Volume 22 Treatises 3: Calendar 2 - Phases of the Month 5, Phases of Convergence 6, Phases of the Five Planets 7

1

步月離第五

Procedures for the Moon's Departure, part five.

2

轉終分：一十四萬四千一百一十，秒六千六十六。

Revolution terminator parts: 144,110, seconds 6,066.

3

轉終日：二十七日，餘二千九百，秒六千六十六。

Revolution terminator days: 27 days, remainder 2,900, seconds 6,066.

4

轉中日：一十三日，餘四千六十五，秒三千三十三。

Revolution mid-cycle days: 13 days, remainder 4,065, seconds 3,033.

5

朔差日：一，餘五千一百四，秒三千九百三十四。

New-moon difference day: 1 day, remainder 5,104, seconds 3,934.

6

象策：七日，餘二千一分，二十二秒半。

Phase stride: 7 days, remainder 2,001 parts, 22 and a half seconds.

7

秒母：一萬。

Second parent: 10,000.

8

上弦：九十一度，三十一分，四十二秒。

First quarter: 91 degrees, 31 parts, 42 seconds.

9

望：一百八十二度，六十二分，八十四秒。

Full moon: 182 degrees, 62 parts, 84 seconds.

10

下弦：二百七十三度，九十四分，二十六秒。

Last quarter: 273 degrees, 94 parts, 26 seconds.

11

月平行度：十三度，三十六分，八十七秒半。

Moon mean motion: 13 degrees, 36 parts, 87 and a half seconds.

12

分、秒母：一百。

Parts-and-seconds parent: 100.

13

七日：初數，四千六百四十八。末數，五百八十二。

Day seven: initial count, 4,648. Final count, 582.

14

十四日：初數，四千六十五。末數，一千一百六十五。

Day fourteen: initial count, 4,065. Final count, 1,165.

15

二十一日：初數，三千四百八十三。末數，一千七百四十七。

Day twenty-one: initial count, 3,483. Final count, 1,747.

16

二十八日：初數，二千九百一。末數，二千三百二十九。

Day twenty-eight: initial count, 2,901. Final count, 2,329.

17

求經朔弦望入轉

To find mean new moon, quarters, and full moon entry into revolution.

18

置天正朔積分，以轉終分及秒去之，不盡，如日法而一，為日，不滿為餘秒，即天正十一月經朔入轉日及餘秒。以象策累加之，去命如前，即得弦、望經日加時入轉日及餘秒。徑求次朔入轉。以朔差加之。

Set the celestial-standard new-moon accumulated parts; cast out full revolution terminator parts and seconds; divide the remainder by the day divisor for days, with the unfilled portion as remainder-seconds—that yields the celestial-standard eleventh month mean new moon's day and remainder-seconds of entry into revolution. Add the phase stride cumulatively; cast out and name as before—to obtain each quarter and full moon mean day-and-hour entry into revolution with its day and remainder-seconds. To find directly the next new moon's entry into revolution. Add the new-moon difference.

19

轉定分及積度朓棵率

Revolution fixed parts and accumulated-degree tiao–chuo rates.

20

求朔弦望入轉朓棵定數

To find the entry-into-revolution tiao–chuo fixed number for new moon, quarters, and full moon.

21

置入轉小餘，以其日算外，損益率乘之，如日法而一，所得，以損益積為定數。其四七日下余，如初數以下，初率乘之，初數而一，以損益朓棵積為定數。如初數以上，初數減之，餘乘末率，末數而一，便為朓棵定數。

Set the entry-into-revolution minor remainder; beyond the count for that day, multiply by the deficit–surplus rate and divide by the day divisor; apply the result to the accumulation to obtain the fixed number. On the fourth and seventh days and below in remainder: if below the initial count, multiply by the initial rate, divide by the initial count, and apply the result to the tiao–chuo accumulation for the fixed number. If at or above the initial count, subtract the initial count; multiply the remainder by the final rate and divide by the final count—that yields the tiao–chuo fixed number.

22

求朔弦望定日

To find fixed new moon, quarters, and full moon days.

23

置經朔、弦、望小餘，朓減朒加入氣入轉朓棵定數，滿與不足，進退大餘，命甲子算外，各得定朔、弦、望日辰及餘。定朔前幹名與後幹名同者，其月大；不同者，其月小。月內無中氣者為閏。視定朔小餘：秋分後，在日法四分之三以上者，進一日。春分後，定朔日出分與春分日出分相減之餘，三約之，用減四分之三，定朔小餘及此數以上者，亦進一日。或有交，虧初在日入前者，不進之。

Set the mean new moon, quarter, and full moon minor remainders; subtract tiao and add chuo, then add the entry-into-qi and entry-into-revolution tiao–chuo fixed number; on fullness or shortfall, advance or retreat the major remainder and name outside jiazi—thus each fixed new moon, quarter, and full moon with its day, double-hour, and remainder. When the stem before the fixed new moon matches the stem after, the month is long; when they differ, the month is short. A month with no mid-qi within it is intercalary. Inspect the fixed new moon minor remainder: after the autumn equinox, if it reaches three-quarters of the day divisor or above, advance one day. After the spring equinox, subtract the spring-equinox sunrise fraction from the fixed-new-moon sunrise fraction, divide the remainder by three, subtract that from three-quarters of the day divisor; if the fixed new moon minor remainder reaches this threshold or above, advance one day as well. If there is a crossing and first contact comes before sunset, do not advance.

24

定弦、望小餘在日出分以下者，退一日。望或有交，虧初在日出前者，小餘雖在日出後，亦退之。如十七日望者，又視定朔小餘在四分之三以下之數，春分後用減定之數。與定望小餘在日出分以上之數相較之；朔少望多者，望不退，而朔猶進之。望少朔多者，朔不進，而望猶退之。日月之行，有盈有縮，遲疾加減之數，或有四大三小；若隨常理，當察其時早晚，隨所近而進退之，使不過三大二小。

When the fixed quarter or full moon minor remainder falls below the sunrise fraction, retreat one day. At full moon, if there is a crossing and first contact comes before sunrise, retreat even when the minor remainder lies after sunrise. For a seventeenth-day full moon, also inspect whether the fixed new moon minor remainder lies at or below three-quarters; after the spring equinox apply the subtracted fixed number. Compare it with the fixed full moon minor remainder at or above the sunrise fraction; if the new moon is less and the full moon greater, do not retreat the full moon but still advance the new moon. if the full moon is less and the new moon greater, do not advance the new moon but still retreat the full moon. Sun and moon in their courses swell and shrink, with slow-and-fast corrections—and sometimes four long months and three short; by ordinary rule one should weigh whether the season runs early or late and advance or retreat toward the nearer case, keeping the count within three long and two short months.

25

求定朔弦望中積

To find the fixed new moon, quarter, and full moon middle accumulations.

26

置定朔、弦、望大小餘與經朔、弦、望大小餘相減之餘，以加減經朔、弦、望入氣日餘，經朔、弦、望少即加之，多即減之。即為定朔、弦、望入氣。以加其氣中積，即為定朔、弦、望中積。其餘以日法退除為分秒。

Subtract each mean new moon, quarter, and full moon major and minor remainder from its fixed counterpart; use the difference to add or subtract the mean entry-into-qi day remainders—add when the mean is less, subtract when greater. That yields each fixed new moon, quarter, and full moon entry into qi. Add this to the qi middle accumulation—that yields each fixed new moon, quarter, and full moon middle accumulation. Reduce the remainder by the day divisor to obtain parts and seconds.

27

求定朔弦望加時日度

To find the solar degree at the time of fixed new moon, first and last quarter, and full moon.

28

置定朔、弦、望約餘，以所入氣日損益率乘，盈縮損益。萬約之，以損益其下盈縮積，乃盈加縮減定朔弦望中積；又以冬至加時日躔黃道宿度加之，依宿次去之，即得定朔、弦、望加時日所在度及分秒。又置定朔、弦、望約餘，副置之。以乘其日盈縮之損益率，萬約之，應益者盈加縮減，應損者盈減縮加其副，滿百為分，分滿百為度，以加其日夜半日度，命之，各得其日加時日躔黃道宿次。若先於曆註定每日夜半日度，即為妙也。

Set the fixed new moon, quarter, and full moon reduced remainders, multiply by the entered qi's daily gain-and-loss rate, and apply excess-and-deficit adjustment. Reduce by 10,000, adjust the excess-and-deficit accumulation below, then by excess add and deficit subtract the fixed new moon, quarter, and full moon mean accumulation; add the winter solstice time-of-occurrence ecliptic solar lodge degree and remove by lodge sequence to obtain the sun's degree and parts-seconds at the fixed new moon, quarter, and full moon times. Again set the fixed new moon, quarter, and full moon reduced remainders and place a duplicate aside. Multiply by that day's excess-and-deficit gain-and-loss rate and reduce by 10,000; where gain is due, excess adds and deficit subtracts; where loss is due, excess subtracts and deficit adds to the duplicate; when parts fill 100 make fen, when fen fill 100 make degrees, add to that day's midnight solar degree and name it—thus each day's added-time ecliptic solar lodge is found. If each day's midnight solar degree is already recorded in the calendar, so much the better.

29

求定朔弦望加時月度

To find the lunar degree at the time of fixed new moon, first and last quarter, and full moon.

30

凡合朔加時日月同度，其定朔加時黃道日度，即為定朔加時黃道月度。弦、望各以弦、望度加定弦、望加時黃道日度，依宿次去之，即得定朔、弦、望加時黃道月度及分秒。

At conjunction the sun and moon share the same degree; the fixed new moon time-of-occurrence ecliptic solar degree is the fixed new moon time-of-occurrence ecliptic lunar degree. For first and last quarter and full moon, add the quarter or full moon arc to the fixed added-time ecliptic solar degree and remove by lodge sequence to obtain the fixed new moon, quarter, and full moon added-time ecliptic lunar degree and parts-seconds.

31

求夜半午中入轉

To find revolution entry at midnight and noon.

32

置經朔入轉，以經朔小餘減之，為經朔夜半入轉。又經朔小餘與半法相減之餘，以加減經朔加時入轉，經朔少，如半法加之；多，如半法減之。為經朔午中入轉。若定朔大餘有進退者，亦加減轉入，否則因經為定。每月累加一日，滿終日及餘秒去命如前，各得每日夜半、午中入轉。求夜半，因定朔夜半入轉累加之。求午中，因定朔午中入轉累加之。求加時入轉者，如求加時入氣術。

Set the mean new moon revolution entry and subtract the mean new moon minor remainder to obtain the mean new moon midnight revolution entry. Also take the remainder of the mean new moon minor remainder minus the half divisor to add or subtract the mean new moon added-time revolution entry; if the mean new moon is less, when like the half divisor add; if more, when like the half divisor subtract. That is the mean new moon noon revolution entry. If the fixed new moon major remainder advances or retreats, also add or subtract revolution entry; otherwise take the fixed from the mean. Each month add one day cumulatively; when full days and remainder seconds are reached, remove and name as before to obtain each day's midnight and noon revolution entry. To find midnight, cumulatively add from the fixed new moon midnight revolution entry. To find noon, cumulatively add from the fixed new moon noon revolution entry. To find added-time revolution entry, use the method for added-time qi entry.

33

求加時及夜半月度

To find the added-time and midnight lunar degrees.

34

置其日入轉算外轉定分，以定朔、弦、望小餘乘之，如日法而一，為加時轉分。分滿百為度。減定朔、弦、望加時月度，為夜半月度。以所得轉定分累加之，即得每日夜半月度。或朔至弦、望，或至後朔，皆可累加之。然近則差少，遠則差多。置所求前後夜半相距月度為行度，計其相距入轉積度，與行度相減，餘以相距日數除為日差，行度多以日差加每日轉定分，行度少以日差減每日轉定分，然後用之可中。或欲速求，用此數，欲究其故，宜用後術。

Set that day's revolution entry beyond the revolution count and the fixed revolution parts outside the count; multiply by the fixed new moon, quarter, or full moon minor remainder and divide by the day divisor to obtain added-time revolution parts. When parts fill 100, that makes degrees. Subtract from the fixed new moon, quarter, or full moon added-time lunar degree to obtain the midnight lunar degree. Cumulatively add the revolution fixed parts obtained to obtain each day's midnight lunar degree. Whether from new moon to quarter or full moon, or to the following new moon, all may be cumulatively added. Yet nearby the discrepancy is small; at a distance it is large. Set the lunar degrees between the sought preceding and following midnights as travel degrees; compute the accumulated revolution degrees between them and subtract from the travel degrees; divide the remainder by the intervening days for the daily difference; if travel degrees are greater, add the daily difference to each day's revolution fixed parts; if less, subtract—then use it and results will center. For a quick result use this method; to probe the underlying reason, use the later procedure.

35

求晨昏月度

To find the dawn and dusk lunar degrees.

36

置其日晨分，乘其日算外轉定分，日法而一，為晨轉分。用減定分，餘為昏轉分。又以朔、弦、望定小餘、乘轉定分，日法而一，為加時分。以減晨、昏轉分，為前；不足，覆減之，為後。乃前加後減加時月度，即晨昏月所在宿度及分秒。

Set that day's dawn parts, multiply the revolution fixed parts outside that day's count, and divide by the day divisor to obtain dawn revolution parts. Subtract from the fixed parts; the remainder is dusk revolution parts. Again multiply the fixed new moon, quarter, or full moon minor remainder by the revolution fixed parts and divide by the day divisor to obtain added-time parts. Subtract from the dawn and dusk revolution parts—that is before; if insufficient, reverse-subtract—that is after. Then add before and subtract after from the added-time lunar degree to obtain the lodge degree and parts-seconds of the moon at dawn and dusk.

37

求朔弦望晨昏定程

To find the fixed dawn-and-dusk intervals for new moon, first and last quarter, and full moon.

38

各以其朔昏定月，減上弦昏定月，餘為朔後昏定程。以上弦昏定月，減望昏定，餘為上弦後昏定程。以望晨定月，減下弦晨定月，餘為望後晨定程。以下弦晨定月，減後朔晨定月，餘為下弦後晨定程。

Each subtract that month's new moon dusk fixed moon from the first quarter dusk fixed moon; the remainder is the fixed dusk interval after new moon. Subtract the full moon dusk fixed moon from the first quarter dusk fixed moon; the remainder is the fixed dusk interval after the first quarter. Subtract the last quarter dawn fixed moon from the full moon dawn fixed moon; the remainder is the fixed dawn interval after full moon. Subtract the following new moon dawn fixed moon from the last quarter dawn fixed moon; the remainder is the fixed dawn interval after the last quarter.

39

求每日轉定度

To find the daily fixed revolution degrees.

40

累計每程相距日下轉積度，與晨昏定程相減，餘以相距日數除之，為日差，定程多加之，定程少減之。以加減每日轉定分，為轉定度。因朔、弦、望晨昏月，每日累加之，滿宿次去之，為每日晨昏月度及分秒。凡注曆：朔日以後注昏月，望後一日注晨月。古曆有九道月度，其數雖繁，亦難削去，具其術如後。

Cumulatively total the revolution accumulated degrees below each interval's intervening days and subtract from the dawn-and-dusk fixed interval; divide the remainder by the intervening days for the daily difference; if the fixed interval is greater, add; if less, subtract. Add or subtract the daily revolution fixed parts thereby to obtain the fixed revolution degrees. From the new moon, quarter, and full moon dawn-and-dusk moons, add each day cumulatively; when full, remove by lodge sequence to obtain each day's dawn-and-dusk lunar degree and parts-seconds. When annotating a calendar: from the new moon day onward record the dusk moon; from the day after full moon record the dawn moon. Ancient calendars had Nine Paths lunar degrees; though the reckoning is cumbersome, it is hard to omit—the method is given below.

41

求平交日辰

To find the mean conjunction date and double-hour.

42

置交終日及餘秒，以其月經朔加時入交泛日及餘秒減之，為平交入其月經朔加時後日及餘秒。以加其月經朔大小餘，其大餘命甲子算外，即平交日辰及餘秒。求次交者，以交終日及餘秒加之，大余滿紀法去之，命如前，即次平交日辰及餘秒。

Set the conjunction-cycle days and remainder seconds; subtract that month's mean new moon added-time generalized conjunction days and remainder seconds to obtain the mean conjunction days and remainder seconds after that month's mean new moon added time. Add them to the month’s mean new moon greater and lesser remainders, name the greater remainder from jiazi outside the count, and obtain the mean crossing day, double-hour, and remainder-seconds. For the next crossing, add the crossing-terminal days and remainder-seconds, cast out full era rules from the greater remainder, name as before, and obtain the next mean crossing day, double-hour, and remainder-seconds.

43

求平交入轉朓棵定數

Procedure to find the mean crossing’s entry into rotation tuoke fixed number.

44

置平交小餘，加其日夜半入轉餘，以乘其日損益率，日法而一，所得，以損益其下朓朒積，為定數。

Set the mean crossing lesser remainder, add midnight entry-into-rotation remainder for that day, multiply by the daily gain-and-loss rate, divide by the day divisor, and apply the result to the underlying tuoke accumulation to obtain the fixed number.

45

求正交日辰

Procedure to find the true crossing day and double-hour.

46

置平交小余，以平交入轉朓棵定數，朓減朒加之，滿與不足，進退日辰，即正交日辰及餘秒。與定朔日辰相距，即所在月日。

Set the mean crossing lesser remainder, apply the mean-crossing entry-into-rotation tuoke fixed number (tuoke subtract, tuoke add), and advance or retreat the day and double-hour as needed to obtain the true crossing day, double-hour, and remainder-seconds. Its interval from the fixed new moon day and double-hour gives the month and day in question.

47

求經朔加時中積

Procedure to find the mean new moon central accumulation at hour-of-addition.

48

各以其月經朔加入氣日及餘，加其氣中積餘，其日命為度，其餘以日法退除為分秒，即其經朔加時中積度及分秒。

For each month, add the mean new moon’s entry-into-qi days and remainder to the qi central accumulation remainder, treat days as degrees, convert the remainder by the day divisor into minutes and seconds, and obtain the mean new moon central accumulation at hour-of-addition.

49

求正交加時黃道月度

Procedure to find the ecliptic lunar degree at true crossing hour-of-addition.

50

置平交入經朔加時後算及餘秒，以日法通日，內餘，進二位，如三萬九千一百二十一分為度，不滿退除為分秒，以加其月經朔加時中積，然後以冬至加時黃道日度加而命之，即其得其月正交加時月離黃道宿度及分秒。如求次交者，以交終度及秒加而命之，即得所求。

Set mean crossing after the month’s mean new moon hour-of-addition (count and remainder-seconds), convert days through the day divisor with the remainder, shift two places, take 39,121 as the degree divisor, reduce the remainder to minutes and seconds, add to the mean new moon hour-of-addition central accumulation, then add to the winter solstice ecliptic solar degree at hour-of-addition and name the lodge—the true-crossing hour-of-addition ecliptic lunar degree and minutes and seconds for that month. As for the next crossing, add crossing-terminal degrees and seconds and name the lodge to obtain the value sought.

51

求黃道宿積度

Procedure to find ecliptic lodge accumulated degrees.

52

置正交時黃道宿全度，以正交加時月離黃道宿度及分秒減之，餘為距後度及分秒，以黃道宿度累加之，即各得正交後黃道宿積度及分秒。

Set the full ecliptic lodge at true crossing, subtract the true-crossing hour-of-addition lunar ecliptic degree and minutes and seconds, take the remainder as the post-distance, and add ecliptic lodge widths cumulatively to obtain each post-true-crossing ecliptic accumulation in degrees and minutes and seconds.

53

求黃道宿積度入初末限

Procedure to find whether an ecliptic lodge accumulation falls in the initial or terminal limit.

54

置黃道宿積度及分秒，滿交象度及分秒去之，如在半交象以下，為初限；以上者，以減交象度及分秒，餘為入末限。入交積度交象度並在交會術中。

Set the ecliptic lodge accumulation in degrees and minutes and seconds, cast out full crossing-image degrees and minutes and seconds; if it is at or below half the crossing image, it is in the initial limit; if above, subtract it from the crossing-image degrees and minutes and seconds; the remainder is entry into the terminal limit. Entry-into-crossing accumulation and crossing-image degree both belong to the crossing-conjunction procedure.

55

求月行九道宿度

Procedure to find the moon’s nine-path lodge degrees.

56

凡月行所交：冬入陰曆，夏入陽曆，月行青道。冬至夏至後，青道半交在春分之宿，當黃道東。立冬立夏後，青道半交在立春之宿，當黃道東南。至所沖之宿亦如之。冬入陽曆，夏入陰曆，月行白道。冬至夏至後，白道半交在秋分之宿，當黃道西。立冬立夏後，白道半交在立秋之宿，當黃道西北。至所沖之宿亦如之。春入陽曆，秋入陰曆，月行硃道。春分秋分後，硃道半交在夏至之宿，當黃道南。立春立秋後，硃道半交在立夏之宿，當黃道西南。至所沖之宿亦如之。春入陰曆，秋入陽曆，月行黑道。春分秋分後，黑道半交在冬至之宿當黃道北。立春立秋後，黑道半交在立冬之宿，當黃道東北。至所沖之宿亦如之。四序離為八節，至陰陽之所交，皆與黃道相會，故月行有九道。各以所入初末限度及分秒，減一百一度，餘以所入初末限度及分乘之，半而退位為分，分滿百為度，命為月道與黃道泛差。凡日以赤道內為陰，外為陽；月以黃道內為陰，外為陽。故月行正交，入夏至後宿度內為同名，入冬至後宿度內為異名。其在同名者，置月行與黃道泛差，九因八約之，為定差，半交後，正交前，以差減；正交後，半交前，以差加。此加減出入六度，正，如黃赤道相交同名之差，若較之漸異，則隨交所在，遷變不同也。仍以正交度距秋分度數，乘定差，如象限而一，所得為月道與赤道定差。前加者為減，減者為加。其中異名者，置月行與黃道泛差，七因八約之，為定差。半交後，以差加；正交後，半交前，以差減。此加減出入六度，異，如黃道赤道相交異名之差，較之漸同，則隨交所遷變不常。仍以正交度距春分度數，乘定差，如象限而一，所得為月道與赤道定差。前加者為減，減者為加。各加減黃道宿積度，為九道宿積度。以前宿九道積度減之，為其宿九道度及分。其分就近約為太半少。論春夏秋冬以四時日所在宿度為正。

In general, when the moon crosses: in winter it enters the yin sequence and in summer the yang sequence, and the moon runs the green path. After winter and summer solstice, the green path’s half-crossing stands at the spring-equinox lodges, east of the ecliptic. After Start of Winter and Start of Summer, the green path’s half-crossing lies at the Start-of-Spring lodges, southeast of the ecliptic. The same holds at the opposite lodges. In winter it enters the yang sequence and in summer the yin sequence, and the moon runs the white path. After winter and summer solstice, the white path’s half-crossing stands at the autumn-equinox lodges, west of the ecliptic. After Start of Winter and Start of Summer, the white path’s half-crossing lies at the Start-of-Autumn lodges, northwest of the ecliptic. The same holds at the opposite lodges. In spring it enters the yang sequence and in autumn the yin sequence, and the moon runs the cinnabar path. After the spring and autumn equinoxes, the cinnabar path’s half-crossing stands at the summer-solstice lodges, south of the ecliptic. After Start of Spring and Start of Autumn, the cinnabar path’s half-crossing lies at the Start-of-Summer lodges, southwest of the ecliptic. The same holds at the opposite lodges. In spring it enters the yin sequence and in autumn the yang sequence, and the moon runs the black path. After the spring and autumn equinoxes, the black path’s half-crossing stands at the winter-solstice lodges, north of the ecliptic. After Start of Spring and Start of Autumn, the black path’s half-crossing lies at the Start-of-Winter lodges, northeast of the ecliptic. The same holds at the opposite lodges. The four seasons split into eight nodes; wherever yin and yang cross they meet the ecliptic, so the moon has nine paths. Subtract each entry’s initial or terminal limit from 101°, multiply by that limit’s degrees and minutes, halve and shift one place for minutes, convert hundreds into degrees, and name the result the moon path’s provisional ecliptic difference. In general the sun treats inside the equator as yin and outside as yang; the moon treats inside the ecliptic as yin and outside as yang. At true crossing, lodge degrees after summer solstice within the ecliptic are same name; those after winter solstice within are different name. In the same-name case, set the moon path’s provisional ecliptic difference, multiply by nine and reduce by eight for the fixed difference; after half-crossing and before true crossing subtract it; after true crossing and before half-crossing add it. This adjustment stays within six degrees; when aligned it matches the same-name latitude difference at the ecliptic–equator crossing, but compared it diverges gradually as the crossing position shifts. Multiply the fixed difference by the true-crossing degree’s distance from the autumn equinox and divide by the image limit to obtain the moon path’s fixed equator difference. Prior additions become subtractions and prior subtractions become additions. In the different-name case, set the moon path’s provisional ecliptic difference, multiply by seven and reduce by eight for the fixed difference. After half-crossing add the difference; after true crossing and before half-crossing subtract it. This adjustment also stays within six degrees; in the different-name case it matches the ecliptic–equator crossing’s different-name difference and gradually converges when compared, though it still shifts with the crossing. Multiply the fixed difference by the true-crossing degree’s distance from the spring equinox and divide by the image limit to obtain the moon path’s fixed equator difference. Where the prior step added, subtract; where it subtracted, add. Apply the additions and subtractions to each ecliptic lodge accumulated degree to obtain the nine-path lodge accumulated degrees. Subtract the preceding lodge's nine-path accumulation to obtain this lodge's nine-path degree and parts. Round its parts to the nearest great, half, or small fraction. Spring, summer, autumn, and winter are judged by the lodge degree where the sun stands in each season's day.

57

求正交加時月離九道宿度

To find the nine-path lodge degree of moon departure at the true-nodal crossing hour.

58

以正交加時黃道日度及分，減一百一度，餘以正交度及分乘之，半而退位為分，分滿百為度，命為月道與黃道泛差。其在同名者，置月行與黃道泛差。九因八約之，為定差，以加；仍以正交度距秋分度數，乘定差，如象限而一，所得為月道與赤道定差，以減，其在異名者，置月行與黃道泛差，七因八約之，為定差，以減；仍以正交度距春分度數，乘定差，如象限而一，所得為月道與赤道定差，以加。置正交加時黃道月度及分，以二差加減之，即為正交加時月離九道宿度及分。

Set the ecliptic solar degree and parts at true-nodal crossing hour, subtract 101°, multiply the remainder by the nodal degree and parts, halve and shift one place for parts, convert hundreds to degrees, and name the result the general difference between lunar path and ecliptic. In the same-name case, set the general difference between lunar motion and the ecliptic. Reduce by nine-eighths to obtain the fixed difference, then add; Still multiply the fixed difference by the nodal degree's distance from the autumn equinox and divide by the quadrant limit for the lunar-path–equator fixed difference, and subtract; in the different-name case, set the lunar-motion–ecliptic general difference, reduce by seven-eighths for the fixed difference, and subtract; Still multiply the fixed difference by the nodal degree's distance from the spring equinox and divide by the quadrant limit for the lunar-path–equator fixed difference, and add. Set the ecliptic lunar degree and parts at true-nodal crossing hour, apply both differences, and obtain the nine-path lodge degree and parts of moon departure at true-nodal crossing hour.

59

求定朔望加時月所在度

To find the lodge degree of the moon at fixed new and full moon hours.

60

置定朔加時日躔黃道宿次，凡合朔加時，月行潛在日下，與太陽同度，是為加時月離宿次。各以弦、望度及分秒，加其所當弦、望加時月躔黃道宿度，滿宿次去之，命如前，各得定朔、弦、望加時月所在黃道宿度及分秒。

Set the ecliptic solar-progression lodge at fixed new moon; at conjunction the moon lies hidden beneath the sun at the same degree, which is the moon-departure lodge at that hour. Add the chord, quarter, and full-moon degrees and parts to each corresponding hour's ecliptic solar-progression lodge degree, discard full lodge counts and assign mansions as before, and obtain the ecliptic lodge degree and parts where the moon stands at each fixed new moon, quarter, and full moon hour.

61

求定朔弦望加時九道月度

To find the nine-path lunar degree at fixed new moon, quarters, and full moon.

62

各以朔、弦、望加時月離黃道宿度及分秒，加前宿正交後黃道積度，為定朔、弦、望加時正交後黃道積度。如前求九道積度，以前宿九道積度減之，餘為定朔、弦、望加時九道月離宿度及分秒。其合朔加時，若非正交，則日在黃道，月在九道，所入宿度，雖多少不同，考其兩極，若應繩准。故雲：月行潛在日下，與太陽同度，即為加時九道月度。其求晨昏夜半月度，並依前術。

Add each new, quarter, and full moon hour's moon-departure ecliptic lodge degree and parts to the preceding lodge's post-true-nodal ecliptic accumulation to obtain the post-nodal ecliptic accumulation at each fixed new moon, quarter, and full moon hour. Find the nine-path accumulation as before, subtract the preceding lodge's nine-path accumulation, and the remainder is the nine-path moon-departure lodge degree and parts at each fixed new moon, quarter, and full moon hour. At conjunction, if not at true nodal crossing, the sun lies on the ecliptic and the moon on the nine paths; entered lodges may differ in degree, yet the two poles align as if by plumb and level. Hence: when the moon moves hidden beneath the sun at the same degree, that is the nine-path lunar degree at the hour. Dawn, dusk, and midnight lunar degrees are all found by the preceding procedure.

63

步交會第六

Step Convergence 6

64

交終分：一十四萬二千三百一十九，秒九千三百六十八。

Convergence termination parts: 142,319, seconds 9,368.

65

交終日：二十七日，餘一千一百九分，秒九千三百六十八。

Convergence termination day: 27 days, remainder 1,109 parts, seconds 9,368.

66

交中日：十三，餘三千一百六十九，秋九千六百八十四。

Convergence middle day: 13, remainder 3,169, seconds 9,684.

67

交朔日：二，餘一千六百六十五，秒六百三十二。

Convergence new-moon day: 2, remainder 1,665, seconds 632.

68

交望日：十四，餘四千二，秒五千。

Convergence full-moon day: 14, remainder 4,002, seconds 5,000.

69

秒母：一萬。

Seconds mother: 10,000.

70

交終：三百六十三度，七十九分，三十六秒。

Convergence termination: 363°79′36″.

71

交中：一百八十一度，八十九分，六十八秒。

Convergence middle: 181°89′68″.

72

交象：九十度，九十四分，八十四秒。

Convergence image: 90°94′84″.

73

半交象：四十五度，四十七分，四十二秒。

Half convergence image: 45°47′42″.

74

日蝕既前限：二千四百。定法：二百四十八。

Solar eclipse totality anterior limit: 2,400. Fixed method: 248.

75

日蝕既後限：三千一百。定法：三百二十。

Solar eclipse totality posterior limit: 3,100. Fixed method: 320.

76

月蝕限：五千一百。

Lunar eclipse limit: 5,100.

77

月蝕既限：一千七百。定法：三百四十。

Lunar eclipse totality limit: 1,700. Fixed method: 340.

78

分秒母：一百。

Parts-seconds mother: 100.

79

求朔望入交

To find new and full moon entry into convergence.

80

置天正朔積分，以交終分去之，不盡，如日法而一，為日，不滿為餘，即天正十一月經朔加時入交泛日及餘秒。交朔加之，得次朔。交望加之，得次望。再加交望，亦得次朔。各為朔、望入交泛日及餘秒

Set the civil new-year new-moon accumulated parts, discard full convergence termination parts, divide the remainder by the day factor for days and parts, and obtain the eleventh-month mean new-moon hour general convergence day, remainder, and seconds. Add the convergence new-moon increment to obtain the next new moon. Add the convergence full-moon increment to obtain the next full moon. Add the convergence full-moon increment again to obtain the next new moon. Each yields the general convergence day, remainder, and seconds for every new and full moon.

81

求定朔每日夜半入交

To find daily midnight convergence entry for each fixed new moon.

82

各置入交泛日及餘秒，減去經朔、望小餘，即為定朔、望夜半入交泛日及餘秒。若定朔、望有進退者，亦進退交日，否則因經為定。大月加二日，小月加一日，餘皆加四千一百二十秒六百三十二，即次朔夜半入交。累加一日，滿交終日及餘秒去之，即每日夜半入交泛日及餘秒。

Set each general convergence day and remainder seconds, subtract the mean new- or full-moon minor remainder, and obtain the fixed new or full moon midnight general convergence day and remainder seconds. If a fixed new or full moon advances or retreats, adjust the convergence day likewise; otherwise take the mean value as fixed. Add two days in a long month and one in a short month; always add 4,120 seconds 632 to the remainder for the next new moon's midnight convergence entry. Add one day cumulatively, cast out full convergence termination days and remainder-seconds, and obtain each day's midnight general convergence day and remainder-seconds.

83

求定朔望加時入交

To find fixed new and full moon hour-of-addition entry into convergence.

84

置經朔、望加時入交泛日及餘秒，以入氣入轉朓棵定數，朓減朒加之，即定朔加時入交泛日及餘秒。

Set the canonical new- and full-moon hour-of-addition general convergence day and remainder-seconds, apply the entry-into-qi and entry-into-rotation tiao–chuo fixed numbers (tiao subtract, chuo add), and obtain the fixed new-moon hour-of-addition general convergence day and remainder-seconds.

85

求定朔望加時入交積度及陰陽曆

To find fixed new- and full-moon hour-of-addition convergence accumulated degrees and yang/yin calendar.

86

置定朔、望加時入交泛日，以日法通之，內餘，進二位，如三萬九千一百二十一而一為度，不滿退除為分秒，即定朔、望加時月行入交積度。以定朔、望加時入轉遲疾度，遲減疾加之，即月行之入交定積度。如交中度以下，入陽曆積度；以上，去之，餘為入陰曆積度。每日夜半，准此求之。

Set the fixed new- or full-moon hour-of-addition general convergence day, convert through the day divisor with the remainder, shift two places, divide by 39,121 for degrees, reduce the remainder to parts and seconds, and obtain the hour-of-addition moon's entry into convergence accumulated degree. Apply the fixed syzygy hour's entry-into-rotation slow-fast degree (slow subtract, fast add) to obtain the moon's fixed entry into convergence accumulated degree. At or below convergence middle counts as yang-calendar accumulated degree; above, subtract it; the remainder is yin-calendar accumulated degree. For each midnight, follow this procedure.

87

求月去黃道度

To find the moon's ecliptic latitude (departure from the yellow path).

88

視月入陰陽曆積度及分，如交象以下，為少象；以上，覆減交中，餘為老象。置所入老少象度於上，列交象度於下，相減相乘，倍而退位為分，滿百為度，用減所入老少象度及分，餘又與交中度相減相乘，八因之，以百一十除為分，分滿百為度，即得月去黃道度。

Inspect the entered yin- or yang-calendar accumulated degree and parts; at or below the convergence image counts as the young image; above, cover and subtract convergence middle; the remainder is the old image. Set the entered young or old image degree above and the convergence image degree below, subtract and multiply, double and shift one place for parts, convert hundreds to degrees, subtract from the entered young or old image degree and parts, then with the remainder subtract and multiply against convergence middle, multiply by eight, divide by 110 for parts, convert hundreds to degrees, and obtain the moon's departure from the yellow path in degrees.

89

求朔望加時入交常日及定日

To find new- and full-moon hour-of-addition general and fixed convergence days.

90

朔望入交泛日，以入氣朓棵定數，朓減朒加之，為入交常日。

Take the new- or full-moon general convergence day, apply the entry-into-qi tiao–chuo fixed number (tiao subtract, chuo add), and obtain the general convergence day.

91

又置入轉朓棵定數，進一位，一百二十七而一，所得朓減朒加入交常日，為入交定日及餘秒。

Set the entry-into-rotation tiao–chuo fixed number, shift one place, divide by 127, apply tiao subtract and chuo add to the general convergence day, and obtain the fixed convergence day and remainder-seconds.

92

求人交陰陽曆前後分

To find human-entry yang/yin calendar before and after parts.

93

視入交定日，如交中以下，為陽曆；以上，去之，為陰曆。如一日上下，以日法通日為分。為交後分。十三日上下，覆減交中，為交前分。

Inspect the fixed entry-into-convergence day; at or below convergence middle counts as yang calendar; above, subtract it — yin calendar. If within about one day, convert the day through the day divisor into parts. That is the after-convergence part. If within about thirteen days, cover and subtract convergence middle to obtain the before-convergence part.

94

求日月蝕其定餘

To find the fixed remainder for solar and lunar eclipse.

95

置朔、望入氣入轉朓棵定數，同名相從，異名相消，以一千三百三十七乘之，定朔、望加時入轉算外轉定分除之，所得，以朓減朒加經朔、望小餘，為泛餘。

Set the new- and full-moon entry-into-qi and entry-into-rotation tiao–chuo fixed numbers, combine same names and cancel different names, multiply by 1,337, divide by the fixed syzygy hour's entry-into-rotation count-outside rotation fixed parts, apply tiao subtract and chuo add to the canonical new- and full-moon minor remainder, and obtain the general remainder.

96

日蝕：視泛餘如半法以下，為中前分；半法以上，去半法，為中後分。置中前後分，與半法相減相乘，倍之，萬約為分，曰時差。中前，以時差減泛余為定餘，覆減半法，餘為午前分。中後，以時差加泛為定餘，減去半法，為午後分。

Solar eclipse: if the general remainder is at or below the half method, it is the before-middle part; at or above the half method, subtract the half method for the after-middle part. Set the before- and after-middle parts, subtract and multiply against the half method, double, and reduce by 10,000 to obtain the time difference. Before middle: subtract the time difference from the general remainder for the fixed remainder; cover and subtract the half method — the remainder is the before-noon part. After middle: add the time difference to the general remainder for the fixed remainder and subtract the half method to obtain the after-noon part.

97

月食：視泛餘在日入後、夜半前者，如日法四分之三以下，減去半法，為酉前分；四分之三以上，覆減日法，餘為酉後分，又視泛餘在夜半後、日出前者，如日法四分之一以下，為卯前分，四分之一以上，覆減半法，餘為卯後分。其卯酉前後分，自相乘。四因，退位，萬約為分，以加泛餘，為定餘。各置定餘，以發斂加時法求之，即得日月所蝕之辰刻。

Lunar eclipse: if the general remainder falls after sunset and before midnight, at or below three-quarters of the day divisor subtract the half method for the before-you part; above three-quarters, cover and subtract the day divisor for the after-you part; if after midnight and before sunrise, at or below one-quarter of the day divisor counts as before-mao, and above one-quarter cover and subtract the half method for after-mao. Self-multiply each of the before- and after-mao and before- and after-you parts. Multiply by four, shift one place back, reduce by 10,000 for parts, add to the general remainder, and obtain the fixed remainder. Set each fixed remainder and apply the aggregation-and-release hour-of-addition method to obtain the double-hour and ke of the eclipse.

98

求日月食甚日行積度

To find the solar motion accumulated degree at eclipse greatest.

99

置定朔、望食甚大小餘，與經朔、望大小餘相減之餘，以加減經朔、望入氣日小餘，經朔、望日少加多減。即為食甚入氣。以加其氣中積，為食甚中積。又置食甚入氣小餘，以所入氣日損益率盈縮之損益乘之，日法而一，以損益其日盈縮積；盈加縮減食甚中積，即為食甚日行積度及分。

Set the fixed syzygy eclipse-greatest major and minor remainders, subtract the canonical syzygy major and minor remainders, and with the difference adjust the canonical syzygy entry-into-qi day minor remainder (when the day count is less, add more and subtract less). That is the entry into qi at eclipse greatest. Add that to the qi mid-accumulation to obtain the mid-accumulation at eclipse greatest. Set the entry-into-qi minor remainder at eclipse greatest, multiply by that qi day's excess-and-deficit gain-and-loss rate, divide by the day divisor, and adjust that day's excess-and-deficit accumulation; excess adds and deficit subtracts from the eclipse-greatest mid-accumulation to obtain the eclipse-greatest daily solar motion accumulated degree and parts.

100

求氣差

To find qi difference.

101

置日食甚日行積度及分，滿中限去之，餘在象限以下，為初限；以上，覆減中限，為末限，皆有相乘，進二位，如四百七十八而一，所得，用減一千七百四十四，餘為氣差恆數。以午前後分乘之，半晝分除之，所得，以減恆數為定數。不及減，覆減之，為定數。應加者減之，減者加之。春分後，陽曆減，陰曆加；秋分後，陽曆加，陰曆減。春分前、秋分後各二日二千一百分為定氣，於此加減之。

Set the solar eclipse-greatest daily solar motion accumulated degree and parts, cast out the central limit, and if the remainder is at or below the image limit it is the initial limit; if above, cover and subtract the central limit for the final limit; subtract and multiply in each case, shift two places, divide by 478, subtract from 1,744, and obtain the qi-difference constant. Multiply by the before- or after-noon part, divide by half the day-length part, subtract from the constant, and obtain the fixed number. If subtraction fails, cover and subtract instead to obtain the fixed number. Where addition is indicated, subtract; where subtraction is indicated, add. After the spring equinox, yang calendar subtracts and yin calendar adds; after the autumn equinox, yang calendar adds and yin calendar subtracts. Before the spring equinox and after the autumn equinox, treat two days and 2,100 parts each as fixed qi and apply these additions and subtractions within that interval.

102

求刻差

To find ke difference.

103

置日食甚日行積度及分，滿中限去之，餘與中限相減相乘，進二位，如四百七十八而一，所得，為刻差恆數。以午前後分乘之，日法四分之一除之，所得為定數。若在恆數以上者，倍恆數，以所得數減之為定數，依其加減。冬至後，午前陽加陰減，午後陽減陰加。夏至後，午前陽減陰加，午後陽加陰減。

Set the solar-eclipse-greatest solar daily accumulated degree and parts, discard full middle limits, multiply the remainder by its distance from the middle limit, advance two places, divide by 478, and obtain the quarter-mark difference constant. Multiply by the before- and after-noon parts and divide by one quarter of the day divisor to obtain the fixed number. If above the constant, double the constant, subtract the obtained number for the fixed number, and apply the indicated additions or subtractions. After winter solstice, before noon add in the yang case and subtract in the yin case; after noon subtract in the yang case and add in the yin case. After summer solstice, before noon subtract in the yang case and add in the yin case; after noon add in the yang case and subtract in the yin case.

104

求日食去前後定分

To find the solar eclipse fixed distance from crossing before and after.

105

氣刻二差定數，同名相從，異名相消，為食差。依其加減去交前後分，為去交前後定分。視其前後定分，如在陽曆，即不食；如在陰曆，即有食之。如交前陰曆不及減，反減之，反減食差。為交後陽曆；交後陰曆不及減，反減之，為交前陽曆；即不食，交前陽曆不及減，反減之，為交後陰曆；交後陽曆，不及減，反減之，為交前陰曆；即日有食之。

Combine the qi- and quarter-mark-difference fixed numbers of the same name and cancel those of different name to obtain the eclipse difference. Apply the additions and subtractions to the crossing before-and-after parts to obtain the fixed crossing before-and-after parts. Inspect the before-and-after fixed parts: in the yang lunation there is no eclipse; in the yin lunation there is an eclipse. If before crossing in the yin lunation subtraction is insufficient, reverse the subtraction and reverse-subtract the eclipse difference. that becomes after-crossing yang lunation; if after crossing in the yin lunation subtraction is insufficient, reverse the subtraction to obtain before-crossing yang lunation; then there is no eclipse; if before crossing in the yang lunation subtraction is insufficient, reverse the subtraction to obtain after-crossing yin lunation; if after crossing in the yang lunation subtraction is insufficient, reverse the subtraction to obtain before-crossing yin lunation; then that day there is a solar eclipse.

106

求日食分

To find the solar eclipse parts.

107

視去交前後定分，如二千四百以下，為既前分，以二百四十八除為大分。二千四百以上，覆減五千五百，不足減者不食。為既後分，以三百二十除為大分。不盡，退除為秒，即得日食之分秒。

Inspect the fixed crossing before-and-after parts: at 2,400 or below take before-totality parts and divide by 248 for great parts. If 2,400 or above, cover and subtract 5,500; if subtraction is insufficient, there is no eclipse. These are after-totality parts; divide by 320 for great parts. Convert any remainder by retreating and dividing for seconds to obtain the solar eclipse parts and seconds.

108

求月食分

To find the lunar eclipse parts.

109

視去交前後分，不用氣刻差者。一千七百以下者，食既。以上，覆減五千一百，不足減者不食。餘以三百四十除為大分，不盡，退除為秒，即為月食之分秒也。去交分在既限以下，覆減既限，亦以三百四十除，為既內之大分。

Use the crossing before-and-after parts without the qi and quarter-mark differences. At 1,700 or below the eclipse is total. If above, cover and subtract 5,100; if subtraction is insufficient, there is no eclipse. Divide the remainder by 340 for great parts, convert any remainder into seconds, and obtain the lunar eclipse parts and seconds. If the distance-from-crossing parts are at or below the totality limit, cover and subtract the totality limit and likewise divide by 340 for the great parts within totality.

110

求日食定用分

To find the solar eclipse fixed usage parts.

111

置日食之大分，與三十分相減相乘，又以二千四百五十乘之，如定朔入轉算外轉定分而一，所得，為定用分。減定余，為初虧分。加定餘，為複圓分。各以發斂加時法求之，即得日食三限辰刻。

Set the solar eclipse great parts, subtract and multiply with 30 parts, multiply by 2,450, divide by the fixed new moon's outside-rotation fixed parts, and obtain the fixed usage parts. Subtract from the fixed remainder to obtain the first-diminishment parts. Add to the fixed remainder to obtain the restoration-of-roundness parts. Apply the expansion-contraction hour-addition method to each to obtain the solar eclipse's three-limit chen and ke.

112

求月食定用分

To find the lunar eclipse fixed usage parts.

113

置月食之大分，與三十分相減相乘，又以二千一百乘之，如定望入轉算外轉定分而一，所得，為定用分。加減定余，為初虧、複圓分。各如發斂加時法求之，即得月食三限辰刻。

Set the lunar eclipse great parts, subtract and multiply with 30 parts, multiply by 2,100, divide by the fixed full moon's outside-rotation fixed parts, and obtain the fixed usage parts. Add and subtract the fixed remainder to obtain the first-diminishment and restoration-of-roundness parts. Apply the expansion-contraction hour-addition method to each to obtain the lunar eclipse's three-limit chen and ke.

114

月食既者，以既內大分與十五相減相乘，又以四千二百乘之，如定望入轉算外轉定分而一，所得，為既內分。用減定用分，為既外分。置月食餘減定用分，為初虧。因加既外分，為食既。又加既內分，為食甚。既定餘分也。再加既內分，為生光。複加既外分，為複圓。各以發斂加時法求之，既得月食五限辰刻。

For a total lunar eclipse, take the within-totality great parts, subtract and multiply with 15, multiply by 4,200, divide by the fixed full moon's outside-rotation fixed parts, and obtain the within-totality parts. Subtract these from the fixed usage parts to obtain the outside-totality parts. Set the lunar eclipse remainder, subtract the fixed usage parts, and obtain first diminishment. Then add the outside-totality parts to obtain eclipse totality. Again add the within-totality parts to obtain greatest eclipse. This is already the fixed remainder parts. Again add the within-totality parts to obtain light generation. Again add the outside-totality parts to obtain restoration of roundness. Apply the expansion-contraction hour-addition method to each to obtain the lunar eclipse's five-limit chen and ke.

115

求月食入更點

To find the lunar eclipse's entry into watch and point.

116

置食甚所入日晨分，倍之，五約為更法。又五約更法，為點法。乃置月食初末諸分，昏分以上減昏分，晨分以下加晨分。如不滿更法為初更。不滿點法為一點。依法以次求之，既各得更點數。

Set the day's dawn parts at greatest eclipse, double them, and five-reduce to obtain the watch method. Again five-reduce the watch method to obtain the point method. Then set the lunar eclipse's beginning-and-end parts: above dusk parts subtract dusk parts; below dawn parts add dawn parts. If insufficient for the watch method, that is the first watch. Any remainder short of the point divisor counts as one point. Apply the method in order to each limit; each then yields its geng and point counts.

117

求日食所起

Procedure to find the solar eclipse’s direction of first contact.

118

食在既前，初起西南，甚于正南，複于東南；食在既後，初起西北，甚於正北，複于東北。其食八分以上，皆起正西，複於正東。此據正午地而論之。

Before full obscuration, first contact lies southwest, maximum due south, and recovery southeast; After full obscuration, first contact lies northwest, maximum due north, and recovery northeast. For eclipses of eight-tenths or greater, first contact is always due west and recovery due east. This assumes an observer on the local meridian (at noon).

119

求月食所起

Procedure to find the lunar eclipse’s direction of first contact.

120

月在陽曆：初起東北，甚於正北，複於西北。月在陰曆：初起東南，甚于正南，複於西南。其食八分以上，皆起正東，複於正西。此亦據午地而論之

With the moon in the yang half of the nodal month, first contact is northeast, maximum due north, and recovery northwest. With the moon in the yin half of the nodal month, first contact is southeast, maximum due south, and recovery southwest. For eclipses of eight-tenths or greater, first contact is always due east and recovery due west. This likewise assumes an observer on the local meridian.

121

求日食出入帶食所見分數

Procedure to find the fraction of a solar eclipse visible when it is caught at sunrise or sunset (horizon partial eclipse).

122

各以食甚小餘，與日出入分相減，餘為帶食差，以乘所食之分，滿定用分而一，月食既者，以既內分減帶食差，餘乘所食分，如既外分而一。不及減者，為帶食既出入。以減所食分，即日月出入帶食所見之分。其食甚在晝，晨為漸進，昏為已退。食甚在夜，晨為已退，昏為漸進。

For each case, subtract the sun’s rise-or-set parts from the eclipse-maximum minor remainder to get the horizon-partial difference; multiply by the eclipsed fraction and divide by the fixed application parts; for total lunar eclipse, subtract the within-totality parts from that difference, multiply the remainder by the eclipsed fraction, and divide by the outside-totality parts. If the subtraction cannot be completed, the eclipse is partially total at rise or set. Subtract the result from the eclipsed fraction to obtain the visible fraction at rise or set for sun or moon with horizon partial eclipse. If greatest eclipse is in daylight, at dawn the eclipse is still advancing and at dusk it has already retreated. If greatest eclipse is at night, at dawn it has already retreated and at dusk it is still advancing.

123

求日月食甚宿次

Procedure to find the lodge at solar or lunar eclipse maximum.

124

置日月食甚日行積度，望即更加半周天。以天正冬至加時黃道日度，加而命之，依黃道宿次去之，即各得日月食甚宿度及分。

Set the daily-motion accumulated degrees at eclipse maximum for sun or moon; at full moon add half a circuit of heaven. Add to the winter solstice ecliptic solar degree at hour-of-addition, name the lodges, and remove by the ecliptic lodge sequence to obtain each eclipse-maximum lodge degree and parts for sun or moon.

125

步五星第七

Procedures for the Five Planets, part seven.

126

周率：二百八萬六千一百四十二，五十四秒。

Cycle rate: 2,086,142, 54 seconds.

127

曆率：二千二百六十五萬五百七。

Calendar rate: 22,650,507.

128

曆度法：六萬二千一十四。

Calendar degree divisor: 62,014.

129

周日：三百九十八日，八十八分。

Circuit day: 398 days, 88 parts.

130

曆度：三百六十五度，二十四分，八十二秒。

Calendar degree: 365 degrees, 24 parts, 82 seconds.

131

曆中：一百八十二度，六十二分，四十一秒。

Calendar mid: 182 degrees, 62 parts, 41 seconds.

132

曆策：一十五度，二十一分，八十七秒。

Calendar stride: 15 degrees, 21 parts, 87 seconds.

133

伏見：一十三度。

Heliacal setting visibility: 13 degrees.

134

以下表格略

The tables below are omitted.

135

周率：四百七萬九千四十一，秒九十七。

Cycle rate: 4,079,041, 97 seconds.

136

曆率：三百五十九萬二千七百五十八，秒三十二。

Calendar rate: 3,592,758, 32 seconds.

137

曆度法：九千八百三十六半。

Calendar degree divisor: 9,836½.

138

周日：七百七十九日，九十三分，一十六秒。

Circuit day: 779 days, 93 parts, 16 seconds.

139

曆度：三百六十五度，二十四分，七十六秒。

Calendar degree: 365 degrees, 24 parts, 76 seconds.

140

曆中：一百八十二度，六十二分，三十八秒。

Calendar mid: 182 degrees, 62 parts, 38 seconds.

141

曆策：一十五度，二十一分，八十六秒。

Calendar stride: 15 degrees, 21 parts, 86 seconds.

142

伏見：一十九度。

Heliacal setting visibility: 19 degrees.

143

以下表格略

The tables below are omitted.

144

周率：一百九十七萬七千四百一十二，秒四十六。

Cycle rate: 1,977,412, 46 seconds.

145

曆率：五千六百二十二萬三千二百一十九。

Calendar rate: 56,222,319.

146

曆度法：一十五萬三千九百二十八。

Calendar degree divisor: 153,928.

147

周日：三百七十八日，九分，三秒。

Circuit day: 378 days, 9 parts, 3 seconds.

148

曆度：三百六十五度，二十五分，六十六秒。

Calendar degree: 365 degrees, 25 parts, 66 seconds.

149

曆中：一百八十二度，六十二分，八十三秒。

Calendar mid: 182 degrees, 62 parts, 83 seconds.

150

曆策：一十五度，二十一分，九十秒。

Calendar stride: 15 degrees, 21 parts, 90 seconds.

151

伏見：一十七度。

Heliacal setting visibility: 17 degrees.

152

以下表格略

The tables below are omitted.

153

周率：三百五萬三千八百四，秒二十三。

Cycle rate: 3,053,804, 23 seconds.

154

曆率：一百九十萬二百四十，秒一十一。

Calendar rate: 1,900,240, seconds 11.

155

曆度法：五千二百三十。

Calendar degree divisor: 5,230.

156

周日：五百八十三日，九十分，一十四秒。

Circuit day: 583 days, 90 parts, 14 seconds.

157

合日：二百九十一日，九十五分，七秒。

Conjunction day: 291 days, 95 parts, 7 seconds.

158

曆度：三百六十五度，二十四分，六十八秒。

Calendar limit: 365°24′68″.

159

曆中：一百八十二度，六十二分，三十四秒。

Calendar middle: 182°62′34″.

160

曆策：一十五度，二十一分，八十六秒。

Calendar policy: 15°21′86″.

161

伏見：一十度半。

Hidden appearance: 10½ degrees.

162

以下表格略

The tables below are omitted.

163

周率：六十萬六千三十一，秒八十四。

Circuit rate: 606,031, seconds 84.

164

曆率：一百九十一萬二百四十二，秒三十五。

Calendar rate: 1,910,242, seconds 35.

165

曆度法：五千二百三十。

Calendar degree divisor: 5,230.

166

周日：一百一十五日，八十七分，六十秒。

Circuit day: 115 days, 87 parts, 60 seconds.

167

合日：五十七日，九十三分，八十秒。

Conjunction day: 57 days, 93 parts, 80 seconds.

168

曆度：三百六十五度，二十四分，七十一秒。

Calendar limit: 365°24′71″.

169

曆中：一百八十二度，六十二分，三十五秒半。

Calendar middle: 182°62′35.5″.

170

曆策：一十五度，二十一分，八十六秒。

Calendar policy: 15°21′86″.

171

晨伏夕見：一十四度。

Morning hidden, evening visible: 14 degrees.

172

夕伏晨見：一十九度。

Evening hidden, morning visible: 19 degrees.

173

以下表格略

The tables below are omitted.

174

求五星天正冬至後平合及諸段中積中星

To find each planet's mean conjunction after heavenly-origin winter solstice and the central accumulation and central star for every segment.

175

置通積分，各以其星周率去之。不盡，為前合分。覆減周率，餘為後合分。如日法而一，不滿退除為分秒，即其星天正冬至後平合中積、中星。命為日，曰中積。命為度，曰中星。以段日累加中積，即為諸段中積。以平度累加中星，經退減之，即為諸段中星。

Set the comprehensive accumulated parts and, for each star, cast out full circuit rates. The remainder is the anterior conjunction parts. Subtract the circuit rate again; the remainder is the posterior conjunction parts. Divide by the day divisor, reduce the remainder to parts and seconds, and obtain that star's mean-conjunction central accumulation and central star after heavenly-origin winter solstice. Name the result as days—this is the central accumulation. Name the result as degrees—this is the central star. Add segment days cumulatively to the central accumulation to obtain each segment's central accumulation. Add mean degrees cumulatively to the central star, casting back when full, to obtain each segment's central star.

176

求五星平合及諸段入曆

To find each planet's mean conjunction and every segment's entry into the calendar cycle.

177

置前通積分，各加其星後合分，以曆率去之，不盡，各以其星曆度法除為度，不滿退為分秒，即為其星平合入曆度及分秒。以諸段限度累加之，即得諸段入曆。

Set the anterior comprehensive accumulated parts, add each star's posterior conjunction parts, cast out full calendar rates, divide the remainder by that star's calendar degree divisor for degrees, reduce the remainder to parts and seconds, and obtain the mean-conjunction entry-into-calendar degree and parts and seconds. Add each segment's limit degree cumulatively to obtain every segment's entry into the calendar cycle.

178

求五星平合及諸盈縮差

To find each planet's mean conjunction and every segment's excess-and-deficit difference.

179

各置其星其段入曆度及分秒，如在曆中以下，為在盈；以上，減去曆中，餘為在縮。以其星曆策除之為策數，不盡為入策度及分，命策數算外，以其策數下損益率乘之，如曆策而一為分，以損益其下盈縮積度，即為其星其段盈縮定差。

Set each star and segment's entry-into-calendar degree and parts and seconds; at or below the calendar middle counts as excess; if above, subtract the calendar middle; the remainder is in deficit. Divide by that star's calendar policy for the policy count, take the remainder as entry-into-policy degree and parts, name from outside the policy count, multiply the tabulated loss-and-gain rate, divide by the calendar policy for parts, apply it to the underlying excess-and-deficit accumulation, and obtain that star and segment's excess-and-deficit fixed difference.

180

求五星平合及諸段定積

To find each planet's mean conjunction and every segment's fixed accumulation.

181

各置其星其段中積，以其盈縮定差盈加減之。即其段定積日及分。以加天正冬至大餘及約分，滿紀法六十去之，不盡，即為定日及加時分秒。不滿命甲子算外，即得日辰。

Set each star and segment's central accumulation and add or subtract the excess-and-deficit fixed difference according to excess or deficit. The result is the segment's fixed-accumulation days and parts. Add to the heavenly-origin winter solstice greater remainder and approximate parts, cast out full era rules of 60, and obtain the fixed day and hour-of-addition parts and seconds. If it does not fill, name from jiazi outside the count to obtain the day and double-hour.

182

求五星及諸段所在日月

To find the sun and moon dates for each planet and segment.

183

各置其段定積日及分，以加天閏日及分，滿朔策及約分除之為月數，不盡，為入月已來日數及分。其月數命天正十一月算外，即得其段入月經朔日數及分，以日辰相距為所在定朔月日。

Set each segment's fixed-accumulation days and parts, add heavenly intercalary days and parts, divide by the new-moon policy and approximate parts for month count, and take the remainder as days and parts elapsed within the month. Name the month count from heavenly-origin month eleven outside the count, obtain the segment's days and parts since mean new moon, and take the day-and-double-hour interval as the fixed new-moon month and day.

184

求五星平合及諸段加時定星

To find each planet's mean conjunction and every segment's hour-of-addition fixed star.

185

各置中星，以盈縮定差盈加縮減之，金星倍之，水星三因之，然後加減。即為五星諸段定星。以加天正冬至加時黃道日度，依宿命之，即其星其段加時所在宿度及分秒。

Set each central star, apply the excess-and-deficit fixed difference (add in excess, subtract in deficit), double for Venus and triple for Mercury, then add or subtract. The result is each planet's fixed star for every segment. Add to the heavenly-origin winter solstice ecliptic solar degree at hour of addition, name the lodge accordingly, and obtain that star and segment's lodge degree and parts and seconds at hour of addition.

186

求五星諸段初日晨前夜半定星

To find each planet's fixed star at dawn before midnight on the first day of each segment.

187

各以其段初行率，乘其段定積日下加時分，百約之，乃順減退加其日加時定星，即為其段初日晨前夜半定星所在宿度。

Multiply each segment's initial motion rate by the hour-of-addition parts under the segment's fixed-accumulation day, reduce by 100, subtract in forward motion and add in retrograde to that day's hour-of-addition fixed star, and obtain the lodge degree of the segment-first-day dawn-before-midnight fixed star.

188

求諸段日率度率

To find each segment's day rate and degree rate.

189

各以其段日辰距後段日辰為日率。以其段夜半宿次與後段夜半宿次相減，餘為夜率。

Take each segment's day-and-double-hour interval to the next segment as the day rate. Subtract this segment's midnight lodge sequence from the next segment's; the remainder is the degree rate.

190

求諸段平行分

To find each segment's parallel motion parts.

191

各置其段度率及分秒，以其段日率除之，即其段平行度及分秒。

Set each segment's degree rate and parts and seconds, divide by that segment's day rate, and obtain its parallel degree and parts and seconds.

192

求諸段總差日差

To find each segment's total difference and daily difference.

193

以本段前後平行分相減，余為其段泛差。假令求木星次疾*差，乃以順疾、順遲平行分相減，餘為次疾泛差。他皆仿此。倍而退位為增減差，加減其段平行分，為初末日行分。前多後少者，加為初，減為末。前少後多者，減為初，加為末。倍增減差為總差，以日率減一除之，為日差。

Subtract this segment's prior and subsequent parallel motion parts; the remainder is its general difference. To find Jupiter's second-fast difference, subtract forward-swift from forward-slow parallel motion parts; the remainder is the second-fast general difference. All other cases follow this pattern. Double the general difference and shift back one place for the increase-decrease difference, then add or subtract it from parallel motion parts to obtain initial and final day motion parts. If the prior exceeds the subsequent, add for the initial day and subtract for the final day. If the prior is less and the subsequent more, subtract for the initial day and add for the final day. Double the increase-decrease difference for the total difference and divide by the day rate minus one to obtain the daily difference.

194

求前後伏遲退段增減差

To find the increase-decrease differences for prior and subsequent concealment, slow, and retrograde segments.

195

前伏者，置後段初日行分，加其日差之半，為末日行分。後伏者，置前段末日行分，加其日差之半，為初日行分。以減伏段平行分，餘為增減差。前遲者，置前段末日行分，倍其日差減之，為初日行分。後遲者，置後段初日行分，倍其日差減之，為末日行分。以遲段平行分減之，餘為增減差。前後近留之遲段。

For prior concealment, set the next segment's initial-day motion parts, add half its daily difference, and obtain final-day motion parts. For subsequent concealment, set the prior segment's final-day motion parts, add half its daily difference, and obtain initial-day motion parts. Subtract from the concealment segment's parallel motion parts; the remainder is the increase-decrease difference. For prior slow motion, set the prior segment's final-day motion parts, subtract double its daily difference, and obtain initial-day motion parts. For subsequent slow motion, set the next segment's initial-day motion parts, subtract double its daily difference, and obtain final-day motion parts. Subtract from the slow segment's parallel motion parts; the remainder is the increase-decrease difference. Slow segments before and after that lie near station.

196

木、火、土三星退行者，六因平行分，退一位，為增減差。

For Jupiter, Mars, and Saturn in retrograde, multiply parallel motion parts by six, shift back one place, and obtain the increase-decrease difference.

197

金星前後伏退，三因平行分，半而退位，為增減差。前退者，置後段初日行分，以其日差減之，為末日行分，後退者，置前段末日行分，以其日差減之，為初日行分。以本段平行分減，餘為增減差。

For Venus at prior and subsequent concealment and retrograde, multiply parallel motion parts by three, halve and shift back one place, and obtain the increase-decrease difference. For prior retrograde, set the next segment's initial-day motion parts, subtract its daily difference, and obtain final-day motion parts; for subsequent retrograde, set the prior segment's final-day motion parts, subtract its daily difference, and obtain initial-day motion parts. Subtract from this segment's parallel motion parts; the remainder is the increase-decrease difference.

198

水星，半平行分為增減差，皆以增減差加減平行分，為初末日行分。前多後少，加初減末；前少後多，減初加末。又倍增減差為總差，以日率減一除之，為日差。

For Mercury, take half the parallel motion parts as the increase-decrease difference, then add or subtract it from parallel motion parts to obtain initial and final day motion parts. If the prior exceeds the subsequent, add for the initial day and subtract for the final day; if the prior is less and the subsequent more, subtract for the initial day and add for the final day. Again double the increase-decrease difference for the total difference and divide by the day rate minus one to obtain the daily difference.

199

求每日晨前夜半星行宿次

To find each day's star lodge sequence at dawn before midnight.

200

各置其段初日行分，以日差累損益之後少則損之，後多則益之。為每日行度及分秒。乃順加退減之，滿宿次去之，即得每日晨前夜半星行宿次。視前段末日、後段初日行分相較之數，不過一二日差為妙。或多日差數倍，或顛倒不倫，當類會前後增減差稍損益之，使其有倫，然後用之。或前後平行俱多俱少，則平注之。或總差之秒，不盈一分，亦平注之。若有不倫而平注之得倫者，亦平注之。

Set each segment's initial-day motion parts and apply the daily difference cumulatively; subtract when the subsequent value is less and add when it is more. The result is each day's motion in degrees, parts, and seconds. Then add in forward motion and subtract in retrograde, cast out full lodge sequences, and obtain each day's star lodge sequence at dawn before midnight. Compare the prior segment's final-day and next segment's initial-day motion parts; a gap of no more than one or two daily differences is ideal. If the daily difference is many times too large or the sequence is inverted and incoherent, reconcile it with the surrounding increase-decrease differences, adjust slightly until coherent, and then apply it. If prior and subsequent parallel motion are both too large or both too small, distribute the correction evenly. If the total difference's seconds do not reach one part, distribute evenly as well. If the values are incoherent but even distribution restores coherence, distribute evenly.

201

求五星平合及見伏入氣

To find each planet's mean conjunction and appearance-and-concealment entry into qi.

202

置定積，以氣策及約分除之，為氣數，不滿為入氣日及分秒，命天正冬至算外，即所求平合及伏見入氣日及分秒。

Set the fixed accumulation, divide by the qi divisor and approximate parts for the qi count, take the remainder as days, parts, and seconds entered into qi, name from heavenly-origin winter solstice outside the count, and obtain the mean conjunction and concealment-and-appearance entry-into-qi days, parts, and seconds.

203

求五星平合及見伏行差

To find each planet's mean conjunction and appearance-and-concealment motion difference.

204

各以其段初日星行分與其太陽行分相減，餘為行差。若金在退行，水在退合者，相並為行差。如水星夕伏晨見者，直以太陽行分為行差。

Subtract the sun's motion parts from each segment's initial-day star motion parts; the remainder is the motion difference. If Venus is retrograde or Mercury at retrograde conjunction, add the two for the motion difference. When Mercury is evening concealed and morning visible, take the sun's motion parts directly as the motion difference.

205

求五星定合見伏泛積

To find each planet's fixed conjunction, appearance, and concealment general accumulation.

206

木、火、土三星，各以平合晨疾夕伏定積，便為定合定見定伏泛積。金、水二星，置其段盈縮差，水星倍之。各以行差除之，為日，不滿退除為分秒。若在平合夕見晨伏者，盈減縮加；如在退合夕伏晨見者，盈加縮減。皆以加減定積，為定合定見定伏泛積。

For Jupiter, Mars, and Saturn, the mean-conjunction morning-swift evening-concealed fixed accumulation is the fixed conjunction, appearance, and concealment general accumulation. For Venus and Mercury, set that segment's excess-and-deficit difference, doubling it for Mercury. Divide each by the motion difference for days; reduce any remainder to parts and seconds. At mean conjunction with evening visibility and morning concealment, subtract in excess and add in deficit; At retrograde conjunction with evening concealment and morning visibility, add in excess and subtract in deficit. Apply the indicated additions and subtractions to the fixed accumulation to obtain the fixed conjunction, appearance, and hiding general accumulation.

207

求五星定合定積定星

To find each planet's fixed conjunction fixed accumulation and fixed star.

208

木、火、土三星，各以平合行差除其日太陽盈縮差，為距合差日。以太陽盈縮差減之，為距合差度。日在盈曆，以差日差度減之。在縮，加之。加減其星定合泛積，為定合定積定星。

For Jupiter, Mars, and Saturn, divide each day's solar excess-and-deficit difference by the mean-conjunction motion difference to obtain the conjunction-distance difference day. Subtract from the solar excess-and-deficit difference to obtain the conjunction-distance difference degree. When the sun is in the excess lunation, subtract the difference day and difference degree. When in deficit, add them. Add or subtract on that star's fixed-conjunction general accumulation to obtain the fixed conjunction fixed accumulation and fixed star.

209

金、水二星順合退合，各以平合退合行差除其日太陽盈縮差，為距合差日。順加退減太陽盈縮差，為距合差度。順在盈曆，以差日差度加之；在縮，減之。退在盈曆，以差日減之，差度加之；在縮，以差日加之，差度減之。皆以加減其星定合及再定合泛積，為定合再定合定積定星。以冬至大餘及約分，加定積，滿紀法去，命，即得定合日辰。以冬至加時黃道日度，加定星，滿宿次去之，即得定合所在宿次。其順退所在盈縮，太陽盈縮也。

For Venus and Mercury in direct or retrograde conjunction, divide that day's solar excess-and-deficit difference by the corresponding mean-conjunction motion difference to obtain the conjunction-distance difference day. Add the solar excess-and-deficit difference in direct motion and subtract it in retrograde motion to obtain the conjunction-distance difference degree. In direct motion when in the excess lunation, add the difference day and difference degree; when in deficit, subtract them. In retrograde motion when in the excess lunation, subtract the difference day and add the difference degree; when in deficit, add the difference day and subtract the difference degree. Apply the indicated additions and subtractions to that star's fixed conjunction and second fixed-conjunction general accumulation to obtain the fixed conjunction, second fixed conjunction, fixed accumulation, and fixed star. Add the fixed accumulation to the winter solstice greater remainder and approximate parts, cast out full era rules, name the count, and obtain the fixed conjunction day and double-hour. Add the fixed star to the winter solstice ecliptic solar degree at hour of addition, remove full lodge sequences, and obtain the lodge of the fixed conjunction. Whether the motion is direct or retrograde, the excess or deficit used is the sun's excess and deficit.

210

求木水土三星定見伏定積日

To find the fixed appearance, hiding, and fixed-accumulation days for Jupiter, Mars, and Saturn.

211

各置其星定見伏泛積，晨加夕減象限日及分秒，半中限為象限，如中限以下，自相乘，以上，覆減歲周日及分秒，餘亦自相乘，滿七十五而一，所得，以其星伏見度乘之，十五除之，為差。其差如其段行差而一，為日，不滿退除為分秒。見加伏減泛積為定積。加命如前，即得日辰也。

Set each star's fixed appearance-and-hiding general accumulation; at dawn add and at dusk subtract the image-limit day and parts and seconds, taking half the middle limit as the image limit; square values at or below the middle limit, and for values above overlay-subtract from the solar circuit day and parts and seconds and square the remainder; divide by 75, multiply by that star's heliacal appearance-and-hiding degree, and divide by 15 to obtain the difference. Divide the difference by that segment's motion difference for days; cast back the remainder into parts and seconds. Add at appearance and subtract at hiding on the general accumulation to obtain the fixed accumulation. Add and name as before to obtain the day and double-hour.

212

求金水二星定見伏定積日

To find the fixed appearance, hiding, and fixed-accumulation days for Venus and Mercury.

213

各以伏見日行差，除其日太陽盈縮差，為日。若晨伏夕見，日在盈曆，加之，在縮，減之。如夕伏晨見，日在盈曆，減之，在縮，加之。加減其星泛積為常積。視常積，如中限以下，為冬至後，以上，去之，餘為夏至後。其二至後，如象限以下，自相乘，以上，覆減中限，亦自相乘，各如法而一，為分。冬至後晨，夏至後夕，以一十八為法。冬至後夕，夏至後晨，以七十五為法。以伏見度乘之，十五除之，為差。差滿行差而一，為日，不滿退除為分秒。加減常積為定積。冬至後晨見夕伏，加之；夕見晨伏，減之。夏至後晨見夕伏，減之；夕見晨伏，加之也。加命如前，即得定見伏日辰。

For each star, divide that day's solar excess-and-deficit difference by the appearance-hiding daily motion difference to obtain days. For morning hiding and evening appearance, add when the sun is in the excess lunation and subtract when in deficit. For evening hiding and morning appearance, subtract when the sun is in the excess lunation and add when in deficit. Add or subtract on that star's general accumulation to obtain the regular accumulation. Inspect the regular accumulation: at or below the middle limit is after winter solstice; above it, cast out the middle limit and take the remainder as after summer solstice. After each solstice, square values at or below the image limit; above it, overlay-subtract from the middle limit and square the remainder; divide each by the appropriate divisor to obtain parts. After winter solstice at dawn and after summer solstice at dusk, use 18 as the divisor. After winter solstice at dusk and after summer solstice at dawn, use 75 as the divisor. Multiply by the heliacal appearance-and-hiding degree and divide by 15 to obtain the difference. When the difference fills the motion difference, take one for days; cast back the remainder into parts and seconds. Add or subtract on the regular accumulation to obtain the fixed accumulation. After winter solstice, for morning appearance and evening hiding, add; for evening appearance and morning hiding, subtract. After summer solstice, for morning appearance and evening hiding, subtract; for evening appearance and morning hiding, add. Add and name as before to obtain the fixed appearance-and-hiding day and double-hour.

214

其水星，夕疾，在大暑氣初日至立冬氣九日三十五分以下者，不見。晨留，在大寒氣初日至立夏氣九日三十五分以下者，春不晨見，秋不夕見者，亦舊有之矣。

For Mercury in evening swift motion from the start of Great Heat through day 9, part 35 of Start of Winter, it is invisible. When in morning station from the start of Great Cold through day 9, part 35 of Start of Summer, it is not seen at dawn in spring or at dusk in autumn—an old observation as well.

215

古之言天者有三家：一曰蓋天，二曰宣夜，三曰渾天。漢靈帝時，蔡邕于朔方上書，言「宣夜之學，絕無師法」；《周髀》術數具存，考驗天狀，多所違失；惟有渾天為近，最得其情，近世太史候台銅儀是也。立八心體圓而具天地之形，以正黃道赤道之表裏，以行日月之度數，步五緯之遲速，察氣候之推遷，精微深妙，百代所不可廢者也。然傳曆久遠，製造者眾，測候占察，互有得失，張衡之制，謂之《靈憲》，史失其傳。魏、晉以來，官有其器，而無本書，故前志亦闕。吳中常侍王蕃雲：「渾天儀者，羲和之舊器，謂之機衡。」積代相傳，沿革不一。宋太平興國中，蜀人張思訓首創其式，造之禁中，逾年而成，詔置文明殿東鼓樓下，曰「太平渾儀」。自思訓死，璣衡斷壞，無複知其法制者。景德中，曆官韓顯符依仿劉曜時、孔挺、晁崇之法，失之簡略。景祐中，冬官正舒易簡乃用唐梁令瓚、僧一行之法，頗為詳備，亦失之於密而難為用。元祐時，尚書右丞蘇頌與昭文館校理沈括奉敕詳定《渾儀法要》，遂奏舉吏部勾當官韓公廉通《九章勾股法》，常以推考天度與張衡、王蕃、僧一行、梁令瓚、張思訓法式，大綱可以尋究。若據算術考案象器，亦能成就，請置局差官製造。詔如所言。奏鄭州原武主簿王沇之，太史局官周日嚴、于太古、張促宣，同行監造。制度既成，詔置之集英殿，總謂之渾天儀。公廉交造儀時，先撰《九章勾股驗測渾天書》一卷，貯之禁中，今失其傳，故世無知者。

Ancient theorists of Heaven recognized three schools: Canopy Heaven, Overnight Heaven, and Spherical Heaven. Under Emperor Ling of Han, Cai Yong wrote from Shuofang that "the Overnight Heaven school has no surviving master lineage"; "the Zhou Gnomon methods are fully preserved, yet testing them against the sky shows wide error"; "only Spherical Heaven comes closest to the truth—the recent Imperial Astronomer's bronze observation-platform instrument is the example." It mounts eight pivot-tiers in a round body embodying Heaven and Earth, fixes the ecliptic and equator inside and out, carries the sun and moon through their degrees, paces the five planets' motion, and tracks seasonal change—subtle and indispensable for every age. Yet over long transmission many makers differed in measurement, observation, and prognostication. Zhang Heng's design, the Ling Xian, was lost from the histories. From Wei and Jin on the court kept the instrument without the original text, so earlier bibliographies omit it as well. Wang Fan, Palace Attendant of Wu, wrote: "The spherical armillary is Xihe's ancient instrument, called the Mechanism and Balance." Handed down through generations, its forms have varied without end." During Song Taiping Xingguo, the Shu artisan Zhang Sixun first devised the design and built it in the inner palace; after more than a year it was finished, and by edict it was installed beneath the east drum tower of the Hall of Cultured Brilliance as the "Taiping Armillary Instrument." When Sixun died, the armillary regulator lay broken, and no one again knew how it had been made. In Jingde, calendar officer Han Xianfu followed the methods of Liu Yao's day, Kong Ting, and Chao Chong, but the result was too spare. In Jingyou, Director of the Winter Office Shu Yijian used the Tang methods of Liang Lingzan and the monk Yixing; though detailed, it was too intricate for practical use. In Yuanyou, Vice Director Su Song and Hall collator Shen Kuo were ordered to fix the Essentials of the Armillary Instrument; they then recommended Ministry clerk Han Gonglian, versed in the Nine Chapters' right-triangle methods, who had long compared celestial degrees with the formulas of Zhang Heng, Wang Fan, Yixing, Liang Lingzan, and Zhang Sixun and could master the main outlines. By arithmetic the instrument could be verified and completed; they asked that a bureau be set up to assign officials to build it. The throne approved as proposed. They named Zhengzhou Yuanwu chief clerk Wang Yanzhi and Astronomy Bureau officers Zhou Riyan, Yu Taigu, and Zhang Cuxuan to supervise the work together. When the mechanism was finished, an edict placed it in the Hall for Assembling Excellence; the whole was called the Armillary Sphere. While building the instrument, Gonglian first wrote a fascicle on verifying the armillary sphere with Nine Chapters right-triangle methods and lodged it in the inner palace; that text is now lost, so the world no longer knows his methods.

216

舊制渾儀，規天矩地，機隱於內，上布經躔，次具日月五星行度，以察其寒暑進退，如張衡渾天、開元水運銅渾儀者，是也。久而不合，乖於施用。公廉之制則為輪三重：一曰六合儀，縱置地渾中，即天經環也，與地渾相結，其體不動；二曰三辰儀，置六合儀內；三曰四遊儀，置三辰儀內。植四龍柱於地渾之下，又置鰲雲於六合儀下。四龍柱下設十字水趺，鑿溝道通水以平高下。別設天常單環於六合儀內，又設黃道赤道二單環，皆置三辰儀內，東西相交，隨天運轉，以驗列舍之行。又為四象環，附三辰儀，相結於天運環，黃赤道兩交為直距二縱置於四遊儀內。北屬六合儀地渾之上，以正北極出地之度。南屬六合儀地渾之下，以正南極入地之度。此屬儀之大形也。直距內夾軒望筒一，於筒之半設關軸，附直距上，使運轉低昂，筒常指日，日體常在筒竅中，天西行一周，日東移一度，仍以窺測四方星度，皆斟酌李淳風、孔挺、韓顯符、舒易簡之制也。三辰儀上設天運環，以水運之。水運之法始于漢張衡，成于唐梁令瓚及僧一行，複于太平興國中張思訓，公廉今又變正其制，設天運環，下以天柱關軸之類上動渾儀，此新制也。

The old armillary squared heaven and rectified earth, mechanisms hidden within, with celestial paths above and the sun, moon, and five planets' courses set out to read cold and heat—like Zhang Heng's armillary heaven or the Kaiyuan water-driven bronze armillary. Kept long, it fell out of alignment and proved hard to use. Gonglian's design used three nested wheels: first, the Six Harmonies Instrument, set upright within the Earth Sphere—the heaven-meridian ring joined to the Earth Sphere, its frame fixed; second, the Three Chronograms Instrument within the Six Harmonies Instrument; third, the Four Paces Instrument within the Three Chronograms Instrument. Four dragon pillars stood beneath the Earth Sphere, with a turtle-and-cloud pedestal under the Six Harmonies Instrument. Below the pillars a cross-shaped water trough was cut with channels to level the instrument. Within the Six Harmonies Instrument he added a celestial constant ring; within the Three Chronograms Instrument he set yellow-path and red-path rings that crossed east and west and turned with heaven to verify the lodges' motion. He also made a Four Images ring on the Three Chronograms Instrument, linked to the heaven-motion ring, with the yellow and red crossings as vertical distances inside the Four Paces Instrument. To the north it joined the Six Harmonies Instrument above the Earth Sphere to fix how far the north pole rises above the ground. To the south it joined the instrument below the Earth Sphere to fix how far the south pole sinks below the ground. Such was the instrument's overall form. Between the vertical distances he clamped a sighting tube with a pivot at its midpoint on the vertical arm so it could tilt; the tube always aimed at the sun within its aperture—as heaven wheeled west once, the sun shifted east one degree—and still served to sight stars in every quarter, drawing on Li Chunfeng, Kong Ting, Han Xianfu, and Shu Yijian. On the Three Chronograms Instrument he mounted a heaven-motion ring powered by water. Water drive began with Zhang Heng, was perfected under Liang Lingzan and Yixing, revived by Zhang Sixun in Taiping Xingguo; Gonglian now revised it again, using a heaven-motion ring and pillar pivots below to move the armillary—a new design.

217

舊制渾象，張衡所謂置密室中者，推步七曜之運，以度曆象昏明之候，校二十四氣，考晝夜刻漏，無出於渾象。《隋志》稱梁秘府中有宋元嘉中所造者，以木為之，其圓如丸，遍體布二十八宿、三家星色、黃赤道、天河等，別為橫規繞於外，上下半之，以象地也。開元中，詔僧一行與梁令瓚更造銅渾象，為圓天之象，上具列宿周天度數，注水激輪，令其自轉，一日一夜天轉一周，又別置日月五星循繞，絡在天外，令得運行。每天西轉一匝，日正東行一度，月行一十三度有奇，凡二十九轉而日月會，三百六十五轉而日行一匝。仍置木櫃以為地平，令象半在地上，半在地下，又立二木偶人于地平之前，置鐘鼓使木人自然撞擊以報辰刻，命之曰《水運渾天俯視圖》。既成，命置之武成殿。

The old celestial globe—what Zhang Heng meant by housing in a sealed chamber—traces the seven luminaries to read dusk and dawn, checks the twenty-four qi, and tests day-and-night clepsydra marks; nothing surpasses it. The Sui Treatises record that the Liang secretariat held a Yuanjia-era globe of wood, round as a ball, covered with the twenty-eight lodges, the three schools' star colors, yellow and red paths, and the River of Heaven, with a transverse ring halving it above and below to stand for earth. In Kaiyuan, Yixing and Liang Lingzan were ordered to cast a bronze globe as the round sky, inscribed with every lodge's degree; poured water turned the wheel so that in one day and night heaven wheeled once, while sun, moon, and five planets were set to circle outside and run their courses. Each day heaven turned west once, the sun marched east one degree, the moon thirteen-odd degrees; after twenty-nine turns sun and moon met, and after three hundred sixty-five the sun finished a circuit. A wooden cabinet served as level earth so the globe stood half above and half below; two wooden men before it struck bells and drums by themselves to mark the hours, and the work was titled Diagram of the Water-Driven Armillary Heaven Viewed from Above. When finished, an edict placed it in the Hall of Martial Accomplishment.

218

宋太史局舊無渾象，太平興國中，張思訓准開元之法，而上以蓋為紫宮，旁為周天度，而東西轉之，出新意也。

The Song Astronomy Bureau had no globe until Taiping Xingguo, when Zhang Sixun followed Kaiyuan practice but capped it with the Purple Palace and set the circuit of heaven beside it to turn east and west—a fresh design.

219

公廉乃增損《隋志》制之，上列二十八宿周天度數，及紫微垣中外官星，以俯窺七政之運轉，納於六合儀天經地渾之內，同以木櫃載之。其中貫以樞軸，南北出渾象外，南長北短，地渾在木櫃面，橫置之，以象地。天經與地渾相結，縱置之，半在地上，半隱地下，以象天。其樞軸北貫天經上杠中，末與杠平，出櫃外三十五度稍弱，以象北極出地。南亦貫天經出下杠外，入櫃內三十五度少弱，以象南極入地。就赤道為牙距，四百七十八牙以銜天輪，隨機輪地轂正東西運轉，昏明中星既應其度，分至節氣亦驗應而不差。

Gonglian then revised the Sui Treatises' globe, inscribing the twenty-eight lodges' degrees and the Purple Forbidden Enclosure's inner and outer stars to watch the seven regulators from above, nested within the Six Harmonies Instrument's heaven meridian and Earth Sphere on the same wooden cabinet. A pivot ran through it, projecting north and south beyond the globe—longer to the south, shorter to the north; the Earth Sphere lay on the cabinet, cross-mounted as earth. The heaven meridian joined the Earth Sphere upright, half above ground and half below, to represent heaven. The pivot's north end pierced the upper frame of the heaven meridian, flush with the frame, and projected thirty-five-odd degrees beyond the cabinet as the north pole above earth. The south end pierced below the lower frame into the cabinet thirty-five-odd degrees shy of level, showing the south pole below earth. At the red path he cut four hundred seventy-eight cog teeth to mesh the heaven wheel; as the drive wheel and earth hub turned east and west, dusk, dawn, and culminating stars matched their degrees and solstices and qi checked true.

220

王蕃雲：「渾象之法，地當在天內，其勢不便，故反觀其形，地為外郭，于已解者無異，詭狀殊體而合於理，可謂奇巧者也。」今地渾說在渾象外，蓋出於王蕃制也。其下則思訓舊制，有樞輪關軸，激水運動，以直神搖鈴扣鐘擊鼓，置時刻十二神司辰像於輪上，時初、正至，則執牌迴圈而出，報隨刻數以定晝夜長短。至冬水凝，運轉遲澀，則以水銀代之。

Wang Fan wrote: "By the globe's logic earth should lie inside heaven, which is awkward, so the form is reversed with earth as the outer shell; for those who understand, nothing changes—it is a strange body yet sound in principle, and may be called ingenious." Today the Earth Sphere stands outside the globe, following Wang Fan's design. Below stood Sixun's older mechanism: pivot wheels and axles, water drive, spirits rocking bells and beating drums, twelve hour-gods on the wheel who at each hour's first and proper marks turned out plaques to report the clepsydra count and fix day and night length. When winter froze the water and the works slowed, mercury replaced it.

221

今公廉所制，共置一台，台中有二隔，渾儀置其上，渾象置其中，激水運轉，樞機輪軸隱於下。內設晝夜時刻機輪五重；第一重曰天輪，以撥渾象赤道牙距；第二重曰撥牙輪，上安牙距，隨天柱中輪轉動，以運上下四輪；第三重曰時刻鐘鼓輪，上安時初、時正百刻撥牙，以扣鐘擊鼓搖鈴；第四重曰日時初正司辰輪，上安時初十二司辰、時正十二司辰；第五重曰報刻司辰輪，上安百刻司辰。以上五輪並貫於一軸，上以天束束之，下以鐵杵臼承之，前以木閣五層蔽之，稍增異其舊制矣。五輪之北，又側設樞輪，其輪以七十二輻為三十六洪，束以三輞，夾持受水三十六壺。轂中橫貫鐵樞軸一，南北出軸為地轂，運撥地輪。天柱中輪動，機輪動渾象，上動渾天儀。又樞輪左設天池、平水壺，平水壺受天池水，注入受水壺，以激樞輪。受水壺落入退水壺。由壺下北竅引水入升水下壺，以升水下輪運水入升水上壺，上壺內升水上輪及河車同轉上下輪，運水入天河，天河複流入天地，每一晝一夜周而復始。此公廉制渾儀、渾象二器而通三用，總而名之曰渾天儀。

Gonglian built one platform with two compartments: the armillary instrument above, the celestial globe within, water driving the motion while wheels and axles lay hidden below. Inside ran five wheels for day, night, hours, and marks: first, the heaven wheel, meshing with the globe's red-path cogs; second, the tooth-engaging wheel, cogged above, turning with the central pillar wheel to drive four wheels; third, the hour-and-mark bell-and-drum wheel, with teeth for each hour's first and proper marks and the hundred marks to ring bells and drums; fourth, the first- and proper-hour mark-keeping wheel, with twelve keepers for the first hour and twelve for the proper hour; fifth, the mark-reporting wheel, bearing the hundred-mark keepers. All five wheels shared one axle, braced above and borne on an iron socket below, screened by a five-tier wooden cabinet—a modest change from older practice. North of the five wheels he added a side pivot wheel of seventy-two spokes forming thirty-six troughs in three rims, holding thirty-six water jars. An iron axle crossed the hub, projecting north and south as the earth hub to drive the earth wheel. When the central pillar wheel turned, the mechanism wheel turned the globe and, above, the armillary instrument. To the pivot wheel's left stood a heaven pool and level-water jar; the level-water jar took heaven-pool water and fed the receiving jar to drive the pivot wheel. From the receiving jar water fell into the return-water jar. A north vent drew water into the lower lift jar; the lower lift wheel raised it to the upper jar, where upper lift wheel and river carriage turned together and poured into the River of Heaven, which returned to heaven and earth and cycled every day and night. Thus Gonglian's armillary instrument and globe—two bodies, three uses—were together called the Armillary Sphere.

222

金既取汴，皆輦致于燕，天輪赤道牙距撥輪懸象鐘鼓司辰刻報天池水壺等器久皆棄毀，惟銅渾儀置之太史局候台。但自汴至燕相去一千餘裏，地勢高下不同，望筒中取極星稍差，移下四度才得窺之。明昌六年秋八月，風雨大作，雷電震擊，龍起渾儀鰲雲水趺下，台忽中裂而摧，渾儀僕落台下，旋命有司營葺之，複置臺上。貞祐南渡，以渾儀熔鑄成物，不忍毀拆，若全體以運則艱於輦載，遂委而去。

When Jin seized Bian, everything was hauled to Yan; heaven wheel, red-path cogs, drive wheel, suspended globe, bells, drums, mark-keepers, clepsydra works, heaven pool, and water jars were long ruined—only the bronze armillary was kept on the Astronomy Bureau terrace. Yet Bian and Yan lay more than a thousand li apart with different terrain; through the sighting tube the pole star sat slightly awry, and the tube had to be lowered four degrees before it could be aimed. In autumn, eighth month, sixth year of Mingchang, a storm broke with thunder; dragons seemed to rise from the instrument's turtle-and-cloud and trough, the platform split and fell, and the armillary toppled; officials were ordered to repair it and restore it to the terrace. At the Zhenyou flight south, the armillary was melted down for metal rather than dismantled, yet moving it whole was too hard, so it was left behind.

223

興定中，司天臺官以台中不置渾儀及測候人數不足，言之於朝，宜鑄儀象，多補生員，庶得盡占考之實。宣宗召禮部尚書楊雲翼問之，雲翼對曰：「國家自來銅禁甚嚴，雖罄公私所有，恐不能給。今調度方殷，財用不足，實未可行。」他日，上又言之，於是止添測候之人數員，鑄儀之議遂寢。

In Xingding, bureau officers told the court that with no armillary on the terrace and too few observers, new instruments should be cast and trainees added so observation could be real again. Emperor Xuanzong asked Minister of Rites Yang Yunyi, who answered: "Since the founding copper has been tightly barred; even draining public and private stores might not suffice. Schedules are tight and funds scarce; it truly cannot be done now." On another day the emperor raised it again; in the end only observer posts were added, and the casting plan died.

224

初，張行簡為禮部尚書提點司天監時，嘗制蓮花、星丸二漏以進，章宗命置蓮花漏二禁中，星丸漏遇車駕巡幸則用之。貞祐南渡，二漏皆遷於汴，汴亡廢毀，無所稽其制矣。

Earlier, as Minister of Rites over the Astronomy Directorate, Zhang Xingjian devised lotus and star-pill clepsydras and presented them; Zhangzong kept two lotus clepsydras in the inner palace and used the star-pill clepsydra on imperial tours. At the Zhenyou flight both were moved to Bian; when Bian fell they were destroyed, and their design is lost.

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