1
○授時曆經上△步氣朔第一
○ The Season-Granting Calendar Classic, Upper Part △ First Procedure: Qi and New Moons
2
至元十八年歲次辛巳為元。 〈（上考往古，下驗將來，皆距立元為算。 周歲消長，百年各一，其諸應等數，隨時推測，不用為元。）〉 日周，一萬。
The calendar epoch is the eighteenth year of the Zhiyuan reign (1281 CE), the cyclical year xinsi. (Whether one looks back into antiquity or tests the future, every calculation measures distance from the established epoch. (The tropical year lengthens or shortens by one unit per century; other corresponding constants are computed for the period in question and are not built into the epoch.) Day divisor: 10,000.
3
歲實，三百六十五萬二千四百二十五分。 通余，五萬二千四百二十五分。 朔實，二十九萬五千三百五分九十三秒。 通閏，十萬八千七百五十三分八十四秒。
Tropical year value: 3,652,425 parts. Common remainder: 52,425 parts. Synodic month length: 295,305 parts and 93 seconds. Common intercalation remainder: 108,753 parts and 84 seconds.
4
歲周，三百六十五日二千四百二十五分。 朔策，二十九日五千三百五分九十三秒。 氣策，十五日二千一百八十四分三十七秒半。 望策，十四日七千六百五十二分九十六秒半。
Year circuit: 365 days and 2,425 parts. New-moon interval: 29 days, 5,305 parts, and 93 seconds. Qi interval: 15 days, 2,184 parts, and 37½ seconds. Full-moon interval: 14 days, 7,652 parts, and 96½ seconds.
5
弦策，七日三千八百二十六分四十八秒少。 氣應，五十五萬六百分。 閏應，二十萬一千八百五十分。 沒限，七千八百一十五分六十二秒半。
Quarter-moon interval: 7 days, 3,826 parts, and slightly less than 48 seconds. Qi epoch offset: 556,000 parts. Intercalation epoch offset: 201,850 parts. Extinction limit: 7,815 parts and 62½ seconds.
6
氣盈，二千一百八十四分三十七秒半。 朔虛，四千六百九十四分七秒。 旬周，六十萬。 紀法，六十。
Qi excess: 2,184 parts and 37½ seconds. New-moon deficit: 4,694 parts and 7 seconds. Sexagenary cycle divisor: 600,000. Sexagenary cycle length: 60.
7
推天正冬至
To compute the winter solstice of the celestial first month
8
置所求距算，以歲實 〈（上推往古，每百年長一； 下算將來，每百年消一。）〉 乘之，為中積。 加氣應，為通積。 滿旬周，去之； 不盡，以日周約之為日，不滿為分。 其日命甲子算外，即所求天正冬至日辰及分。 〈（如上考者，以氣應減中積，滿旬周，去之； 不盡，以減旬周。 余同上。）〉
Take the distance count for the date sought and multiply it by the tropical year value (When projecting into the past, add one per century; (when calculating the future, subtract one per century.) The product is the mean accumulation. Add the qi epoch offset to obtain the general accumulation. Remove multiples of the sexagenary cycle divisor; divide the remainder by the day divisor to get days; the remainder is fractional parts. Count the day from jiazi outside the reckoning; this gives the winter solstice of the celestial first month—the day, double-hour, and fractional parts sought. (For backward projection, subtract the qi epoch offset from the mean accumulation, then remove multiples of the sexagenary cycle divisor; subtract the remainder from the sexagenary cycle divisor. (Proceed with the remainder as above.)
9
求次氣
To compute the next qi
10
置天正冬至日分，以氣策累加之，其日滿紀法，去之，外命如前，各得次氣日辰及分秒。 推天正經朔
Take the winter solstice day and fractional parts; add the qi interval repeatedly; when days reach 60, discard the cycle; count from outside as before to obtain each successive qi—its day, double-hour, and parts and seconds. To compute the standard new moon of the celestial first month
11
置中積，加閏應，為閏積。 滿朔實，去之不盡，為閏余，以減通積，為朔積。 滿旬周，去之； 不盡，以日周約之，為日，不滿為分，即所求天正經朔日及分秒。 〈（上考者，以閏應減中積，滿朔實，去之不盡，以減朔實，為閏余。 以日周約之為日，不滿為分，以減冬至日及分，不及減者，加紀法減之，命如上。）〉
Take the mean accumulation, add the intercalation epoch offset, and call the result the intercalation accumulation. Divide by the synodic month length and keep the remainder as the intercalation remainder; subtract this from the general accumulation to obtain the new-moon accumulation. Remove multiples of the sexagenary cycle divisor; divide the remainder by the day divisor for days and keep the leftover as parts—this yields the standard new moon of the celestial first month: its day and fractional parts and seconds. (For backward projection, subtract the intercalation epoch offset from the mean accumulation; divide by the synodic month length and keep the remainder; subtract this from the synodic month length to obtain the intercalation remainder. (Convert to days via the day divisor, with the remainder as parts; subtract from the winter solstice day and parts; if the subtraction underflows, add 60 and subtract again; count as above.)
12
求弦望及次朔
To compute quarter-moons, full moons, and successive new moons
13
置天正經朔日及分秒，以弦策累加之，其日滿紀法，去之，各得弦望及次朔日及分秒。 推沒日
Take the standard new moon day with its parts and seconds; add the quarter-moon interval repeatedly, discarding cycles of 60 days; each step yields a quarter-moon, full moon, or successive new moon with its day and parts and seconds. To compute extinction days
14
置有沒之氣分秒， 〈（如沒限已上為有沒之氣。）〉 以十五乘之，用減氣策，余滿氣盈而一，為日，並恒氣日，命為沒日。 推滅日
Take the qi whose fractional parts and seconds qualify for extinction, (A qi whose remainder is at or above the extinction limit qualifies.) Multiply by 15, subtract from the qi interval, divide the remainder by the qi excess to get days, add these to the fixed qi day, and name the result the extinction day. To compute annihilation days
15
置有滅之朔分秒， 〈（在朔虛分已下為有滅之朔。）〉 以三十乘之，滿朔虛而一，為日，並經朔日，命為滅日。 △步發斂第二土王策，三日四百三十六分八十七秒半。
Take the new moon whose fractional parts and seconds qualify for annihilation, (A new moon whose remainder is at or below the new-moon deficit qualifies.) Multiply by 30, divide by the new-moon deficit to get days, add these to the standard new-moon day, and name the result the annihilation day. △ Second Procedure: Ebb and Flow. Earth-sovereign interval: 3 days, 436 parts, and 87½ seconds.
16
月閏，九千六十二分八十二秒。 辰法，一萬。 半辰法，五千。 刻法，一千二百。
Monthly intercalation offset: 9,062 parts and 82 seconds. Double-hour divisor: 10,000. Half double-hour divisor: 5,000. Quarter (ke) divisor: 1,200.
17
推五行用事
To compute when each of the Five Phases holds dominion
18
各以四立之節，為春木、夏火、秋金、冬水首用事日。 以土王策減四季中氣，各得其季土始用事日。 氣候正月
The four seasonal establishment nodes mark the first dominion days of spring (wood), summer (fire), autumn (metal), and winter (water) respectively. Subtract the earth-sovereign interval from each season's mid-qi to obtain when earth first holds dominion in that season. Climatic pentads: the first month
19
立春，正月節東風解凍蟄蟲始振魚陟負冰
Beginning of Spring, first-month minor term: the east wind melts the ice; hibernating insects begin to stir; fish rise; ice floats with them.
20
雨水，正月中獺祭魚候雁北草木萌動二月
Rain Water, first-month mid-term: otters offer fish; migrating geese head north; grass and trees sprout. Second month.
21
驚蟄，二月節桃始華倉鹒鳴鷹化為鳩春分，二月中玄鳥至雷乃發聲始電三月
Awakening of Insects, second-month minor term: peach blossoms open; orioles call; hawks turn into turtledoves. Spring Equinox, second-month mid-term: swallows arrive; thunder first rolls; lightning first flashes. Third month.
22
清明，三月節桐始華田鼠化為釐虹始見
Clear and Bright, third-month minor term: paulownia flowers; field mice become quails; rainbows first appear.
23
谷雨，三月中萍始生鳴鳩拂其羽戴勝降於桑四月立夏，四月節螻蟈鳴蚯蚓出王瓜生小滿，四月中苦萊秀靡草死麥秋至
Grain Rain, third-month mid-term: duckweed sprouts; turtledoves preen; hoopoes alight on mulberry trees. Fourth month. Beginning of Summer, fourth-month minor term: tree frogs sing; earthworms surface; snake-gourds sprout. Lesser Fullness, fourth-month mid-term: bitter herbs flourish; tender grasses wither; wheat ripens.
24
芒種，五月節螳螂生鵙始鳴反舌無聲夏至，五月中鹿角解蜩始鳴半夏生六月
Bearded Grain, fifth-month minor term: mantises hatch; shrikes begin to call; mockingbirds go quiet. Summer Solstice, fifth-month mid-term: deer shed their antlers; cicadas first sing; pinellia flourishes. Sixth month.
25
小暑，六月節溫風至蟋蟀居壁鷹始摯
Lesser Heat, sixth-month minor term: warm breezes blow; crickets move into walls; hawks begin to hunt.
26
大暑，六月中腐草為螢土潤溽暑大雨時行七月立秋，七月節涼風至白露降寒蟬鳴處暑，七月中鷹乃祭鳥天地始肅禾乃登
Greater Heat, sixth-month mid-term: rotting grass turns into fireflies; the soil is damp; sultry heat; heavy rains come and go. Seventh month. Beginning of Autumn, seventh-month minor term: cool winds blow; white dew falls; cold cicadas chirp. End of Heat, seventh-month mid-term: hawks offer birds; heaven and earth turn stern; grain comes to harvest.
27
白露，八月節鴻雁來玄鳥歸群鳥養羞秋分，八月中雷始收聲蟄蟲壞戶水始涸九月
White Dew, eighth-month minor term: wild geese arrive; swallows depart; flocking birds lay up food. Autumn Equinox, eighth-month mid-term: thunder falls silent; hibernating insects seal their burrows; waters begin to recede. Ninth month.
28
寒露，九月節鴻雁來賓雀入大水為蛤菊有黃華
Cold Dew, ninth-month minor term: guest wild geese arrive; sparrows dive into the waters and turn into clams; chrysanthemums bloom yellow.
29
霜降，九月中豺乃祭獸草木黃落蟄蟲咸俯十月
Frost's Descent, ninth-month mid-term: jackals offer their prey; grass and trees yellow and shed; hibernating creatures all lie low. Tenth month.
30
立冬，十月節水始冰地始凍雉入大水為蜃
Beginning of Winter, tenth-month minor term: waters begin to freeze; the ground hardens; pheasants dive into the waters and become clams.
31
小雪，十月中虹藏不見天氣上升，地氣下降閉塞而成冬十一月大雪，十一月節鹖鴠不鳴虎始交荔挺出冬至，十一月中蚯蚓結麋角解水泉動
Lesser Snow, tenth-month mid-term: rainbows vanish; yang qi ascends; yin qi descends; the world seals into winter. Eleventh month. Greater Snow, eleventh-month minor term: cold-weather birds fall silent; tigers begin to mate; winter wheat sprouts. Winter Solstice, eleventh-month mid-term: earthworms coil; elk shed antlers; springs begin to flow.
32
十二月小寒，十二月節雁北鄉鵲始巢雉雊
Twelfth month. Lesser Cold, twelfth-month minor term: wild geese head for the north; magpies begin to nest; pheasants call.
33
大寒，十二月中雞乳征鳥厲疾水澤腹堅推中氣去經朔
Greater Cold, twelfth-month mid-term: hens hatch chicks; migratory birds fly hard; ice thickens at the center of marshes. To compute mid-qi offset from the standard new moon
34
置天正閏余，以日周約之，為日，命之，得冬至去經朔。 以月閏累加之，各得中氣去經朔日算。 〈（滿朔策，去之，乃全置閏，然俟定朔無中氣者裁之。）〉
Take the celestial first month's intercalation remainder, convert it to days via the day divisor, and count them to obtain the winter solstice's offset from the standard new moon. Add the monthly intercalation offset repeatedly to obtain each mid-qi's day-count distance from the standard new moon. (Discard full synodic-month cycles; only then insert an intercalary month—but trim any month whose true new moon contains no mid-qi.)
35
推發斂加時
To compute ebb-and-flow with added double-hours
36
置所求分秒，以十二乘之，滿辰法而一，為辰數； 余以刻法收之，為刻； 命子正算外，即所在辰刻。 〈（如滿半辰法，通作一辰，命起子初。）〉 △步日躔第三
Take the parts and seconds sought, multiply by 12, and divide by the double-hour divisor to obtain the double-hour count; collect the remainder with the quarter divisor to obtain quarters; count from midnight (zi) outside the reckoning to locate the double-hour and quarter. (If the remainder reaches half a double-hour divisor, carry one full double-hour and count from the start of zi.) △ Third Procedure: The Sun's Daily Motion
37
周天分，三百六十五萬二千五百七十五分。 周天，三百六十五度二十五分七十五秒。 半周天，一百八十二度六十二分八十七秒半。 象限，九十一度三十一分四十三秒太。
Celestial circuit in parts: 3,652,575 parts. Celestial circuit: 365 degrees, 25 parts, and 75 seconds. Half celestial circuit: 182 degrees, 62 parts, and 87½ seconds. Quadrant limit: 91 degrees, 31 parts, and slightly more than 43 seconds.
38
歲差，一分五十秒。 周應，三百一十五萬一千七十五分。 半歲周，一百八十二日六千二百一十二分半。 盈初縮末限，八十八日九千九十二分少。
Precession rate: 1 part and 50 seconds per year. Circuit epoch offset: 3,151,075 parts. Half-year circuit: 182 days and 6,212½ parts. Expansion-initial / contraction-final limit: 88 days and slightly less than 9,092 parts.
39
縮初盈末限，九十三日七千一百二十分少。 推天正經朔弦望入盈縮歷
Contraction-initial / expansion-final limit: 93 days and slightly less than 7,120 parts. To compute when the celestial first month's standard new moon, quarter-moons, and full moons enter the expansion-contraction sequence
40
置半歲周，以閏余日及分減之，即得天正經朔入縮歷。 〈（冬至後盈，夏至後縮。）〉 以弦策累加之，各得弦望及次朔入盈縮歷日及分秒。 〈（滿半歲周去之，即交盈縮。）〉
Take the half-year circuit and subtract the intercalation remainder in days and parts to obtain the celestial first month's standard new moon entering the contraction sequence. (After the winter solstice the sun slows; after the summer solstice it speeds up.) Add the quarter-moon interval repeatedly to obtain each quarter-moon, full moon, and successive new moon entering the expansion-contraction sequence, with its days, parts, and seconds. (Discard full half-year circuits; this marks the transition between expansion and contraction.)
41
求盈縮差
To compute the expansion-contraction correction
42
視入歷盈者，在盈初縮末限已下，為初限，已上，反減半歲周，余為末限； 縮者，在縮初盈末限已下，為初限，已上，反減半歲周，余為末限。 其盈初縮末者，置立差三十一，以初末限乘之，加平差二萬四千六百，又以初末限乘之，用減定差五百一十三萬三千二百，余再以初末限乘之，滿億為度，不滿退除為分秒。 縮初盈末者，置立差二十七，以初末限乘之，加平差二萬二千一百，又以初末限乘之，用減定差四百八十七萬六百，余再以初末限乘之，滿億為度，不滿退除為分秒，即所求盈縮差。
If the entry is in the expansion phase: below the expansion-initial limit counts as the initial segment; at or above, subtract from the half-year circuit to get the final segment; If the entry is in the contraction phase: below the contraction-initial limit counts as the initial segment; at or above, subtract from the half-year circuit to get the final segment. For the expansion-initial / contraction-final case: set the base difference to 31, multiply by the segment limit, add the mean difference 24,600, multiply again, subtract from the fixed difference 5,133,200, multiply the remainder once more, divide by 100 million for degrees, and reduce the remainder to parts and seconds. For the contraction-initial / expansion-final case: set the base difference to 27, multiply by the segment limit, add the mean difference 22,100, multiply again, subtract from the fixed difference 4,870,600, multiply the remainder once more, convert to degrees and parts and seconds—this is the expansion-contraction correction sought.
43
又術：置入限分，以其日盈縮分乘之，萬約為分，以加其下盈縮積，萬約為度，不滿為分秒，亦得所求盈縮差。 赤道宿度
Alternate method: take the segment entry in parts, multiply by that day's expansion-contraction rate, scale by 10,000, add to the tabulated accumulation below, convert to degrees and parts and seconds—this also yields the correction sought. Equatorial lodge degrees
44
角十二一十亢九二十氐十六三十房五六十心六五十尾十九一十箕十四十右東方七宿，七十九度二十分。 鬥二十五二十牛七二十女十一三十五虛八九十五太
Horn 12°10′, Neck 9°20′, Root 16°30′, Chamber 5°60′, Heart 6°50′, Tail 19°10′, Winnowing-Basket 14°10′—the seven eastern lodges total 79°20′. Dipper 25°20′, Ox 7°20′, Maid 11°35′, Void 8°95′ slightly more
45
危十五四十室十七一十壁八六十右北方七宿，九十三度八十分太。 奎十六六十婁十一八十胃十五六十昴十一三十畢十七四十觜初五參十一一十
Rooftop 15°40′, Encampment 17°10′, Wall 8°60′—the seven northern lodges total slightly more than 93°80′. Stride 16°60′, Bond 11°80′, Stomach 15°60′, Hairy Head 11°30′, Net 17°40′, Turtle Beak 0°5′, Three Stars 11°10′
46
右西方七宿，八十三度八十五分。 井三十三三十鬼二二十柳十三三十星六三十張十七二十五翼十八七十五軫十七三十右南方七宿，一百八度四十分。
The seven western lodges total 83°85′. Well 33°30′, Ghost 2°20′, Willow 13°30′, Star 6°30′, Extended Net 17°25′, Wings 18°75′, Chariot Crossboard 17°30′—the seven southern lodges total 108°40′.
47
右赤道宿次，並依新制渾儀測定，用為常數，校天為密。 若考往古，即用當時宿度為準。 推冬至赤道日度
These equatorial lodge positions were all measured with the new armillary sphere and adopted as fixed constants, verified against the sky for maximum accuracy. When examining past ages, use the lodge degrees current in that period as the standard. To compute the winter solstice solar degree on the equator
48
置中積，以加周應為通積，滿周天分， 〈（上推往古，每百年消一； 下算將來，每百年長一。）〉 去之，不盡，以日周約之為度，不滿，退約為分秒。 命起赤道虛宿六度外，去之，至不滿宿，即所求天正冬至加時日躔赤道宿度及分秒。 〈（上考者，以周應減中積，滿周天，去之； 不盡，以減周天，余以日周約之為度； 余同上。 如當時有宿度者，止依當時宿度命之。）〉
Take the mean accumulation, add the circuit epoch offset to form the general accumulation, and when it reaches the celestial circuit in parts, (When projecting into the past, subtract one per century; (when calculating the future, add one per century.) discard the full circuit; divide the remainder by the day divisor for degrees, then reduce the leftover to parts and seconds. Count from Void lodge 6° on the equator, subtract whole lodges, and the remainder gives the winter solstice solar position on the equator at the epoch hour—lodge, degrees, and seconds. (For backward projection, subtract the circuit epoch offset from the mean accumulation and remove full celestial circuits; subtract the remainder from the full circuit, then divide by the day divisor for degrees; Proceed with the remainder as above. (If contemporary lodge degrees are known, count only from those contemporary positions.)
49
求四正赤道日度
To compute the four seasonal equatorial solar degrees
50
置天正冬至加時赤道日度，累加象限，滿赤道宿次，去之，各得春夏秋正日所在宿度及分秒。 求四正赤道宿積度
Take the celestial first month's winter solstice equatorial position and add the quadrant limit repeatedly, discarding full lodge cycles, to obtain the equatorial lodge positions for spring, summer, and autumn. To compute accumulated equatorial degrees after each seasonal node
51
置四正赤道宿全度，以四正赤道日度及分減之，余為距後度； 以赤道宿度累加之，各得四正後赤道宿積度及分。 黃赤道率
Take each seasonal lodge's full span and subtract the solar degree at that node; the remainder is the distance past the node; Add equatorial lodge spans cumulatively to obtain the accumulated equatorial degrees after each seasonal node. Ecliptic–equator conversion rates
52
〈（表略）〉 推黃道宿度
(Table omitted.) To compute ecliptic lodge degrees
53
置四正後赤道宿積度，以其赤道積度減之，余以黃道率乘之，如赤道率而一； 所得，以加黃道積度，為二十八宿黃道積度； 以前宿黃道積度減之，為其宿黃道度及分。 〈（其秒就近為分。）〉
Take the accumulated equatorial degree after a seasonal node, subtract the node's equatorial accumulation, multiply the difference by the ecliptic rate, and divide by the equatorial rate; add the result to the ecliptic accumulation to obtain the ecliptic accumulated degree for all twenty-eight lodges; Subtract the previous lodge's ecliptic accumulation to obtain that lodge's ecliptic span in degrees and parts. (Round seconds to the nearest minute.)
54
黃道宿度
Ecliptic lodge degrees
55
角十二八十七亢九五十六氐十六四十房五四十八心六二十七尾十七九十五箕九五十九右東方七宿，七十八度一十二分。
Horn 12°87′, Neck 9°56′, Root 16°40′, Chamber 5°48′, Heart 6°27′, Tail 17°95′, Winnowing-Basket 9°59′—the seven eastern lodges total 78°12′.
56
鬥二十三四十七牛六九十女十一十二虛九分空太危十五九十五室十八三十二壁九三十四右北方七宿，九十四度一十分太。
Dipper 23°47′, Ox 6°90′, Maid 11°12′, Void 9′ (blank) slightly more, Rooftop 15°95′, Encampment 18°32′, Wall 9°34′—the seven northern lodges total slightly more than 94°10′.
57
奎十七八十七婁十二三十六胃十五八十一昴十一○八畢十六五十觜初○五參十二十八右西方七宿，八十三度九十五分。
Stride 17°87′, Bond 12°36′, Stomach 15°81′, Hairy Head 11°08′, Net 16°50′, Turtle Beak 0°05′, Three Stars 10°28′—the seven western lodges total 83°95′.
58
井三十一○三鬼二一十一柳十三星六三十一張十七七十九翼二十○九軫十八七十五右南方七宿，一百九度八分。
Well 31°03′, Ghost 2°11′, Willow 13°, Star 6°31′, Extended Net 17°79′, Wings 20°09′, Chariot Crossboard 18°75′—the seven southern lodges total 109°08′.
59
右黃道宿度，依今歷所測赤道準冬至歲差所在算定，以憑推步。 若上下考驗，據歲差每移一度，依術推變，各得當時宿度。 推冬至加時黃道日度
These ecliptic lodge degrees are derived from the equatorial positions measured for the present calendar, adjusted for precession at the winter solstice, and serve as the basis for all computations. When verifying past or future dates, shift one degree per precession step and apply the formulas to obtain the lodge degrees valid for that era. To compute the winter solstice solar degree on the ecliptic at the epoch hour
60
置天正冬至加時赤道日度，以其赤道積度減之，余以黃道率乘之，如赤道率而一； 所得，以加黃道積度，即所求年天正冬至加時黃道日度及分秒。 求四正加時黃道日度
Take the celestial first month's winter solstice equatorial position, subtract its equatorial accumulation, multiply by the ecliptic rate, and divide by the equatorial rate; add the result to the ecliptic accumulation to obtain the winter solstice ecliptic position for the year sought, in degrees and seconds. To compute the four seasonal ecliptic solar degrees at the epoch hour
61
置所求年冬至日躔黃赤道差，與次年黃赤道差相減，余四而一，所得，加象限，為四正定象度。 置冬至加時黃道日度，以四正定象度累加之，滿黃道宿次，去之，各得四正定氣加時黃道度及分。
Take the ecliptic–equator correction at the winter solstice for the year sought and the following year, subtract, divide by four, add the quadrant limit, and obtain the fixed quadrant offset for the four seasonal nodes. Take the winter solstice ecliptic position and add the fixed quadrant offset repeatedly, discarding full lodge cycles, to obtain each seasonal node's ecliptic position in degrees and parts.
62
求四正晨前夜半日度
To compute the four seasonal solar degrees at dawn before midnight
63
置四正恒氣日及分秒， 〈（冬夏二至，盈縮之端，以恒為定。）〉 以盈縮差命為日分，盈減縮加之，即為四正定氣日及分。 置日下分，以其日行度乘之，如日周而一； 所得，以減四正加時黃道日度，各得四正定氣晨前夜半日度及分秒。
Take the four seasonal nodes' mean qi days and parts and seconds, (At the two solstices—the turning points of expansion and contraction—use the mean value as the fixed correction.) Apply the expansion-contraction correction as fractional days—subtract for expansion, add for contraction—to obtain the true dates of the four seasonal nodes. Take the fractional parts of the day, multiply by the sun's daily motion, and divide by the day divisor; subtract the result from each seasonal node's ecliptic position at the epoch hour to obtain the solar degree at dawn before midnight for that node.
64
求四正後每日晨前夜半黃道日度
To compute the daily ecliptic solar degree at dawn before midnight after each seasonal node
65
以四正定氣日距後正定氣日為相距日，以四正定氣晨前夜半日度距後正定氣晨前夜半日度為相距度，累計相距日之行定度，與相距度相減； 余如相距日而一，為日差； 〈（相距度多為加，相距度少為減。）〉 以加減四正每日行度率，為每日行定度； 累加四正晨前夜半黃道日度，滿宿次，去之，為每日晨前夜半黃道日度及分秒。
Let the interval in days run from one seasonal node's true date to the next, and the interval in degrees from one dawn-before-midnight position to the next; accumulate the sun's fixed daily motion over those days and subtract from the degree interval; divide the remainder by the day interval to obtain the daily correction; (Add the daily correction when the degree interval is larger; subtract when it is smaller.) adjust the mean daily motion rate by this correction to obtain the true daily motion; Accumulate from each seasonal node's dawn-before-midnight position, discarding full lodge cycles, to obtain the ecliptic solar degree at dawn before midnight for every day.
66
求每日午中黃道日度
To compute the daily ecliptic solar degree at noon
67
置其日行定度，半之，以加其日晨前夜半黃道日度，得午中黃道日度及分秒。 求每日午中黃道積度
Halve the true daily motion and add it to that day's dawn-before-midnight ecliptic position to obtain the noon ecliptic position. To compute the accumulated ecliptic degree at noon for each day
68
以二至加時黃道日度距所求日午中黃道日度，為二至後黃道積度及分秒。 求每日午中赤道日度
Measure from the solstice ecliptic position at the epoch hour to the sought day's noon position to obtain the accumulated ecliptic degrees since the solstice. To compute the daily equatorial solar degree at noon
69
置所求日午中黃道積度，滿象限，去之，余為分後； 內減黃道積度，以赤道率乘之，如黃道率而一； 所得，以加赤道積度及所去象限，為所求赤道積度及分秒； 以二至赤道日度加而命之，即每日午中赤道日度及分秒。
Take the noon ecliptic accumulation for the day sought, discard full quadrant limits, and keep the remainder within the quadrant; subtract from the ecliptic accumulation, multiply by the equatorial rate, and divide by the ecliptic rate; add the result to the equatorial accumulation and the discarded quadrant to obtain the equatorial accumulation sought; Count from the solstice equatorial position to obtain the equatorial solar degree at noon for each day.
70
黃道十二次宿度危，十二度六十四分九十一秒。 入娵訾之次，辰在亥。 奎，一度七十三分六十三秒。 入降婁之次，辰在戌。 胃，三度七十四分五十六秒。 入大梁之次，辰在酉。
Ecliptic degrees of the twelve celestial stations: Rooftop, 12°64′91″. Entering the Ziwei station; the chronogram branch is hai (Pig). Stride: 1°73′63″. Entering the Jianglou station; the chronogram branch is xu (Dog). Stomach: 3°74′56″. Entering the Daliang station; the chronogram branch is you (Rooster).
71
畢，六度八十八分五秒。 入實沈之次，辰在申。 井，八度三十四分九十四秒。 入鶉首之次，辰在未。 柳，三度八十六分八十秒。 入鶉火之次，辰在午。
Net: 6°88′5″. Entering the Shichen station; the chronogram branch is shen (Monkey). Well: 8°34′94″. Entering the Chunshou station; the chronogram branch is wei (Goat). Willow: 3°86′80″. Entering the Chunhuo station; the chronogram branch is wu (Horse).
72
張，十五度二十六分六秒。 入鶉尾之次，辰在巳。 軫，十度七分九十七秒。 入壽星之次，辰在辰。 氐，一度一十四分五十二秒。 入大火之次，辰在卯。
Extended Net: 15°26′6″. Entering the Chunwei station; the chronogram branch is si (Snake). Chariot Crossboard: 10°7′97″. Entering the Shouxing station; the chronogram branch is chen (Dragon). Root: 1°14′52″. Entering the Dahuo station; the chronogram branch is mao (Hare).
73
尾，三度一分一十五秒。 入析木之次，辰在寅。 鬥，三度七十六分八十五秒。 入星紀之次，辰在醜。 女，二度六分三十八秒。 入玄枵之次，辰在子。
Tail: 3°1′15″. Entering the Ximu station; the chronogram branch is yin (Tiger). Dipper: 3°76′85″. Entering the Xingji station; the chronogram branch is chou (Ox). Maid: 2°6′38″. Entering the Xuanxiao station; the chronogram branch is zi (Rat).
74
求入十二次時刻
To compute the moment of entry into each of the twelve celestial stations
75
各置入次宿度及分秒，以其日晨前夜半日度減之，余以日周乘之，為實； 以其日行定度為法； 實如法而一，所得，依發斂加時求之，即入次時刻。 △步月離第四
For each station, take its entry lodge position and subtract that day's dawn-before-midnight solar degree; multiply the remainder by the day divisor to form the dividend; use the sun's true daily motion as the divisor; divide dividend by divisor and convert the result via the ebb-and-flow added-double-hour procedure to obtain the moment of station entry. △ Fourth Procedure: Lunar Motion
76
轉終分，二十七萬五千五百四十六分。 轉終，二十七日五千五百四十六分。 轉中，十三日七千七百七十三分。 初限，八十四。
Rotation-end divisor: 275,546 parts. Rotation end: 27 days and 5,546 parts. Rotation midpoint: 13 days and 7,773 parts. Initial limit: 84.
77
中限，一百六十八。 周限，三百三十六。 月平行，十三度三十六分八十七秒半。 轉差，一日九千七百五十九分九十三秒。
Middle limit: 168. Circuit limit: 336. Mean lunar motion: 13°36′87½″. Rotation increment: 1 day, 9,759 parts, and 93 seconds.
78
弦策，七日三千八百二十六分四十八秒少。 上弦，九十一度三十一分四十三秒太。 望，一百八十二度六十二分八十七秒半。 下弦，二百七十三度九十四分三十一秒少。
Quarter-moon interval: 7 days, 3,826 parts, and slightly less than 48 seconds. First quarter: 91°31′ and slightly more than 43″. Full moon: 182°62′87½″. Last quarter: 273°94′ and slightly less than 31″.
79
轉應，一十三萬一千九百四分。 推天正經朔入轉
Rotation epoch offset: 131,904 parts. To compute when the celestial first month's standard new moon enters the rotation sequence
80
置中積，加轉應，減閏余，滿轉終分，去之，不盡，以日周約之為日，不滿為分，即天正經朔入轉日及分。 〈（上考者，中積內加所求閏余，減轉應，滿轉終，去之，不盡，以減轉終，余同上。）〉
Take the mean accumulation, add the rotation epoch offset, subtract the intercalation remainder, discard full rotation-end cycles, divide the remainder by the day divisor for days and keep the leftover as parts—this yields the celestial first month's standard new moon entering rotation, in days and parts. (For backward projection, add the intercalation remainder to the mean accumulation, subtract the rotation epoch offset, discard full rotation-end cycles, subtract the remainder from the rotation end, and proceed as above.)
81
求弦望及次朔入轉
To compute when each quarter-moon, full moon, and successive new moon enters the rotation sequence
82
置天正經朔入轉日及分，以弦策累加之，滿轉終，去之，即弦望及次朔入轉日及分秒。 如徑求次朔，以轉差加之。 求經朔弦望入遲疾歷
Start from the celestial first month's standard new moon entering rotation and add the quarter-moon interval repeatedly, discarding full rotation-end cycles, to obtain each quarter-moon, full moon, and successive new moon entering rotation with its days, parts, and seconds. To jump directly to the next new moon, add the rotation increment. To determine each standard new moon, quarter-moon, and full moon's position in the slow-fast sequence
83
各視入轉日及分秒，在轉中已下，為疾歷； 已上，減去轉中，為遲歷。 遲疾轉定及積度 〈（表略）〉 求遲疾差
For each event, take its rotation entry in days, parts, and seconds; if at or below the rotation midpoint, it falls in the fast sequence; if above the midpoint, subtract the rotation midpoint to obtain the slow-sequence position. Fixed slow-fast rotation values and accumulated degrees (Table omitted.) To compute the slow-fast correction
84
置遲疾歷日及分，以十二限二十分乘之，在初限已下為初限，已上覆減中限，余為末限。 置立差三百二十五，以初末限乘之，加平差二萬八千一百，又以初末限乘之，用減定差一千一百一十一萬，余再以初末限乘之，滿億為度，不滿退除為分秒，即遲疾差。
Take the position in the slow-fast sequence and multiply by the twelve-limit scaling factor plus twenty parts; values below the initial limit stay in the initial segment; otherwise subtract from the middle limit to obtain the final segment. Set the base difference to 325, multiply by the segment limit, add the mean difference 28,100, multiply again, subtract from the fixed difference 11,110,000, multiply once more, and convert to degrees and parts and seconds—this is the slow-fast correction.
85
又術：置遲疾歷日及分，以遲疾歷日率減之，余以其下損益分乘之，如八百二十而一，益加損減其下遲疾度，亦為所求遲疾差。 求朔弦望定日
Alternate method: take the slow-fast sequence position, subtract the tabulated day-rate, multiply the remainder by the increase-decrease rate below, divide by 820, and add or subtract from the tabulated slow-fast degree below—this also yields the correction sought. To compute the true dates of new moon, quarter-moons, and full moon
86
以經朔弦望盈縮差與遲疾差，同名相從，異名相消， 〈（盈遲縮疾為同名，盈疾縮遲為異名。）〉 以八百二十乘之，以所入遲疾限下行度除之，即為加減差， 〈（盈遲為加，縮疾為減。）〉 以加減經朔弦望日及分，即定朔弦望日及分。 若定弦望分在日出分已下者，退一日，其日命甲子算外，各得定朔弦望日辰。 定朔幹名與後朔幹同者，其月大； 不同者，其月小； 內無中氣者，為閏月。
Combine the expansion-contraction correction and slow-fast correction for each standard new moon, quarter-moon, and full moon—add when they share the same sign, subtract when they oppose, (Expansion with slow and contraction with fast count as same sign; expansion with fast and contraction with slow count as opposite signs.) multiply by 820 and divide by the tabulated daily motion for that slow-fast segment to obtain the add-subtract correction, (Add for expansion-slow; subtract for contraction-fast.) apply this correction to the standard new moon, quarter-moon, or full moon day and parts to obtain the true date and fractional parts. If a true quarter-moon or full moon falls before sunrise on its fractional day, move back one day; count from jiazi beyond to obtain each true new moon, quarter-moon, and full moon with its day and chronogram branch. If the true new moon's stem name matches the following new moon's stem, the month is long (30 days); if they differ, the month is short (29 days); a month containing no mid-qi is intercalary.
87
推定朔弦望加時日月宿度
To compute the sun and moon lodge positions at the epoch hour for each true new moon, quarter-moon, and full moon
88
置經朔弦望入盈縮歷日及分，以加減差加減之，為定朔弦望入歷，在盈，便為中積，在縮，加半歲周，為中積； 命日為度，以盈縮差盈加縮減之，為加時定積度； 以冬至加時日躔黃道宿度加而命之，各得定朔弦望加時日度。
Take each standard event's entry into the expansion-contraction sequence and adjust by the add-subtract correction to obtain its true sequence entry; if in expansion, use it directly as mean accumulation; if in contraction, add the half-year circuit; convert days to degrees and apply the expansion-contraction correction—adding for expansion, subtracting for contraction—to obtain the true accumulated degree at the epoch hour; count forward from the winter solstice ecliptic position at the epoch hour to obtain each true new moon, quarter-moon, and full moon solar degree at the epoch hour.
89
凡合朔加時，日月同度，便為定朔加時月度，其弦望各以弦望度加定積，為定弦望月行定積度，依上加而命之，各得定弦望加時黃道月度。 推定朔弦望加時赤道月度
At conjunction the sun and moon share the same degree, giving the true new moon's lunar position at the epoch hour; for quarter-moons and full moon, add the quarter or full-moon arc to the fixed accumulation and count forward from the winter solstice position to obtain each true quarter-moon or full moon ecliptic lunar degree at the epoch hour. To compute the equatorial lunar positions at the epoch hour for each true new moon, quarter-moon, and full moon
90
各置定朔弦望加時黃道月行定積度，滿象限，去之，以其黃道積度減之，余以赤道率乘之，如黃道率而一，用加其下赤道積度及所去象限，各為赤道加時定積度； 以冬至加時赤道日度加而命之，各為定朔弦望加時赤道月度及分秒。 〈（象限已下及半周，去之，為至後； 滿象限及三象，去之，為分後。）〉
For each event, take the true ecliptic lunar accumulation at the epoch hour, discard full quadrant limits, subtract the ecliptic accumulation, multiply by the equatorial rate and divide by the ecliptic rate, then add to the tabulated equatorial accumulation and discarded quadrant to obtain the equatorial fixed accumulation at the epoch hour; count forward from the winter solstice equatorial position at the epoch hour to obtain each true new moon, quarter-moon, and full moon equatorial lunar position in degrees, parts, and seconds. (Discard the quadrant limit and half circuit to mark the post-solstice segment; (discard the quadrant limit and three quadrants to mark the post-equinox segment.)
91
推朔後平交入轉遲疾歷
To compute the post-new-moon mean node entry into the rotation slow-fast sequence
92
置交終日及分，內減經朔入交日及分，為朔後平交日； 以加經朔入轉，為朔後平交入轉； 在轉中已下，為疾歷； 已上，去之，為遲歷。 求正交日辰
Take the nodical period in days and parts and subtract the standard new moon's node entry to obtain the post-new-moon mean node interval; add this to the standard new moon's rotation entry to obtain the post-new-moon mean node rotation entry; if at or below the rotation midpoint, it falls in the fast sequence; if above, subtract the midpoint to obtain the slow-sequence position. To compute the true node crossing day and chronogram branch
93
置經朔，加朔後平交日，以遲疾歷依前求到遲疾差，遲加疾減之，為正交日及分，其日命甲子算外，即正交日辰。 推正交加時黃道月度
Take the standard new moon, add the post-new-moon mean node interval, compute the slow-fast correction as before, add for slow and subtract for fast, and count from jiazi beyond to obtain the true node crossing day and chronogram branch. To compute the ecliptic lunar degree at the true node crossing epoch hour
94
置朔後平交日，以月平行度乘之，為距後度； 以加經朔中積，為冬至距正交定積度； 以冬至日躔黃道宿度加而命之，為正交加時月離黃道宿度及分秒。
Take the post-new-moon mean node interval and multiply by the mean lunar motion to obtain the post-interval arc; add this to the standard new moon mean accumulation to obtain the fixed accumulated degree from winter solstice to the true node; count forward from the winter solstice ecliptic position to obtain the lunar departure ecliptic lodge position at the true node epoch hour in degrees, parts, and seconds.
95
求正交在二至後初末限
To determine the initial or final limit segment after the solstice for the true node
96
置冬至距正交積度及分，在半歲周已下，為冬至後； 已上，去之，為夏至後。 其二至後，在象限已下，為初限，已上，減去半歲周，為末限。 求定差距差定限度
Take the accumulated degree from winter solstice to the true node; if below the half-year circuit, the node falls after the winter solstice; if above, subtract the half-year circuit to mark the post-summer-solstice segment. Within each solstice half-year: below the quadrant limit counts as the initial segment; above it, subtract the half-year circuit to obtain the final segment. To compute the fixed difference, distance difference, and fixed limit degree
97
置初末限度，以十四度六十六分乘之，如象限而一，為定差； 反減十四度六十六分，余為距差。 以二十四乘定差，如十四度六十六分而一； 所得，交在冬至後名減，夏至後名加，皆加減九十八度，為定限度及分秒。
Take the segment limit, multiply by 14°66′, divide by the quadrant limit, and obtain the fixed difference; subtract from 14°66′ to obtain the distance difference. Multiply the fixed difference by 24 and divide by 14°66′; Apply the result: subtract for nodes after the winter solstice, add for nodes after the summer solstice, each time adjusting by 98° to obtain the fixed limit degree in degrees, parts, and seconds.
98
求四正赤道宿度
To compute the equatorial lodge positions of the four seasonal nodes
99
置冬至加時赤道度，命為冬至正度； 以象限累加之，各得春分、夏至、秋分正積度； 各命赤道宿次去之，為四正赤道宿度及分秒。 求月離赤道正交宿度
Take the winter solstice equatorial position at the epoch hour and designate it as the winter solstice cardinal degree; add the quadrant limit repeatedly to obtain the cardinal accumulated degrees for spring equinox, summer solstice, and autumn equinox; for each, count through the equatorial lodge sequence and discard full cycles to obtain the four seasonal nodes' equatorial lodge positions in degrees, parts, and seconds. To compute the equatorial lodge position of the moon's true node crossing
100
以距差加減春秋二正赤道宿度，為月離赤道正交宿度及分秒。 冬至後，初限加，末限減，視春正； 夏至後，初限減，末限加，視秋正。 求正交後赤道積度入初末限
Apply the distance difference to the spring and autumn equinox equatorial lodge positions to obtain the moon's true node equatorial lodge position in degrees, parts, and seconds. After the winter solstice: add for the initial segment, subtract for the final segment, using the spring equinox position as reference; after the summer solstice: subtract for the initial segment, add for the final segment, using the autumn equinox position as reference. To determine the initial or final limit segment for the equatorial accumulation after the true node
101
各置春秋二正赤道所當宿全度及分，以月離赤道正交宿度及分減之，余為正交後積度； 以赤道宿次累加之，滿象限去之，為半交後； 又去之，為中交後； 再去之，為半交後； 視各交積度在半象已下，為初限； 已上，用減象限，余為末限。
For each equinox, take the full span of the occupied equatorial lodge and subtract the moon's true node equatorial position; the remainder is the accumulated degree after the true node; add equatorial lodge spans cumulatively, discarding full quadrant limits, to mark the post-half-node segment; discard another quadrant limit to mark the post-middle-node segment; discard another quadrant limit to mark the post-half-node segment; for each node segment, if the accumulated degree falls below half a quadrant, count it as the initial limit; if above, subtract the quadrant limit to obtain the final limit segment.
102
求月離赤道正交後半交白道舊名九道出入赤道內外度及定差
To compute the white-path latitude—entry and exit relative to the equator, formerly called the nine paths—after the equatorial true and half nodes, and the fixed difference
103
置各交定差度及分，以二十五乘之，如六十一而一； 所得，視月離黃道正交在冬至後宿度為減，夏至後宿度為加，皆加減二十三度九十分，為月離赤道後半交白道出入赤道內外度及分； 以周天六之一，六十度八十七分六十二秒半，除之，為定差。） 月離赤道正交後為外，中交後為內。
Take each node's fixed difference in degrees and parts, multiply by 25, and divide by 61; Apply the result: subtract when the ecliptic true node falls after the winter solstice, add when it falls after the summer solstice, each time adjusting by 23°90′, to obtain the white-path entry and exit relative to the equator after the equatorial half node, in degrees and parts; divide by one-sixth of the circuit of heaven, 60°87′62.5″, to obtain the fixed difference.) After the equatorial true node the moon is north of the equator; after the middle node it is south.
104
求月離出入赤道內外白道去極度
To compute the white-path polar distance as the moon crosses north or south of the equator
105
置每日月離赤道交後初末限，用減象限，余為白道積； 用其積度減之，余以其差率乘之； 所得，百約之，以加其下積差，為每日積差； 用減周天六之一，余以定差乘之，為每日月離赤道內外度； 內減外加象限，為每日月離白道去極度及分秒。
For each day, take the initial or final limit after the equatorial node and subtract the quadrant limit to obtain the white-path accumulation; subtract the tabulated accumulated degree and multiply the remainder by the difference rate; divide the result by 100, add it to the next lower tabulated accumulated difference, and obtain each day's accumulated difference; subtract one-sixth of the circuit of heaven and multiply the remainder by the fixed difference to obtain each day's equatorial latitude; subtract the quadrant limit when inner, add it when outer, to obtain each day's white-path polar distance in degrees, parts, and seconds.
106
求每交月離白道積度及宿次
To compute the white-path accumulated degree and lodge position for each node crossing
107
置定限度，與初末限相減相乘，退位為分，為定差； 〈（正交、中交後為加，半交後為減。）〉 以差加減正交後赤道積度，為月離白道定積度； 以前宿白道定積度減之，各得月離白道宿次及分。
Take the fixed limit degree, multiply it by the difference between the initial and final limits, shift the decimal place to obtain parts, and derive the fixed difference; (Add after the true and middle nodes; subtract after the half node.) apply the correction to the equatorial accumulation after the true node to obtain the fixed white-path accumulated degree; subtract each preceding lodge's fixed white-path accumulation to obtain the moon's white-path lodge position and fractional parts.
108
推定朔弦望加時月離白道宿度
To compute the white-path lodge degree at the epoch hour for each true new moon, quarter-moon, and full moon
109
各以月離赤道正交宿度距所求定朔弦望加時月離赤道宿度，為正交後積度； 滿象限，去之，為半交後； 又去之，為中交後； 再去之，為半交後； 視交後積度在半象已下，為初限； 已上，用減象限，為末限； 以初末限與定限度相減相乘，退位為分，分滿百為度，為定差；） 正交中交後為加，半交後為減。 以差加減月離赤道正交後積度，為定積度，以正交宿度加之，以其所當月離白道宿次去之，各得定朔弦望加時月離白道宿度及分秒。
For each event, measure from the equatorial true-node position to the sought equatorial lunar position at the epoch hour; the result is the accumulated degree after the true node; discard a full quadrant limit to mark the post-half-node segment; discard another quadrant limit to mark the post-middle-node segment; discard a third quadrant limit to mark the second post-half-node segment; if the post-node accumulation falls below half a quadrant, count it as the initial limit; if above, subtract the quadrant limit to obtain the final limit segment; multiply the fixed limit degree by the difference between the initial and final limits, shift the decimal place to parts, convert parts to degrees when they reach 100, and obtain the fixed difference;） Add after the true and middle nodes; subtract after the half node. Apply the correction to the equatorial accumulation after the true node to obtain the fixed accumulation, add the true-node position, subtract the corresponding white-path lodge span, and obtain each true new moon, quarter-moon, and full moon white-path position at the epoch hour in degrees, parts, and seconds.
110
求定朔弦望加時及夜半晨昏入轉
To compute rotation entry at the epoch hour, midnight, dawn, and dusk for each true new moon, quarter-moon, and full moon
111
置經朔弦望入轉日及分，以定朔弦望加減差加減之，為定朔弦望加時入轉； 以定朔弦望日下分減之，為夜半入轉； 以晨分加之，為晨轉； 昏分加之，為昏轉。
Take each standard event's rotation entry in days and parts and adjust by the true new-moon, quarter-moon, and full-moon correction to obtain rotation entry at the epoch hour; subtract the fractional day at the true event to obtain midnight rotation entry; add the dawn fraction to obtain dawn rotation entry; add the dusk fraction to obtain dusk rotation entry.
112
求夜半月度
To compute the lunar position at midnight
113
置定朔弦望日下分，以其入轉日轉定度乘之，萬約為加時轉度，以減加時定積度，余為夜半定積度； 依前加而命之，各得夜半月離宿度及分秒。 求晨昏月度
Take the fractional day at each true event, multiply by the tabulated daily rotation degree, divide by 10,000 to obtain the rotation arc at the epoch hour, subtract it from the fixed accumulation at the epoch hour, and obtain the fixed accumulation at midnight; count forward as before to obtain each midnight lunar position in degrees, parts, and seconds. To compute the lunar position at dawn and dusk
114
置其日晨昏分，以夜半入轉日轉定度乘之，萬約為晨昏轉度； 各加夜半定積度，為晨昏定積度； 加命如前，各得晨昏月離宿度及分秒。 求每日晨昏月離白道宿次
Take that day's dawn or dusk fraction, multiply by the tabulated daily rotation degree from midnight entry, and divide by 10,000 to obtain the dawn-dusk rotation arc; add each to the midnight fixed accumulation to obtain the fixed accumulation at dawn or dusk; count forward as before to obtain each dawn and dusk lunar position in degrees, parts, and seconds. To compute the daily dawn and dusk white-path lodge positions
115
累計相距日數轉定度，為轉積度； 與定朔弦望晨昏宿次前後相距度相減，余以相距日數除之，為日差； 〈（距度多為加，距度少為減。）〉 以加減每日轉定度，為行定度； 以累加定朔弦望晨昏月度，加命如前，即每日晨昏月離白道宿次。 〈（朔後用晨，望後用昏，朔望晨昏俱用。）〉
sum the daily rotation degrees over the intervening days to obtain the accumulated rotation arc; subtract the dawn-dusk lodge positions before and after each true event and divide the difference by the number of intervening days to obtain the daily correction; (Add when the separation arc is larger; subtract when it is smaller.) apply the correction to each day's tabulated rotation degree to obtain the true daily motion; accumulate from each true event's dawn-dusk lunar position, count forward as before, and obtain the daily dawn and dusk white-path lodge positions. (After new moon use the dawn position; after full moon use the dusk position; at new moon and full moon use both dawn and dusk.)