun 6 fen, width 6 cun, thickness 6 fen. Set upright north and south, each face marked with the full circuit of 365 and a fraction degrees, and the north celestial pole 35 and a fraction degrees above the horizon.
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陰緯單環：週邊、徑、闊與陽經雙環等，外厚二寸五分，內厚一寸九分。 上列十幹、十二支、八封方位，以正地形。 上有池沿環流轉，以定平准。
Yin latitude single ring: circumference, diameter, and width match the yang meridian double ring; outer thickness 2 cun 5 fen, inner thickness 1 cun 9 fen. It bears the ten heavenly stems, twelve earthly branches, and eight trigram directions to orient the instrument to the ground. A channel rim above circulates water to establish true level.
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天常單環：週邊二丈四寸六分，直徑六尺八寸二分，闊、厚一寸二分。 上列十幹、十二支、四維時刻之數，以測辰刻，與陽經、陰緯環相固，如卵之殼幕然。
Heaven's constant single ring: circumference 2 zhang 4 cun 6 fen, diameter 6 chi 8 cun 2 fen, width and thickness each 1 cun 2 fen. It bears the ten stems, twelve branches, and corner-hour divisions for measuring double-hours and clepsydra marks, fixed to the yang meridian and yin latitude rings like a shell around an egg.
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第二重，名三辰儀。
The second layer is called the Three Luminaries Instrument.
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璿璣雙環：週邊一丈九尺五寸六分，直徑六尺五寸二分，闊一寸四分，厚一寸。 兩面各均周天三百六十五度少強，作二樞對兩極。
Xuanji double ring: circumference 1 zhang 9 chi 5 cun 6 fen, diameter 6 chi 5 cun 2 fen, width 1 cun 4 fen, thickness 1 cun. Each face is evenly graduated for the full circuit of 365 and a fraction degrees, with two pivots set opposite the celestial poles.
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赤道單環：週邊一丈九尺六寸八分，直徑六尺五寸六分，闊一寸一分，厚六分。 上列二十八宿距度、周天三百六十五度少強，附於璿璣之上。
Equatorial single ring: circumference 1 zhang 9 chi 6 cun 8 fen, diameter 6 chi 5 cun 6 fen, width 1 cun 1 fen, thickness 6 fen. It bears the interval degrees of the twenty-eight lunar lodges and the full circuit of 365 and a fraction degrees, mounted on the Xuanji ring.
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黃道單環：週邊一丈九尺二分，直徑六尺三寸四分，闊一寸二分，厚一寸。 上列周天三百六十五度少強，均分二十四氣、七十二候、六十四卦、三百六十策。 出入赤道二十四度，與赤道相交，每歲退差一分有餘。
Ecliptic single ring: circumference 1 zhang 9 chi 2 fen, diameter 6 chi 3 cun 4 fen, width 1 cun 2 fen, thickness 1 cun. It bears the full circuit of 365 and a fraction degrees, evenly divided into the twenty-four solar terms, seventy-two pentads, sixty-four hexagrams, and three hundred sixty yarrow stalks. It inclines twenty-four degrees to the equator and crosses it, precessing backward by more than one degree each year.
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白道單環：週邊一丈八尺六寸三分，直徑六尺二寸一分，闊一寸一分，厚五分。 上列交度，置於黃道環中，入黃道六度，每一交終，退行黃道一度半弱，皆旋轉於六合之內。
Lunar path single ring: circumference 1 zhang 8 chi 6 cun 3 fen, diameter 6 chi 2 cun 1 fen, width 1 cun 1 fen, thickness 5 fen. It bears the nodal degrees, set inside the ecliptic ring, inclined six degrees to the ecliptic; at the end of each nodal cycle it regresses along the ecliptic by one and a fraction degrees; all these rings turn within the Six Harmonies layer.
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第三重，名四遊儀。
The third layer is called the Four Pivots Instrument.
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璿樞雙環：週邊一丈八尺二寸一分，直徑六尺七分，闊二寸，厚七分。 兩面各列周天三百六十五度少強，挾直距以對樞軸，東西運轉於三辰儀內，以格星度。
Xuanshu double ring: circumference 1 zhang 8 chi 2 cun 1 fen, diameter 6 chi 7 fen, width 2 cun, thickness 7 fen. Each face is graduated for the full circuit of 365 and a fraction degrees; gripping the sight arm against the pivot axis, it rotates east and west within the Three Luminaries layer to take star positions.
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橫簫望筒：長五尺七寸，外方內圓，中通望孔，直徑六分，周於日輪，在璿樞直距之中，使南北遊仰，以窺辰宿，無所不至。
Transverse sight-tube: five chi seven cun long, square outside and round inside, with a central sight hole six fen across, aligned with the solar disc; mounted in the Xuanshu sight arm, it pivots north-south and tilts up and down to sight any star or lodge.
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十字水準槽：長九尺四寸八分，首闊一尺二寸七分，身闊九寸二分、高七尺。 水槽一寸，深八分，四柱各長六尺七寸八分，植于水槽之末，以輔天體，皆以銅為之。 乃格七曜遠近盈縮，以知晝夜長短之效。 其所測二十八舍距度，著於後； 其周天星入宿去極所主吉凶，則具在《天文志》。
Cross-shaped leveling trough: length 9 chi 4 cun 8 fen, head width 1 chi 2 cun 7 fen, body width 9 cun 2 fen, height 7 chi. The trough is one cun wide and eight fen deep; four pillars, each six chi seven cun eight fen long, stand at the ends of the trough to bear the celestial assembly; all are cast in bronze. In this way it gauges the seven luminaries' distances and phases of waxing and waning, revealing how day and night lengthen and shorten. The interval degrees of the twenty-eight lodges measured with it are recorded below; The stars' lodge entries, polar distances, and the fortunes they govern are set out fully in the Treatise on Astronomy.
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角十二度，亢九度，氐十六度，房五度，心四度，尾十九度，箕十度，鬥二十五度，牛七度，女十一度，虛十度，危十六度，室十七度，壁九度，奎十六度，婁十二度，胃十五度，昴十一度，畢十八度，觜一度，參十度，井三十四度，鬼二度，柳十四度，星七度，張十八度，翼十八度，軫十七度。
Horn: 12°; Neck: 9°; Root: 16°; Chamber: 5°; Heart: 4°; Tail: 19°; Winnowing-basket: 10°; Dipper: 25°; Ox: 7°; Maid: 11°; Void: 10°; Rooftop: 16°; Encampment: 17°; Wall: 9°; Legs: 16°; Bond: 12°; Stomach: 15°; Hairy Head: 11°; Net: 18°; Turtle Beak: 1°; Three Stars: 10°; Well: 34°; Ghost: 2°; Willow: 14°; Star: 7°; Extended Net: 18°; Wings: 18°; Chariot Axle: 17°.
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皇祐漏刻
The Huangyou Clepsydra Graduations
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自黃帝觀漏水，制器取則，三代因以命官，則挈壺氏其職也。 後之作者，或下漏，或浮漏，或輪漏，或權衡，製作不一。 宋舊有刻漏及以水為權衡，置文德殿之東廡。 景祐三年，再加考定，而水有遲疾，用有司之請，增平水壺一、渴烏二、晝夜箭二十一。 然常以四時日出傳卯正一刻，又每時正已傳一刻，至八刻已傳次時，即二時初末相侵殆半。 皇祐初，詔舒易簡、于淵、周琮更造，其法用平水重壺均調水勢，使無遲疾。 分百刻於晝夜； 冬至晝漏四十刻，夜漏六十刻； 夏至晝漏六十刻，夜漏四十刻； 春秋二分晝夜各五十刻。 日未出前二刻半為曉，日沒後二刻半為昏，減夜五刻以益晝漏，謂之昏旦漏刻。 皆隨氣增損焉。 冬至、夏至之間，晝夜長短凡差二十刻，每差一刻，別為一箭，冬至互起其首，凡有四十一箭。 晝有朝、有禺、有中、有晡、有夕，夜有甲、乙、丙、丁、戊，昏旦有星中，每箭各異其數。 凡黃道升降差二度四十分，則隨曆增減改箭。 每時初行一刻至四刻六分之一為時正，終八刻六分之二則交次時。 今列二十四氣、晝夜日出入辰刻、昏曉中星，以備參合。
From the Yellow Emperor's observation of dripping water and devising instruments to set standards, the Three Dynasties appointed officials to the task—the Qiehu clan held that charge. Later makers devised descending clepsydras, floating clepsydras, wheel clepsydras, or balance-and-weight clocks—the designs varied widely. The Song court already possessed graduated clepsydras and water-balance clocks, kept in the east wing of Wende Hall. In the third year of Jingyou (1036) it was recalibrated, yet the water still ran unevenly; at the responsible office's request, one leveling jar, two siphon birds, and twenty-one day-and-night arrows were added. Yet it routinely took sunrise in each season to signal the first mark of mao proper, and by each double-hour's proper mark one mark had already been struck; by the eighth mark the next double-hour was sounding—so the start and end of successive double-hours overlapped by nearly half. Early in Huangyou, an edict directed Shu Yijian, Yu Yuan, and Zhou Cong to rebuild the clock; their method used a leveling weighted jar to steady the flow and eliminate uneven dripping. The day and night were divided into one hundred marks; at winter solstice, forty marks by day and sixty by night; at summer solstice, sixty marks by day and forty by night; at the spring and autumn equinoxes, day and night each received fifty marks. Two and a half marks before sunrise counted as dawn, and two and a half after sunset as dusk; five marks were shifted from night to day—this was called the dusk-and-dawn clepsydra adjustment. All varied with the solar terms. Between winter and summer solstice the day-night span differed by twenty marks in all; each mark's difference had its own arrow, the sequence starting alternately from winter solstice—forty-one arrows in total. Day was divided into morning, mid-morning, noon, afternoon, and evening; night into the five watches jia through wu; dusk and dawn had culminating stars—each arrow bore its own set of figures. Whenever the ecliptic's elevation changed by two degrees forty minutes, the arrows were adjusted to match the calendar. Each double-hour began with one mark and reached its proper mark at four and one-sixth marks; at eight and two-sixths marks it passed into the next double-hour. Below are tabulated the twenty-four solar terms, day and night lengths with sunrise and sunset marks, and dusk and dawn culminating stars for cross-reference.
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皇祐圭表
The Huangyou Gnomon Table
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觀天地陰陽之體，以正位辨方、定時考閏，莫近乎圭表。 宋何承天始立表候日景，十年間，知冬至比舊用《景初曆》常後天三日。 又唐一行造《大衍曆》，用圭表測知舊曆氣節常後天一日。 今司天監圭表乃石晉時天文參謀趙延乂所建，表既欹傾，圭亦墊陷，其於天度無所取正。 皇祐初，詔周琮、于淵、舒易簡改制之，乃考古法，立八尺銅表，厚二寸，博四寸，下連石圭一丈三尺，以盡冬至景長之數，面有雙水溝為平准，於溝雙刻尺寸分數，又刻二十四氣岳台晷景所得尺寸，置於司天監。 候之三年，知氣節比舊曆後天半日。 因而成書三卷，命曰《岳台晷景新書》論前代測候是非、步算之法頗詳。 既上奏，詔翰林學士范鎮為序以識。 琮以謂二十四氣所得尺寸，比顯德《欽天曆》王樸算為密。 今載氣之盈縮，備採用焉。
To read heaven and earth's yin-yang pattern, fix position and direction, and set seasons and intercalation, nothing serves better than the gnomon table. Under the Song, He Chengtian first erected a gnomon to observe solar shadows; within ten years he found that winter solstice by the old Jingchu Calendar regularly lagged the heavens by three days. Later, in Tang times Yixing compiled the Dayan Calendar and, measuring with the gnomon table, found the old calendar's solar terms regularly lagged the heavens by one day. The Directorate's present gnomon table was built by Later Jin astronomical adviser Zhao Yanyi; the table had tilted and the gnomon platform had sunk and shifted, so it could no longer fix celestial degrees accurately. Early in Huangyou, Zhou Cong, Yu Yuan, and Shu Yijian were ordered to rebuild it; studying ancient methods, they erected an eight-chi bronze gnomon two cun thick and four cun wide, joined to a stone platform one zhang three chi long to capture the full winter-solstice shadow, with twin water channels on the face for leveling and graduated inches, cun, and fen on both sides, plus the twenty-four solar terms' shadow lengths measured at Yue Terrace; the instrument was installed at the Directorate of Astronomy. After three years of observation, they found the solar terms lagged the heavens by half a day compared with the old calendar. They then compiled a work in three scrolls entitled New Book of Yue Terrace Gnomon Shadows, reviewing earlier measurements' errors and setting out computational methods in detail. After it was presented to the throne, Hanlin Academician Fan Zhen was directed to write a preface for the record. Zhou Cong held that the shadow dimensions obtained for the twenty-four solar terms agreed more closely with Wang Pu's figures in the Xiande Qintian Calendar than earlier reckonings had. The seasonal variations in shadow length are recorded below for reference and use.
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小雪，皇祐元年己丑十月十九日戊寅
Light Snow: Huangyou year 1 (1049), cyclical day wuyin, tenth month, day 19.
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新表測景長一丈一尺三寸五分，王朴算景長一丈一尺三寸九分，新法算景長一丈一尺三寸四分小分四十八。
Measured with the new table: shadow 1 zhang 1 chi 3 cun 5 fen; Wang Pu's calculation: 1 zhang 1 chi 3 cun 9 fen; new method: 1 zhang 1 chi 3 cun 4 fen 48 small fen.
23
二年庚寅十月二十九日癸未雲陰不測。
Year 2 (1050), day guiwei, tenth month, day 29: overcast—no measurement taken.
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三年辛卯十月十日戊子
Year 3 (1051), cyclical day wuzi, tenth month, day 10.
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新表測景長一丈一尺三寸，王朴算景長一丈一尺四寸七分，新法算景長一丈一尺二寸九分小分九十八。
Measured with the new table: 1 zhang 1 chi 3 cun; Wang Pu's calculation: 1 zhang 1 chi 4 cun 7 fen; new method: 1 zhang 1 chi 2 cun 9 fen 98 small fen.
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大雪，元年己丑十一月四日癸巳。 雲陰不測。
Heavy Snow: Huangyou year 1, cyclical day guisi, eleventh month, day 4. Overcast—no measurement taken.
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二年庚寅十一月十五日戊戌
Year 2, cyclical day wuxu, eleventh month, day 15.
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新表測景長一丈二尺四寸五分半，王朴算景長一丈二尺四寸五分，新法算景長一丈二尺四寸四分小分二十五。
Measured with the new table: 1 zhang 2 chi 4 cun 5 fen and a half; Wang Pu's calculation: 1 zhang 2 chi 4 cun 5 fen; new method: 1 zhang 2 chi 4 cun 4 fen 25 small fen.
29
冬至，元年己丑十一月十九日戊申
Winter Solstice: Huangyou year 1, cyclical day wushen, eleventh month, day 19.
30
新表測景長一丈二尺八寸五分，王朴算景長一丈二尺八寸六分，新法算景長一丈二尺八寸五分。
Measured with the new table: 1 zhang 2 chi 8 cun 5 fen; Wang Pu's calculation: 1 zhang 2 chi 8 cun 6 fen; new method: 1 zhang 2 chi 8 cun 5 fen.
31
二年庚寅十一月三十日癸丑
Year 2, cyclical day guichou, eleventh month, day 30.
32
新表測景長一丈二尺八寸四分，王朴算景長一丈二尺八寸六分，新法算景長一丈二尺八寸五分。
Measured with the new table: 1 zhang 2 chi 8 cun 4 fen; Wang Pu's calculation: 1 zhang 2 chi 8 cun 6 fen; new method: 1 zhang 2 chi 8 cun 5 fen.
33
三年辛卯十一月十二日己未雲陰不測。
Year 3, day jiwei, eleventh month, day 12: overcast—no measurement taken.
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小寒，元年己丑十二月四日癸亥
Slight Cold: Huangyou year 1, cyclical day guihai, twelfth month, day 4.
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新表測景長一丈二尺四寸，王朴算景長一丈二尺四寸八分，新法算景長一丈二尺四寸小分十五。
Measured with the new table: 1 zhang 2 chi 4 cun; Wang Pu's calculation: 1 zhang 2 chi 4 cun 8 fen; new method: 1 zhang 2 chi 4 cun 15 small fen.
36
二年庚寅閏十一月十五日戊辰雲陰不測。
Year 2, intercalary eleventh month, day 15, cyclical day wuchen: overcast—no measurement taken.
37
三年辛卯十一月二十七日甲戌
Year 3, cyclical day jiaxu, eleventh month, day 27.
38
新表測景長一丈二尺三寸七分，王朴算景長一丈二尺四寸八分小分二十六。
Measured with the new table: 1 zhang 2 chi 3 cun 7 fen; Wang Pu's calculation: 1 zhang 2 chi 4 cun 8 fen 26 small fen.
39
大寒，元年己丑十二月十九日戊寅雲陰不測。
Great Cold: Huangyou year 1, day wuyin, twelfth month, day 19: overcast—no measurement taken.
40
二年庚寅十二月一日甲申
Year 2, cyclical day jiashen, twelfth month, day 1.
41
新表測景長一丈一尺一寸七分，王朴算景長一丈一尺四寸四分，新法算景長一丈一尺一寸八分小分四十。
Measured with the new table: 1 zhang 1 chi 1 cun 7 fen; Wang Pu's calculation: 1 zhang 1 chi 4 cun 4 fen; new method: 1 zhang 1 chi 1 cun 8 fen 40 small fen.
42
三年辛卯十二月十二日己丑雲陰不測。
Year 3, day jichou, twelfth month, day 12: overcast—no measurement taken.
43
立春，二年庚寅正月六日甲午雲陰不測。
Beginning of Spring: Year 2, day jiawu, first month, day 6: overcast—no measurement taken.
44
三年辛卯十二月十六日己亥雲陰不測。
Year 3, day jihai, twelfth month, day 16: overcast—no measurement taken.
45
四年壬辰十二月二十七日甲辰
Year 4 (1052), cyclical day jiachen, twelfth month, day 27.
46
新表測景長九尺六寸七分半，王朴算景長一丈一寸五分，新法算景長一丈六寸八分小分七
Measured with the new table: 9 chi 6 cun 7 fen and a half; Wang Pu's calculation: 1 zhang 1 chi 5 fen; new method: 1 zhang 6 cun 8 fen 7 small fen.
47
雨水，二年庚寅正月二十一日己酉雲陰不測
Rain Water: Year 2, day jiyou, first month, day 21: overcast—no measurement taken.
48
三年辛卯正月二日甲寅
Year 3, cyclical day jiayin, first month, day 2.
49
新表測景長八尺一寸半分，王朴算景長八尺五寸，新法算景長八尺九寸小分七十六
Measured with the new table: 8 chi 1 cun and a half fen; Wang Pu's calculation: 8 chi 5 cun; new method: 8 chi 9 cun 76 small fen.
50
四年壬辰正月十二日己未
Year 4 (1052), cyclical day jiwei, first month, day 12.
51
新表測景長八尺一寸二分半，王朴算景長八尺六寸一分，新法算景長八尺一寸二分小分一十八。
Measured with the new table: 8 chi 1 cun 2 fen and a half; Wang Pu's calculation: 8 chi 6 cun 1 fen; new method: 8 chi 1 cun 2 fen 18 small fen.
52
驚蟄，二年庚寅二月七日甲子
Awakening of Insects: Year 2, cyclical day jiazi, second month, day 7.
53
新表測景長六尺六寸三分，王朴算景長六尺八寸五分，新法算景長六尺六寸三分小分三十九。
Measured with the new table: 6 chi 6 cun 3 fen; Wang Pu's calculation: 6 chi 8 cun 5 fen; new method: 6 chi 6 cun 3 fen 39 small fen.
54
三年辛卯正月十七日己巳
Year 3, cyclical day jisi, first month, day 17.
55
新表測景長六尺六寸五分，王朴算景長六尺八寸五分，新法算景長六尺六寸五分小分六十八
Measured with the new table: 6 chi 6 cun 5 fen; Wang Pu's calculation: 6 chi 8 cun 5 fen; new method: 6 chi 6 cun 5 fen 68 small fen.
56
四年壬辰正月二十八日乙亥雲陰不測
Year 4 (1052), cyclical day yihai, first month, day 28: overcast—no measurement taken.
57
春分，二年庚寅二月二十三日己卯
Spring Equinox: Year 2, cyclical day jimao, second month, day 23.
58
新表測景長五尺三寸五分，王朴算景長五尺二寸七分，新法算景長五尺三寸四分小分七十七
Measured with the new table: 5 chi 3 cun 5 fen; Wang Pu's calculation: 5 chi 2 cun 7 fen; new method: 5 chi 3 cun 4 fen 77 small fen.
59
三年辛卯二月四日乙酉雲陰不測
Year 3, cyclical day yiyou, second month, day 4: overcast—no measurement taken.
60
四年壬辰二月十四日庚寅
Year 4 (1052), cyclical day gengyin, second month, day 14.
61
新表測景長五尺三寸一分，五朴算景長五尺二寸七分，新法算景長五尺三寸小分七十二。
Measured with the new table: 5 chi 3 cun 1 fen; Wang Pu's calculation: 5 chi 2 cun 7 fen; new method: 5 chi 3 cun 72 small fen.
62
清明，二年庚寅三月八日乙未
Clear and Bright: Year 2, cyclical day yiwei, third month, day 8.
63
新表測景長四尺二寸，王朴算景長三尺八寸九分，新法算景長四尺一寸八分小分六十一。
Measured with the new table: 4 chi 2 cun; Wang Pu's calculation: 3 chi 8 cun 9 fen; new method: 4 chi 1 cun 8 fen 61 small fen.
64
三年辛卯二月十九日庚子雲陰不測。
Year 3, cyclical day gengzi, second month, day 19: overcast—no measurement taken.
65
四年壬辰二月二十九日乙巳
Year 4 (1052), cyclical day yisi, second month, day 29.
66
新表測景長四尺二寸二分，王朴算景長三尺九寸六分，新法算景長四尺二寸一分小分八十五。
Measured with the new table: 4 chi 2 cun 2 fen; Wang Pu's calculation: 3 chi 9 cun 6 fen; new method: 4 chi 2 cun 1 fen 85 small fen.
67
穀雨，二年庚寅三月二十三日庚戌雲陰不測
Grain Rain: Year 2, day gengxu, third month, day 23: overcast—no measurement taken.
68
三年辛卯三月四日乙卯
Year 3, cyclical day yimao, third month, day 4.
69
新表測景長三尺三寸，王朴算景長二尺九寸六分，新法算景長三尺二寸九分小分八十六。
Measured with the new table: 3 chi 3 cun; Wang Pu's calculation: 2 chi 9 cun 6 fen; new method: 3 chi 2 cun 9 fen 86 small fen.
70
四年壬辰三月十五日庚申
Year 4 (1052), cyclical day gengshen, third month, day 15.
71
新表測景長三尺三寸一分半，王朴算景長三尺一寸，新法算景長三尺三寸一分小分一十六。
Measured with the new table: 3 chi 3 cun 1 fen and a half; Wang Pu's calculation: 3 chi 1 cun; new method: 3 chi 3 cun 1 fen 16 small fen.
72
立夏，二年庚寅四月九日乙丑
Beginning of Summer: Year 2, cyclical day yichou, fourth month, day 9.
73
新表測景長二尺五寸七分，王朴算景長二尺三寸，新法算景長二尺五寸六分小分二十八。
Measured with the new table: 2 chi 5 cun 7 fen; Wang Pu's calculation: 2 chi 3 cun; new method: 2 chi 5 cun 6 fen 28 small fen.
74
三年辛卯三月十九日庚午
Year 3, cyclical day gengwu, third month, day 19.
75
新表測景長二尺五寸七分半，王朴算景長二尺三寸，新法算景長二尺五寸七分小分四十二。
Measured with the new table: 2 chi 5 cun 7 fen and a half; Wang Pu's calculation: 2 chi 3 cun; new method: 2 chi 5 cun 7 fen 42 small fen.
76
四年壬辰三月三十日乙亥
Year 4 (1052), cyclical day yihai, third month, day 30.
77
新表測景長二尺五寸八分半，王朴算景長二尺三寸四分，新法算景長二尺五寸八分小分四十四。
Measured with the new table: 2 chi 5 cun 8 fen and a half; Wang Pu's calculation: 2 chi 3 cun 4 fen; new method: 2 chi 5 cun 8 fen 44 small fen.
78
小滿，二年庚寅四月二十四日庚辰
Lesser Fullness: Year 2, cyclical day gengchen, fourth month, day 24.
79
新表測景長二尺三分，王朴算景長一尺八寸六分，新法算景長二尺三分小分五十一。
Measured with the new table: 2 chi 3 fen; Wang Pu's calculation: 1 chi 8 cun 6 fen; new method: 2 chi 3 fen 51 small fen.
80
三年辛卯四月五日乙酉
Year 3, cyclical day yiyou, fourth month, day 5.
81
新表測景長二尺三分半，王朴算景長一尺八寸六分，新法算景長二尺三分小分五十一。
Measured with the new table: 2 chi 3 fen and a half; Wang Pu's calculation: 1 chi 8 cun 6 fen; new method: 2 chi 3 fen 51 small fen.
82
四年壬辰四月十六日辛卯雲陰不測。
Year 4 (1052), cyclical day xinmao, fourth month, day 16: overcast—no measurement taken.
83
芒種，二年庚寅五月九日乙未
Grain in Ear: Year 2, cyclical day yiwei, fifth month, day 9.
84
新表測景長一尺六寸九分，王朴算景長一尺六寸，新法算景長一尺六寸半分小分九十七。
Measured with the new table: 1 chi 6 cun 9 fen; Wang Pu's calculation: 1 chi 6 cun; new method: 1 chi 6 cun and a half fen, 97 small fen.
85
三年辛卯四月二十一日辛丑
Year 3, cyclical day xinchou, fourth month, day 21.
86
新表測景長一尺六寸七分，王朴算景長一尺五寸九分，新法算景長一尺六寸七分小分八十四。
Measured with the new table: 1 chi 6 cun 7 fen; Wang Pu's calculation: 1 chi 5 cun 9 fen; new method: 1 chi 6 cun 7 fen 84 small fen.
87
四年壬辰五月二日丙午
Year 4 (1052), cyclical day bingwu, fifth month, day 2.
88
新表測景長一尺六寸八分半，王朴算景長一尺六寸，新法算景長一尺六寸八分小分二十。
Measured with the new table: 1 chi 6 cun 8 fen and a half; Wang Pu's calculation: 1 chi 6 cun; new method: 1 chi 6 cun 8 fen 20 small fen.
89
夏至，二年庚寅五月二十五日辛亥
Summer Solstice: Year 2, cyclical day xinhai, fifth month, day 25.
90
新表測景長一尺五寸七分半，王朴算景長一尺五寸一分，新法算景長一尺五寸七分。
Measured with the new table: 1 chi 5 cun 7 fen and a half; Wang Pu's calculation: 1 chi 5 cun 1 fen; new method: 1 chi 5 cun 7 fen.
91
三年辛卯五月七日丙辰雲陰不測。
Year 3, day bingchen, fifth month, day 7: overcast—no measurement taken.
92
四年壬辰五月十七日辛酉
Year 4 (1052), cyclical day xinyou, fifth month, day 17.
93
新表測景長一尺五寸七分，王朴算景長一尺五寸一分，新法算景長一尺五寸七分。
Measured with the new table: 1 chi 5 cun 7 fen; Wang Pu's calculation: 1 chi 5 cun 1 fen; new method: 1 chi 5 cun 7 fen.
94
小暑，二年庚寅六月十一日丙寅雲陰不測
Lesser Heat: Year 2, day bingyin, sixth month, day 11: overcast—no measurement taken.
95
三年辛卯五月二十二日辛未
Year 3, cyclical day xinwei, fifth month, day 22.
96
新表測景長一尺六寸九分半，王朴算景長一尺六寸，新法算景長一尺六寸九分小分七十五。
Measured with the new table: 1 chi 6 cun 9 fen and a half; Wang Pu's calculation: 1 chi 6 cun; new method: 1 chi 6 cun 9 fen 75 small fen.
97
四年壬辰六月三日丙子雲陰不測。
Year 4 (1052), day bingzi, sixth month, day 3: overcast—no measurement taken.
98
大暑，二年庚寅六月二十六日辛巳
Greater Heat: Year 2, cyclical day xinsi, sixth month, day 26.
99
新表測景長二尺四寸，王朴算景長一尺八寸五分，新法算景長二尺四分小分九十七。
Measured with the new table: 2 chi 4 cun; Wang Pu's calculation: 1 chi 8 cun 5 fen; new method: 2 chi 4 fen 97 small fen.
100
三年辛卯六月七日丙戌。
Year 3, cyclical day bingxu, sixth month, day 7.
101
新表測景長二尺二分太，王朴算景長一尺八寸五分，新法算景長二尺四分小分二十四。
Measured with the new table: 2 chi 2 fen excess; Wang Pu's calculation: 1 chi 8 cun 5 fen; new method: 2 chi 4 fen 24 small fen.
102
四年壬辰六月十九日壬辰
Year 4 (1052), cyclical day renchen, sixth month, day 19.
103
新表測景長二尺五分，王朴算景長一尺八寸七分，新法算景長二尺六分小分五十三。
Measured with the new table: 2 chi 5 fen; Wang Pu's calculation: 1 chi 8 cun 7 fen; new method: 2 chi 6 fen 53 small fen.
104
立秋，二年庚寅七月十一日丙申
Beginning of Autumn: Year 2, cyclical day bingshen, seventh month, day 11.
105
新表測景長二尺五寸九分，王朴算景長二尺二寸九分，新法算景長二尺五寸九分小分五十一。
Measured with the new table: 2 chi 5 cun 9 fen; Wang Pu's calculation: 2 chi 2 cun 9 fen; new method: 2 chi 5 cun 9 fen 51 small fen.
106
三年辛卯六月二十三日壬寅
Year 3, cyclical day renyin, sixth month, day 23.
107
新表測景長二尺六寸一分半，王朴算景長二尺三寸三分，新法算景長二尺六寸二分小分七十三。
Measured with the new table: 2 chi 6 cun 1 fen and a half; Wang Pu's calculation: 2 chi 3 cun 3 fen; new method: 2 chi 6 cun 2 fen 73 small fen.
108
處暑，二年庚寅七月二十七日壬子雲陰不測。
End of Heat: Year 2, day renzi, seventh month, day 27: overcast—no measurement taken.
109
三年辛卯七月九日丁巳
Year 3, cyclical day dingsi, seventh month, day 9.
110
新表測景長三尺三寸六分，王朴算景長三尺，新法算景長三尺三寸六分小分六十五。
Measured with the new table: 3 chi 3 cun 6 fen; Wang Pu's calculation: 3 chi; new method: 3 chi 3 cun 6 fen 65 small fen.
111
四年壬辰七月十九日壬戌雲陰不測。
Year 4 (1052), day renxu, seventh month, day 19: overcast—no measurement taken.
112
白露，二年庚寅八月十三日丁卯雲陰不測
White Dew: Year 2, day dingmao, eighth month, day 13: overcast—no measurement taken.
113
三年辛卯七月二十四日壬申雲陰不測
Year 3, day renshen, seventh month, day 24: overcast—no measurement taken.
114
四年壬辰八月五日丁丑雲陰不測
Year 4 (1052), day dingchou, eighth month, day 5: overcast—no measurement taken.
115
秋分，二年庚寅八月二十八日壬午雲陰不測
Autumn Equinox: Year 2, day renwu, eighth month, day 28: overcast—no measurement taken.
116
三年辛卯八月九日丁亥
Year 3, cyclical day dinghai, eighth month, day 9.
117
新表測景長五尺三寸八分，王朴算景長五尺二寸一分，新法算景長五尺三寸八分小分六十九。
Measured with the new table: 5 chi 3 cun 8 fen; Wang Pu's calculation: 5 chi 2 cun 1 fen; new method: 5 chi 3 cun 8 fen 69 small fen.
118
四年壬辰八月二十日壬辰雲陰不測
Year 4 (1052), day renchen, eighth month, day 20: overcast—no measurement taken.
119
寒露，二年庚寅九月十三日丁酉雲陰不測
Cold Dew: Year 2, day dingyou, ninth month, day 13: overcast—no measurement taken.
120
三年辛卯九月二十四日壬寅
Year 3, cyclical day renyin, ninth month, day 24.
121
新表測景長六尺六寸七分，王朴算景長六尺八分，新法算景長六尺六寸七分小分八十八。
Measured with the new table: 6 chi 6 cun 7 fen; Wang Pu's calculation: 6 chi 8 fen; new method: 6 chi 6 cun 7 fen 88 small fen.
122
四年壬辰九月六日戊申
Year 4 (1052), cyclical day wushen, ninth month, day 6.
123
新表測景長六尺七寸三分半，王朴算景長六尺九寸一分，新法算景長六尺七寸四分小分八十四。
Measured with the new table: 6 chi 7 cun 3 fen and a half; Wang Pu's calculation: 6 chi 9 cun 1 fen; new method: 6 chi 7 cun 4 fen 84 small fen.
124
霜降，二年庚寅九月二十八日壬子
Frost Descent: Year 2, cyclical day renzi, ninth month, day 28.
125
新表測景長八尺一寸六分，王朴算景長八尺四寸五分，新法算景長八尺一寸四分小分七十。
Measured with the new table: 8 chi 1 cun 6 fen; Wang Pu's calculation: 8 chi 4 cun 5 fen; new method: 8 chi 1 cun 4 fen 70 small fen.
126
三年辛卯九月十日戊午雲陰不測
Year 3, day wuwu, ninth month, day 10: overcast—no measurement taken.
127
四年壬辰九月二十一日癸亥
Year 4 (1052), cyclical day guihai, ninth month, day 21.
128
新表測景長八尺二寸，王朴算景長八尺五寸六分，新法算景長八尺一寸九分小分六十六。
Measured with the new table: 8 chi 2 cun; Wang Pu's calculation: 8 chi 5 cun 6 fen; new method: 8 chi 1 cun 9 fen 66 small fen.
129
立冬，二年庚寅十月十四日戊辰
Beginning of Winter: Year 2, cyclical day wuchen, tenth month, day 14.
130
新表測景長九尺八寸半分，王朴算景長一丈一寸，新法算景長九尺八寸一分小分二十五。
Measured with the new table: 9 chi 8 cun and a half fen; Wang Pu's calculation: 1 zhang 1 chi; new method: 9 chi 8 cun 1 fen 25 small fen.
131
三年辛卯九月二十五日癸酉
Year 3, cyclical day guiyou, ninth month, day 25.
132
新表測景長九尺七寸九分，王朴算景長一丈一寸，新法算景長九尺七寸八分小分六十三。
Measured with the new table: 9 chi 7 cun 9 fen; Wang Pu's calculation: 1 zhang 1 chi; new method: 9 chi 7 cun 8 fen 63 small fen.
133
四年壬辰十月六日戊寅
Year 4 (1052), cyclical day wuyin, tenth month, day 6.
134
新表測景長九尺七寸六分，王朴算景長一丈一寸，新法算景長九尺七寸六分小分一十。
Measured with the new table: 9 chi 7 cun 6 fen; Wang Pu's calculation: 1 zhang 1 chi; new method: 9 chi 7 cun 6 fen 10 small fen.
135
測景正加時早晚
Correcting the Time of Shadow Measurement (Early and Late)
136
後漢熹平三年，《四分曆》志立冬中景長一丈，立春中景長九尺六寸。 尋冬至南極，日晷最長，二氣去至日數既同，則中景應等，而前長後短，頓差四寸。 此曆景冬至後天之驗也。 二氣中景日差九分半弱，進退均調，略無盈縮，以率計之，二氣各退二日十二刻，則晷景之數，立冬更短，立春更長，並差二寸，二氣中景俱長九尺八寸矣，即立冬、立春之正日也。 以此推之，曆置冬至後天亦二日十二刻也。 熹平三年，時曆丁丑冬至，加時正在日中。 以二日十二刻減之，定以乙亥冬至，加時在夜半後二十八刻。 《宋志》大明五年十月十日，景一丈七寸七分半； 十一月二十五日，景一丈八寸一分太。 二十六日，一丈七寸五分強。 折取其中，則中天冬至應在十一月三日求其早晚。 令後二日景相減，則一日差率也，倍之為法。 前二日減，以百刻乘之，為實。 以法除實，得冬至加時在夜半後三十一刻，在《元嘉曆》後一日，天數之正也。 量檢彌年，則加減均同。 異歲相課，則遠近應率。 觀二家之說，略而未通。 熹平乃要取其中，而失於至前、至後之餘。 大明則左右率，而失於為實、為法之數。 若夫較景、定氣，曆家最為急務。 觀古較驗，止以冬至前後數日之間，以定加時早晚。 且景之差行，當二至前後，進退在微芒之間。 又日有變行，盈縮稍異，若以為准，則加時相背。 又晉、漢曆術，多以前後所測晷要取其中，此亦差過半日。 今比歲較驗，在立冬、立春景移過寸，若較取加時，則宜以其相近者通計，半之為距至泛日； 乃以其晷數相減，餘者以法乘之，滿其日晷差而一，為刻； 乃以差刻求冬至，視其前晷，多則為減，少則為加，求夏至返之。
In the third year of the Xiaping era of Later Han, the Monograph on the Quarter-remainder Calendar records the mid-term shadow at Start of Winter as 1 zhang and at Beginning of Spring as 9 chi 6 cun. At the winter solstice the sun stands at its southern limit and the gnomon shadow is longest. The two qi lie equally many days from the solstice, so their mid-term shadows should match; instead the earlier is longer and the later shorter—a sudden discrepancy of 4 cun. This is proof from the calendar's shadow records that the winter solstice was placed late relative to Heaven. The two mid-term shadows differ by a daily rate of nine and a half fen minus; with even advance and retreat there is little surplus or shortfall. By that rate, if each qi is moved back two days and twelve ke, the shadow lengths would shorten at Start of Winter and lengthen at Beginning of Spring by 2 cun each—both mid-term shadows becoming 9 chi 8 cun, which would be the true days of Start of Winter and Beginning of Spring. By the same reasoning, the calendar's winter solstice also lagged Heaven by two days and twelve ke. In Xiaping year 3, the calendar then in use fixed the winter solstice on dingchou, with the added hour exactly at midday. Subtract two days and twelve ke, and the winter solstice falls on yihai, with the added hour twenty-eight ke after midnight. The Monograph of Song records that in the fifth year of Daming, tenth month, day 10, the shadow was 1 zhang 7 cun 7 fen and a half; on the twenty-fifth day of the eleventh month, 1 zhang 8 cun 1 fen excess; and on the twenty-sixth, 1 zhang 7 cun 5 fen strong. Averaging these, the true midwinter solstice should fall on the third day of the eleventh month—whence one may determine how early or late it was. Subtract the shadow of the two preceding days from that of the two following days; the difference is the daily rate, which doubled gives the divisor. Subtract the shadow of the two earlier days, multiply by one hundred ke, and the product is the dividend. Dividing the dividend by the divisor yields winter solstice's added hour at thirty-one ke after midnight—one day later than the Yuankai Calendar, which matches the true count of celestial days. Checked over many years, the additions and subtractions balance out evenly. Compared across different years, near and far dates align with the expected rate. Reviewing the two schools' theories shows them to be sketchy and incomplete. The Xiping method aimed at a mean but missed the surplus before and after the solstices. The Daming method balanced the left and right rates but erred in setting the dividend and divisor. Comparing shadows and fixing the solar terms are the calendrist's most pressing duties. Earlier comparisons relied only on a few days before and after winter solstice to fix whether the added hour came early or late. Moreover, the shadow's rate of change around the two solstices advances and retreats within the finest gradations. The sun's motion also varies, waxing and waning slightly; if that variation is taken as the standard, the added hour will be misaligned. Jin and Han calendar methods often averaged gnomon shadows measured before and after a term—an error of more than half a day. Year-by-year comparison shows that at Beginning of Winter and Beginning of Spring the shadow shifts by more than an inch; to fix the added hour, the nearest observations should be combined and halved to obtain the general day from solstice; subtract the two gnomon readings; multiply the remainder by the divisor, and when it fills the day's gnomon difference, the quotient is one mark; use the mark-difference to find winter solstice: if the preceding gnomon reading is greater, subtract; if less, add; reverse the procedure for summer solstice.
137
加減距至泛日，為定日； 仍加半日之刻，命從前距日辰，算外，即二至加時日辰及刻分。 如此推求，則二至加時早晚可驗矣。
Add or subtract the general day from solstice to obtain the fixed day; add half a day's marks, assign from the preceding interval day's double-hour, and count outward to obtain the two solstices' added hour, double-hour, and marks. Pursued in this way, whether the two solstices' added hour comes early or late can be verified.
138
皇祐岳台晷景法
The Huangyou Yue Terrace Gnomon Shadow Method
139
按《大衍》載日及《崇天》定差之率，雖號通密，然未能盡上下交應之理，則晷度無由合契。 今立新法，使上符盈縮之行，下參句股之數，所算尺寸與天測驗，無有先後。 其術曰：計二至後日數，乃減去二至約餘，仍加半日之分，即所求日午中積數，而置之以求進退差分，求進退差分者，置中積之數，如一象九十一日三十二分以下為在前； 如一象以上，返減二至限一百八十二日六十一分，餘為在後。 置前後度於上，列二百於下，以上減下，餘以下乘上，滿四千一百三十五除之為分，不滿，退除為小分。 在冬至後即為進差，在夏至後即為退差。
The Dayan Calendar's sun carriage and the Chongtian Calendar's fixed-difference rate, though called thoroughly precise, could not fully capture the principle of mutual correspondence above and below, so gnomon degrees could not be made to agree. A new method is now established to match the sun's waxing and waning above and the gougu ratios below, so calculated dimensions and celestial measurements agree without discrepancy. The method runs: count the days after the two solstices, subtract the solstice approximate remainder, and add half a day's fraction to obtain the sought day's noon accumulated number; set this to find the advance-retreat difference—if the accumulated number is ninety-one days thirty-two fen or less (one quadrant), it falls in the front section; If it exceeds one quadrant, subtract it from the two-solstice limit of one hundred eighty-two days sixty-one fen; the remainder falls in the rear section. Place the front and rear degrees above and two hundred below; subtract the lower from the upper, multiply the remainder by both values, divide by four thousand one hundred thirty-five to obtain fen, and any remainder becomes small fen. After winter solstice the result is an advance difference; after summer solstice, a retreat difference.
140
仍列初、末二限，求入初、末限者，置所求日午中積數，日在冬至後初限、夏至後末限之數四十五日六十二分以下，即為所求在初限； 如在已上者，乃返減二至限，餘即為所求入末限。 其冬至後末限、夏至後初限，以一百三十七日為率。
Next set the initial and final limits: if the sought day's noon accumulated number, counted from winter solstice in the initial limit or summer solstice in the final limit, is forty-five days sixty-two fen or less, it falls within the initial limit; If it exceeds that, subtract from the two-solstice limit; the remainder enters the final limit. For the period after winter solstice's final limit and after summer solstice's initial limit, use one hundred thirty-seven days as the rate.
141
用求午中晷數。 求午中晷數者，視所求。 如入冬至後初限、夏至後末限者，以入限日減一千九百三十七半，餘為泛差； 仍以限日分乘其進退差，五因百約之，用減泛差，為定差； 乃以入限日分自相乘，以乘定差，滿一百萬為尺，不滿為寸、為分及小分，以減冬至常晷一丈二尺八寸五分，餘為其日午中晷數。 若所求入冬至後末限、夏至後初限者，乃三約入限日分，以減四百八十五少，餘為泛差； 仍以進退差減極數，餘者若在春分後、秋分前者，直以四約之，以加泛差，為定差； 若在春分前、秋分後者，乃以去二分日數及分乘之，滿六百而一，以減泛差，餘為定差，用以入限日分自相乘，以乘定差，滿一百萬為尺，不滿為寸、為分及小分，以加夏至常晷一尺五寸七分，即為其日午中晷數。 若用周歲曆，直以其日晷景損益差分乘其日午中之余，滿法約之，乃損益其下晷數，即其日午中定晷。
This is used to find the noon gnomon reading. To find the noon gnomon reading, consider which limit applies. If within winter solstice's initial limit or summer solstice's final limit, subtract the limit-entry days from one thousand nine hundred thirty-seven and a half; the remainder is the general difference; multiply the limit days and fractional parts by the advance-retreat difference, reduce by the five-times-one-hundred factor, and subtract from the general difference to obtain the fixed difference; square the limit-entry days and parts, multiply by the fixed difference, and convert at one million per chi (with remainder as cun, fen, and small fen); subtract from the winter solstice constant gnomon of one zhang two chi eight cun five fen to obtain that day's noon gnomon reading. If the date falls in winter solstice's final limit or summer solstice's initial limit, reduce the limit-entry days and parts by one-third, subtract from four hundred eighty-five and a fraction, and the remainder is the general difference; subtract the advance-retreat difference from the extreme number; if the date falls after spring equinox and before autumn equinox, quarter the remainder and add it to the general difference to obtain the fixed difference; If before spring equinox or after autumn equinox, multiply the days and parts from the equinox, divide by six hundred, subtract from the general difference to get the fixed difference; square the limit-entry days and parts, multiply by the fixed difference, convert at one million per chi, and add to the summer solstice constant gnomon of one chi five cun seven fen to obtain that day's noon gnomon reading. With the annual-cycle calendar, multiply that day's gnomon wax-wane difference by the noon remainder, reduce by the divisor, and adjust the base gnomon reading to obtain the day's fixed noon gnomon.
142
如此推求，則上下通應之理，句股斜射之原，皆可視驗，乃具岳台晷景周歲算數。
Pursued in this way, the principle of mutual correspondence above and below and the basis of gougu oblique projection can all be checked against observation; the full annual-cycle calculation tables for Yue Terrace gnomon shadows follow.