Volume 70 Treatises 23: Measures and Calendar 3

1

步五星

Calculating the Five Planets.

2

歲星總：七十九萬七千九百三十一、秒五。《乾元》率二十三萬四千五百三十五、秒五千七百二十五，《儀天》木星周率四百二萬八千五百八十七、秒七千五百六十。

The Jupiter aggregate is 797,931, with 5 fractional parts (seconds). Under the Qianyuan calendar the rate is 234,535 with 5,725 seconds; under the Yitian calendar Jupiter's circuit rate is 4,028,587 with 7,560 seconds.

3

平合：三百九十八日、八千八百五十七、秒二十八。《乾元》餘二千五百五十五、秒八千六百二十五，約分八十七。《儀天》餘八千七百八十七、秒七千五百六十。二曆平合皆謂之周日，數同《應天》。

The mean conjunction is 398 days, 8,857 parts, and 28 seconds. For Qianyuan the remainder is 2,555 with 8,625 seconds and a reduced fraction of 87. Under Yitian the remainder is 8,787 with 7,560 seconds. In both calendars the mean conjunction is termed the synodic day, and the figures match those of the Yingtian calendar.

4

變差：空、秒一十六。《乾元》差二十八、秒九千四百二十二半，秒母一萬。《儀天》歲差九十八、秒九千五百。上限二百五度，下限一百六十度、二十五分、秒六十三。

Variation difference: zero, with 16 seconds. For Qianyuan the difference is 28 with 9,422½ seconds, using 10,000 as the second denominator. Under Yitian the annual difference is 98 with 9,500 seconds. The upper limit is 205 degrees; the lower limit is 160 degrees, 25 parts, and 63 seconds.

5

熒惑總：一百五十六萬一百五十二、秒三。《乾元》率四十五萬八千五百九十二、秒九千一百八十三、十四。《儀天》火星周率七百八十七萬七千一百九十一、秒一千一百。

The Mars aggregate is 1,561,152, with 3 fractional parts (seconds). Under the Qianyuan calendar the rate is 458,592 with 9,183 seconds and 14 additional parts. Under the Yitian calendar Mars's circuit rate is 7,877,191 with 1,100 seconds.

6

平合：七百七十九日、九千二百二、秒一十八。《乾元》餘二千七百四、秒五千九百一十七，約分九十二。《儀天》餘九千二百九十一、秒一千一百。二曆平合皆謂之周日，數同《應天》。

The mean conjunction is 779 days, 9,202 parts, and 18 seconds. For Qianyuan the remainder is 2,704 with 5,917 seconds and a reduced fraction of 92. Under Yitian the remainder is 9,291 with 1,100 seconds. In both calendars the mean conjunction is termed the synodic day, and the figures match those of the Yingtian calendar.

7

變差：三、秒空。《乾元》差二十九、秒一千一百三十五。《儀天》歲差九十八、餘三千八百。上限一百九十六度八十，下限一百六十八度四十五、秒六十三。

Variation difference: 3, with zero seconds. For Qianyuan the difference is 29 with 1,135 seconds. Under Yitian the annual difference is 98 with a remainder of 3,800. The upper limit is 196 degrees and 80 parts; the lower limit is 168 degrees, 45 parts, and 63 seconds.

8

鎮星總：七十五萬六千三百一十一、秒八十五。《乾元》率二十二萬二千三百一十一、秒二千一百六十四、二十。《儀天》土星周率三百八十一萬八千六百八、秒三千五百。

The Saturn aggregate is 756,311, with 85 fractional parts (seconds). Under the Qianyuan calendar the rate is 222,311 with 2,164 seconds and 20 additional parts. Under the Yitian calendar Saturn's circuit rate is 3,818,608 with 3,500 seconds.

9

平合：三百七十八日、八百六、秒五十一。《乾元》餘二百三十六、秒八百三十一，約分八。《儀天》餘八百八、秒三千五百。二曆平合皆謂之周日，數同《應天》。

The mean conjunction is 378 days, 806 parts, and 51 seconds. For Qianyuan the remainder is 236 with 831 seconds and a reduced fraction of 8. Under Yitian the remainder is 808 with 3,500 seconds. In both calendars the mean conjunction is termed the synodic day, and the figures match those of the Yingtian calendar.

10

變差：五、秒七十九。《乾元》差二十八、秒九千五百三。《儀天》歲差一百、秒一千一百，上限一百八十二度、六十三分、秒八十一，下限同上限。

Variation difference: 5, with 79 seconds. For Qianyuan the difference is 28 with 9,503 seconds. Under Yitian the annual difference is 100 with 1,100 seconds; the upper limit is 182 degrees, 63 parts, and 81 seconds, and the lower limit matches the upper limit.

11

太白總：一百一十六萬八千二十二、秒四十二。《乾元》率三十四萬三千三百三十九、秒一千五百四十七。《儀天》金星周率五百八十九萬七千四百八十九、秒五千四百。

The Venus aggregate is 1,168,022, with 42 fractional parts (seconds). Under the Qianyuan calendar the rate is 343,339 with 1,547 seconds. Under the Yitian calendar Venus's circuit rate is 5,897,489 with 5,400 seconds.

12

平合：五百八十三日、八千九百九十六、秒一十。《乾元》餘二千六百七十六、秒一千七百三十五，約分九十一。《儀天》餘九千一百八十九、秒五千四百。二曆平合皆謂之周日，數同《應天》。

The mean conjunction is 583 days, 8,896 parts, and 10 seconds. For Qianyuan the remainder is 2,676 with 1,735 seconds and a reduced fraction of 91. Under Yitian the remainder is 9,189 with 5,400 seconds. In both calendars the mean conjunction is termed the synodic day, and the figures match those of the Yingtian calendar.

13

再合：二百九十一日、九千四百九十九、秒五。《乾元》、《儀天》不立此法。

The second conjunction is 291 days, 9,499 parts, and 5 seconds. Neither the Qianyuan nor the Yitian calendar provides for this procedure.

14

變差：二、秒三十六。《乾元》差二十九、秒一千七百九十八。《儀天》歲差一百二十、餘八千三百九，上限一百九十七度一十六，下限一百六十八度、秒六十三。

Variation difference: 2, with 36 seconds. For Qianyuan the difference is 29 with 1,798 seconds. Under Yitian the annual difference is 120 with a remainder of 8,309; the upper limit is 197 degrees and 16 parts, and the lower limit is 168 degrees and 63 seconds.

15

辰星總：二十三萬一千八百六、秒四十二、八十。《乾元》率八萬八千一百三十七、秒四千四百一十，八十。《儀天》水星周率一百一十七萬三百八十七、秒二千八百。

The Mercury aggregate is 231,806, with 42 seconds and 80 additional parts. Under the Qianyuan calendar the rate is 88,137 with 4,410 seconds and 80 additional parts. Under the Yitian calendar Mercury's circuit rate is 1,170,387 with 2,800 seconds.

16

平合：一百一十五日，八千八百二、秒三十。《乾元》餘二千五百八十七、秒二千九十四，約分八十八。《儀天》餘八千八百八十七、秒二千八百。二曆平合皆謂之周日，數同《應天》。

The mean conjunction is 115 days, 8,802 parts, and 30 seconds. For Qianyuan the remainder is 2,587 with 2,094 seconds and a reduced fraction of 88. Under Yitian the remainder is 8,887 with 2,800 seconds. In both calendars the mean conjunction is termed the synodic day, and the figures match those of the Yingtian calendar.

17

再合：五十七日、九千四百二、秒一十五。《乾元》、《儀天》不立此法。

The second conjunction is 57 days, 9,402 parts, and 15 seconds. Neither the Qianyuan nor the Yitian calendar provides for this procedure.

18

變差：三、秒七十八。《乾元》差二十九、秒一千一百三十八。《儀天》歲差九十八、秒三十，上限一百八十三度、六十二分，下限一百八十二度、六十二分、秒六十三。

Variation difference: 3, with 78 seconds. For Qianyuan the difference is 29 with 1,138 seconds. Under Yitian the annual difference is 98 with 30 seconds; the upper limit is 183 degrees and 62 parts, and the lower limit is 182 degrees, 62 parts, and 63 seconds.

19

求五星天正冬至後加時平合日度分秒：《乾元》謂之五星平合變日，《儀天》謂之常合中日中度。

To determine each planet's mean-conjunction day, degree, parts, and seconds at the added hour after the winter solstice of the celestial first month: the Qianyuan calendar terms this the "five planets' mean-conjunction variation day," while the Yitian calendar calls it the "regular conjunction mid-day and mid-degree."

20

各以星總除元積為總數，不盡者，返減星總，餘，半而進位；又置總數，木、火三之，土如其數，皆百而從之，以元法收之，為天正冬至後平合日度及分。《乾元》置歲積分，各以星率去之，不盡，用減星率，餘以五因之，滿元率收為日，不滿，退除為分。《儀天》各以其星周率去歲積分，不滿者，返減其周率，余以宗法收為日，不盡，退除為分。

For each planet, divide the origin accumulation by the planet's total to get an aggregate count; any remainder is subtracted back from the planet total, and halves are rounded upward; Then take the aggregate number: triple it for Jupiter and Mars, leave Saturn's unchanged, carry at one hundred in each case, and reduce by the origin divisor to obtain the mean-conjunction day, degree, and parts after the winter solstice of the celestial first month. Under Qianyuan, set the year's accumulated parts; for each planet subtract the star rate; if a remainder remains, subtract it from the star rate; multiply the remainder by five; convert to days wherever the origin rate is filled, and retreat-divide the remainder into parts. Under Yitian, subtract each planet's circuit rate from the year's accumulated parts; if the result is insufficient, subtract back from the circuit rate; reduce the remainder to days by the clan divisor, and retreat-divide any leftover into parts.

21

求平合入曆分：《乾元》謂之入曆，《儀天》謂之推五星常合入曆度分。

To find the parts for a mean conjunction's entry into the epact: the Qianyuan calendar calls this "entering the epact," while the Yitian calendar calls it "computing the five planets' regular-conjunction entering-epact degrees and parts."

22

各以其星變差展所求積年，滿三百六十五萬三千二百九十三、秒一十九，去之不盡，以元法收為度，不滿為分，以減平合日，為入曆度分。《乾元》以積年乘星差，以周天策去之，不盡，以元率收為度，不滿，退除為分，用減平合變日，為入曆分。《儀天》各置其星歲差，以積年乘之，滿三百六十八萬九千八百八、秒九千九百去之，不盡，以宗法收為度，不滿，退收為分。

For each planet, multiply the accumulated years sought by that planet's variation difference; remove multiples of 3,653,293 and 19 seconds; reduce the remainder to degrees by the origin divisor, treat the unfilled portion as parts, subtract from the mean-conjunction day, and thus obtain the entering-epact degrees and parts. Under Qianyuan, multiply the accumulated years by the star's difference and subtract the full-circuit tally; reduce the remainder to degrees by the origin rate, retreat-divide any leftover into parts, subtract from the mean-conjunction variation day, and obtain the entering-epact parts. Under Yitian, set each planet's annual difference, multiply by the accumulated years, and remove multiples of 3,689,808 and 9,900 seconds; reduce the remainder to degrees by the clan divisor, and retreat-collect any leftover into parts.

23

求入陰陽變分：在陽末變分以下為入陽曆；以上去之，餘為入陰曆。置入陰、陽曆分，以陰、陽變數去之，不盡，為入陰、陽數及變分。

To find the entering yin-yang variation parts: if the value falls below the yang-terminal variation parts, the planet is entering the yang epact; If it lies above that threshold, subtract it away; the remainder is the entry into the yin epact. Set the entering yin or yang epact parts and subtract the yin or yang variation count; the remainder gives the entering yin or yang number and variation parts.

24

《乾元》歲星前限二萬五百五，中限一萬二百四十八，後限一萬六千二十；熒惑前限一萬九千六百八十二，中限六千五百六十四，後限一萬六千八百四十四；鎮星前限一萬八千二百六十二，中限九千一百二十六，後限同前限，前、後、中皆半周天；太白前限一萬九千七百一十六，中限九千八百五十八，後限一萬六千八百九；辰星前、中、後與鎮星同。又歲星前法一千七百八，後法一千三百三十四，熒惑前法一千六百四十一，後法一千四百三；鎮星、辰星前後法皆一千五百二十二；太白前法一千六百四十三，後法一千四百二。《儀天》各置常合入曆度分，如在上限末數已下者為增數；以上者減去上限末數下度分，餘為入下限減數。又各置所入上、下限度分，以上、下限度分相近者減之，餘為入次限、下限度及分。

Under Qianyuan, Jupiter's front limit is 20,505, its middle limit 10,248, and its rear limit 16,020; Mars's front limit is 19,682, its middle limit 6,564, and its rear limit 16,844; Saturn's front limit is 18,262, its middle limit 9,126, and its rear limit equals its front limit; the front, middle, and rear limits each span half the full circuit; Venus's front limit is 19,716, its middle limit 9,858, and its rear limit 16,809; Mercury's front, middle, and rear limits are the same as Saturn's. Further, Jupiter's front divisor is 1,708 and its rear divisor 1,334; Mars's front divisor is 1,641 and its rear divisor 1,403; Saturn's and Mercury's front and rear divisors are each 1,522; Venus's front divisor is 1,643 and its rear divisor 1,402. Under Yitian, set each planet's regular-conjunction entering-epact degrees and parts; if the value is at or below the upper limit's terminal number, it counts as an increment; If it lies above, subtract the degrees and parts below the upper limit's terminal number; the remainder is the entering lower-limit decrement. Then set the entered upper- or lower-limit degrees and parts for each planet, subtract whichever upper or lower limit degrees and parts are nearer, and the remainder is the entry into the secondary limit's lower limit and parts.

25

《乾元》五星辰星陰、陽差分並陰、陽差度並同初、末。入陰陽定分：《乾元》謂之入諸曆變分，《儀天》謂之求五星常合入增減定數。

Under Qianyuan, Mercury alone among the five planets has yin-yang difference parts and difference degrees that are identical at both the initial and final limits. To find the entering yin-yang fixed parts: the Qianyuan calendar calls this "entering the variation parts of the various epacts," while the Yitian calendar calls it "finding the five planets' regular-conjunction entering increment-and-decrement fixed numbers."

26

以入變分各減初變分，餘卻以其變下損益率展之，百而一為分；損益次變下陰、陽積為定分。《乾元》置平合入曆分，以其星入段前、後限分加減之，如不足，加周天以減之，餘卻依入曆分入初末限；各置其段入曆分，前限以下為在前，以上者去之，為後限分；在中限以下為初限，以上去之，為末限分；置初、末，以前、後限星分除之為限數，不滿，為初末限日；各以其限差分約之，為差；初限以加、末限以減，用加減前、後限度為定度。《儀天》各置常合所入限下度數及分，以其限下損益率乘之，退一等，以百約之為度，不滿為分，以損益其限下增、減積度及分。若求諸變增、減定度者，置其變入上下限，准此求之。

Subtract the initial variation part from each entering variation part; spread the remainder by the lower increase-decrease rate of that variation, taking one part per hundred; Increase or decrease the yin or yang accumulation below the next variation to obtain the fixed parts. Under Qianyuan, set the mean-conjunction entering-epact parts and add or subtract that planet's front or rear segment limit parts; if the result is insufficient, add a full circuit before subtracting; with the remainder, enter the initial or final limit according to the entering-epact parts; Set the entering-epact parts for each segment: if at or below the front limit the planet is in the front segment; if above, subtract the front limit to obtain the rear-limit parts; If at or below the middle limit it falls within the initial limit; if above, subtract the middle limit to obtain the final-limit parts; Set the initial or final value and divide by the front or rear limit star parts to get the limit number; the remainder gives the initial or final limit days; Reduce each by its limit difference parts to obtain the difference; Add for the initial limit and subtract for the final limit, then apply this to the front or rear limit degrees to obtain the fixed degrees. Under Yitian, set the degrees and parts below the limit into which the regular conjunction has entered, multiply by the increase-decrease rate below that limit, drop one place, reduce by one hundred to degrees and treat the remainder as parts, and thus increase or decrease the accumulated degrees and parts below the limit. To find the increment-and-decrement fixed degrees for the various variations, set the upper or lower limits into which each variation has entered and apply this same procedure.

27

定合積日：《乾元》謂之求定日，《儀天》謂之求五星定合積日。

Fixed-conjunction accumulated days: the Qianyuan calendar calls this "finding the fixed day," while the Yitian calendar calls it "finding the five planets' fixed-conjunction accumulated days."

28

日除陰、陽定分，為二；陽加陰減平合日，為定積日及分。《乾元》置變日，以前、後限度前加後減，為定日。《儀天》各置其星常合中日及餘，以入曆增減度增者增之、減者減之，金、水返而加減之，以日躔定差先減後加之，金、水則先加後減，即得定合積日及分。又《儀天》求入盈縮初末限，皆以半周天為准。

Divide the yin-yang fixed parts by the day unit to obtain two components; Add the yang portion and subtract the yin portion from the mean-conjunction day to obtain the fixed accumulated days and parts. Under Qianyuan, set the variation day, add for the front limit and subtract for the rear limit, and obtain the fixed day. Under Yitian, set each planet's regular-conjunction mid-day and remainder; apply entering-epact increment-and-decrement degrees as increases or decreases; for Venus and Mercury reverse the addition and subtraction; apply the sun's-march fixed difference by subtracting first and then adding—for Venus and Mercury, add first and subtract afterward—and thus obtain the fixed-conjunction accumulated days and parts. Furthermore, when the Yitian calendar finds the entering excess-and-shortage initial and final limits, it always uses half the full circuit as the standard.

29

入氣盈縮度分：《乾元》謂之入氣，《儀天》謂之求入盈縮初末限。

Entering-qi excess-and-shortage degrees and parts: the Qianyuan calendar calls this "entering the qi," while the Yitian calendar calls it "finding the entering excess-and-shortage initial and final limits."

30

置定積，以常數去之，不盡者，為入氣日分；置入氣日分，如求朔望盈縮術入之，即得入氣盈縮度分。《乾元》置定日，以氣策去之為氣數，不盡，為入氣日；命以冬至，算外，即得入氣日及分。《儀天》各置定合積日，在半周天以下者去之，餘為在縮，乃視在盈縮初限日及約餘以下者，便為在盈縮初限；以上者，減去盈縮初限日約餘，為在盈縮末限日及餘。

Set the fixed accumulation and subtract the constant number; the remainder gives the entering-qi day-parts; Set the entering-qi day-parts and apply the same procedure used to find new- and full-moon excess and shortage; the result is the entering-qi excess-and-shortage degrees and parts. Under Qianyuan, set the fixed day and divide by the qi tally to obtain the qi number; the remainder gives the entering-qi day; Count outward from the winter solstice; the result gives the entering-qi days and parts. Under Yitian, set the fixed-conjunction accumulated days for each planet; subtract if below half the full circuit, and the remainder falls in shortening; if it is at or below the initial-limit days and reduced remainder of excess and shortage, it lies within the initial excess-and-shortage limit; If above that threshold, subtract the initial-limit days and reduced remainder of excess and shortage to obtain the final excess-and-shortage limit days and remainder.

31

定合日辰：《乾元》謂之日辰，《儀天》同《應天》。

Fixed-conjunction day and time: the Qianyuan calendar calls this "day and time," while the Yitian calendar follows the Yingtian calendar.

32

以其大、小餘加入氣日，命從甲子，算外，即得所求。《乾元》、《儀天》以冬至大、小餘加定日，各滿紀法去之，餘並同《應天》。《乾元》冬至小餘以元率退收，百為母；又有日躔陰陽度，置其氣陰陽分，如求朔日度分術入之，即得所求。

Add the large and small remainders to the entering-qi day, count outward from jiazi, and obtain the desired result. In both Qianyuan and Yitian, add the winter solstice large and small remainders to the fixed day, subtract multiples of the era rule when full, and the remainders match those in the Yingtian calendar. Under Qianyuan, convert the winter solstice small remainder by retreating and collecting with the origin rate, taking one hundred as the denominator; There are also sun's-march yin-yang degrees: set the qi's yin-yang parts and apply the same procedure used to find the new-moon day degrees and parts.

33

求入月日數：《儀天》謂之求定合在何月日。

Finding the entering-month day count: the Yitian calendar calls this "finding on which month and day the fixed conjunction falls."

34

置定合日辰大餘，以定朔大餘減之，餘命算外，即得所求。二曆法同。

Set the large remainder of the fixed-conjunction day and time, subtract the fixed-new-moon large remainder, and count outward from the remainder to obtain the desired result. The two calendars use the same method.

35

定合定星：《乾元》同。《儀天》謂之求日躔先後定數、求五星定合定度及分。

Fixed-conjunction fixed star: under Qianyuan, the method is the same. The Yitian calendar calls this "finding the sun's-march front-and-rear fixed number" and "finding the five planets' fixed-conjunction fixed degrees and parts."

36

各以其星入氣盈縮度分盈加縮減之，又以百除陰、陽定分，為度分；陽加陰減，皆加減平合，為定星；用加天正黃道日度，滿宿去之，不滿宿，即得所求。《乾元》各置其星平合中星，以日躔陰陽度陰減陽加之；又以其星入曆限度前加後減之，即為其星定合定星。余同《應天》。《儀天》置所入限日下小餘，以其日盈縮率乘，以宗法除為分，以盈縮其日下先後定分，為日躔先後定度及分；又各置其星常合中度及分，以入限增定度及分增減之。金、水二星增者減，減者增；又以日躔先後定度及分，木、火、土即先減後加，金、水先加後減其日躔差，木星二因，退位，火星除二，土星退位，從下加三，金、水倍，用即得定度及分。余同《應天》。

For each planet, adjust by its entering-qi excess-and-shortage degrees and parts, adding for excess and subtracting for shortage; then divide the yin-yang fixed parts by one hundred to obtain degree-parts; Add the yang portion and subtract the yin portion, apply these to the mean conjunction, and obtain the fixed star; Add the result to the winter solstice ecliptic solar degree, subtract full lodges, and the remainder within a lodge gives the desired position. Under Qianyuan, set each planet's mean-conjunction central star and adjust by the sun's-march yin-yang degrees, subtracting yin and adding yang; Then add for the front limit and subtract for the rear limit according to the planet's entering-epact limit degrees; the result is that planet's fixed-conjunction fixed star. The remainder follows the Yingtian calendar. Under Yitian, set the small remainder below the entered limit day, multiply by that day's excess-and-shortage rate, divide by the fundamental rule to obtain parts, and adjust the front-and-rear fixed parts below that day by excess and shortage to obtain the sun's-march front-and-rear fixed degrees and parts; Then set each planet's regular-conjunction mid-degrees and parts and adjust them by the increment fixed degrees and parts for the entered limit. For Venus and Mercury, reverse the adjustment: subtract where there is an increase and add where there is a decrease; Then apply the sun's-march front-and-rear fixed degrees and parts: for Jupiter, Mars, and Saturn, subtract first and add afterward; for Venus and Mercury, add first and subtract afterward to the sun's-march difference—double Jupiter's value and drop one place; divide Mars by two; drop Saturn one place and add three from below; double Venus and Mercury—and thus obtain the fixed degrees and parts. The remainder follows the Yingtian calendar.

37

歲星入段亦名入變

When Jupiter enters a segment, this is also called entering a variation.

38

熒惑入段鎮星入段太白入段辰星入段

Mars entering a segment; Saturn entering a segment; Venus entering a segment; Mercury entering a segment.

39

諸段平日平度：《乾元》謂之諸星變定積，《儀天》謂之五星諸變中日中度。

Mean days and mean degrees for the various segments: the Qianyuan calendar calls these the fixed accumulations of the planets' variations, while the Yitian calendar calls them the mid-days and mid-degrees of the five planets' various variations.

40

置平合日度，以諸段下平日平度加之，即得所求。《乾元》各置其星變日，以所求入曆前加後度前加後減之。其太白辰星夕見變及晨疾變，皆以返用加減。熒惑晨見變定，置定差，以進一位滿十一除之為定差，各依加減，即得所求；在留變者，置其變定積，以前變前後度前後減之。其火星三因之，後退者倍之。《儀天》各置其星常合中日中度及分，以其星諸變段下常加合中日變度加減中星，即得諸變中日中度及分。

Set the mean-conjunction day and degrees, add the mean days and mean degrees listed under each segment, and obtain the desired result. Under Qianyuan, set each planet's variation day and adjust by the entering-epact, adding before the limit and subtracting after it, and adding before the degrees and subtracting after them. For Venus and Mercury evening-visibility variations and morning-rapid variations, apply addition and subtraction in reverse order. For the Mars morning-visibility variation fixed value, set the fixed difference, advance one place, divide by eleven when full to obtain the fixed difference, and apply addition or subtraction accordingly; For planets in a stationary variation, set their variation fixed accumulation and subtract according to the previous variation's front and rear degrees. For Mars, triple the value; for rear retrograde motion, double it. Under Yitian, set each planet's regular-conjunction mid-day, mid-degrees, and parts; adjust them with the regular added conjunction mid-day variation degrees listed under each variation segment to obtain the mid-days, mid-degrees, and parts for all variations.

41

諸段入曆：《儀天》謂之求五星諸變入限及增減定度。

Segments entering the epact: the Yitian calendar calls this "finding the five planets' various variations' entering limits and increment-and-decrement fixed degrees."

42

置平合入陰陽曆分，各以逐段陰陽曆分加之，為諸段入曆分。《乾元》以在諸變曆分中入曆名曰限變度。《儀天》各置其星常合入曆度分，以其星諸變段下上下限度分累加之，滿周天去之，余依常合術入之，各得增減定度。其金星在晨疾、晨合、夕見變者，置增減定度及分，以四乘三除，為金星變定差。其火星在晨見變者，以九乘，增減定度及分，退一位，為晨星變定差。

Set the mean-conjunction entering yin-yang epact parts, add the yin-yang epact parts of each successive segment, and obtain the entering-epact parts for the various segments. Under Qianyuan, entering the epact within the various variation epact parts is called the limit variation degrees. Under Yitian, set each planet's regular-conjunction entering-epact degrees and parts, cumulatively add the upper and lower limit degrees and parts listed under each variation segment, subtract full circuits when full, and with the remainder apply the regular-conjunction procedure to obtain the increment-and-decrement fixed degrees for each. For Venus in its morning-rapid, morning-conjunction, and evening-visibility variations, set the increment-and-decrement fixed degrees and parts, multiply by four and divide by three, and obtain Venus's variation fixed difference. For Mars in its morning-visibility variation, multiply the increment-and-decrement fixed degrees and parts by nine, drop one place, and obtain the morning-star variation fixed difference.

43

諸段入變分：置入曆分，各以變分去之，餘為入變分。求陰陽定分，依平合術入之。《乾元》諸段變分在入變前述。《儀天》即同《應天》。

Entering-variation parts for the various segments: set the entering-epact parts, subtract each variation part, and the remainder gives the entering-variation parts. To find the yin-yang fixed parts, apply the mean-conjunction procedure. Under Qianyuan, the variation parts for the various segments were given above, before entering variation. Under Yitian, the method is the same as in the Yingtian calendar.

44

五星諸段定積曰：《乾元》謂之求五星諸變定日。

Fixed accumulated days for the five planets' various segments: the Qianyuan calendar calls this "finding the fixed days of the five planets' various variations."

45

置其入陰陽定分，百除，為日分；陽減陰減諸段平日。其金水夕見、晨疾返為之定積。其金星晨次、晨遲，更用盈縮度縮加盈減定積為定。求其入氣月日，如平合術入之。又熒惑前遲定積，置平合入陰陽曆分，加二萬一千六百七十五，盈三萬六千五百二十五半去之。餘與見求入陰陽曆同者，更不求之，如不同曆者，即依平合術入，所得，用加前遲留退、後退留平日為定積，入氣月日如前。又五星定用盈縮差及陰陽定分：歲熒惑鎮星晨見、夕疾、定合，太白定合、夕見、夕退、再合、晨見及後、晨疾，皆用盈縮定差，太白定合晨、夕見及後疾，皆用盈縮定差。內歲星後疾不用盈縮定差，辰星諸段總用盈縮定差盈加縮減。熒惑晨見陰陽定分身外加一，前疾陽定分再析，各為定分。《乾元》諸變定日在入變前。《儀天》各置其星入變中日，以其星所入變限增減定度及分，增者增之，減者減之。其金星定合、夕見、夕順疾、夕次疾、晨次疾，水星定合、夕見、晨疾變，皆以增減定度及分，增者減之，減者增之，各得定日。合用日躔差者，乃以日躔先後定差先減後加，乃為定日及分。其日躔差，金水定合、夕見、晨疾，以日躔差先加後減，乃為定日及分天之度數。

Set its entering yin-yang fixed parts, divide by one hundred to obtain day-parts; Subtract the yang portion and the yin portion from the various segments' mean days. For Venus and Mercury evening-visibility and morning-rapid variations, reverse the procedure to obtain their fixed accumulation. For Venus in its morning secondary and morning slow variations, further adjust the fixed accumulation with excess-and-shortage degrees, adding for shortage and subtracting for excess, to obtain the fixed value. To find its entering-qi month and day, apply the mean-conjunction procedure. For Mars front-slow fixed accumulation, set the mean-conjunction entering yin-yang epact parts, add 21,675, and subtract 36,525 and a half when full. If the remainder matches the entering yin-yang epact already found, do not recalculate; if the epact differs, apply the mean-conjunction procedure, add the result to the front-slow stationary-retreat and rear-retreat stationary mean days to obtain the fixed accumulation, and find the entering-qi month and day as before. Furthermore, the five planets' fixed values use excess-and-shortage differences and yin-yang fixed parts: for Jupiter, Mars, and Saturn at morning visibility, evening rapid motion, and fixed conjunction; and for Venus at fixed conjunction, evening visibility, evening retrograde, second conjunction, morning visibility and afterward, and morning rapid motion—all use the excess-and-shortage fixed difference; for Venus at fixed conjunction in the morning, evening visibility, and afterward rapid motion, all likewise use the excess-and-shortage fixed difference. Jupiter's later rapid motion does not use the excess-and-shortage fixed difference; Mercury's various segments always use the excess-and-shortage fixed difference, adding for excess and subtracting for shortage. For Mars morning visibility, add one to the yin-yang fixed parts outside the body; for front rapid motion, divide the yang fixed parts again, and each portion becomes a fixed part. Under Qianyuan, the fixed days for the various variations are given before entering variation. Under Yitian, set each planet's entering-variation mid-day and adjust by the increment-and-decrement fixed degrees and parts for the entered variation limit, adding increases and subtracting decreases. For Venus at fixed conjunction, evening visibility, evening direct rapid motion, evening secondary rapid motion, and morning secondary rapid motion; and for Mercury at fixed conjunction, evening visibility, and morning rapid variation—apply the increment-and-decrement fixed degrees and parts, subtracting increases and adding decreases, to obtain each fixed day. Where the sun's-march difference is also required, apply the sun's-march front-and-rear fixed difference by subtracting first and adding afterward to obtain the fixed day and parts. For the sun's-march difference at Venus and Mercury fixed conjunction, evening visibility, and morning rapid motion, apply the sun's-march difference by adding first and subtracting afterward to obtain the fixed day, parts, and celestial degrees.

46

定星：《乾元》謂之求五星諸變定星，《儀天》謂之求五星諸變定度。

Fixed star: the Qianyuan calendar calls this "finding the five planets' various variations' fixed stars," while the Yitian calendar calls it "finding the five planets' various variations' fixed degrees."

47

以合用盈縮定差加減平度分，又以陰陽定分陽加陰減。其金水夕見、晨疾返用為定星，求宿度，加平合入之。熒惑前遲、後退差度以二百三十六度加前遲定星，二百五十七度加後退定星，如半周天以下為陽度；以上者去之，余為陰度；前遲陰陽度在一百一十度以上者，返減半周天，餘以五因之，後退入陰陽度在七十四度以下者，亦五因之，皆滿百為度分，陽減陰加定星，為前遲、後退定星；求宿度，加平合入之。《乾元》置其星其變中星，以入曆前後度前加後減之，又合用陰陽度者，陰減陽加之，為定星；以冬至黃道日度加之，命從鬥宿，算外，即其變所入宿次也。若在留變者，更不求定星也，只用前變定星為留變定星。又熒惑留差，以一百一十九度減前遲定星，以一百三十四度減後退定星，在一百八十二度半以下為前，以上者去之為後，置前後度，在七十三度以下為在前，以上者返減一百八十三度半，餘為後度，皆倍之，百除為度，命曰留差度及分也。又前退定星度，以一百二十三度減前退定星，又以一百三十一度減後退定星，在一百八十二度半以下者為前，以上者去之為後，視前後度在七十三度以下為前，以上者返減一百八十二度半為後：皆以倍之，百除為度，即得前後退差度及分也；用前減後加其段定星為定星。又五星用陰陽度：歲星熒惑鎮星晨見，後疾，夕合；太白夕見、退，夕合，晨見，後疾，平合皆用日躔、陰陽度，其辰星諸段皆用之。《儀天》各置其星其變中度及分，以其變入限增減定度及分，增者增之，減者減之。其金星定合、夕見、夕定度及分，增者減之，減者增之，各得定日、次定日，各加減訖後，合用日躔先後定差者，以日躔先後定差及分先減後加之，即各得定度及分。其日躔差，木星定合，五因，半而退位，晨見先二因，退位，後五因，半而退位；後定疾先差五因，半而退位，定差二因退位；火星定合，身外除二，晨見先差七因，退位，後差身外除二，後差七因退位；土星定合，退位從下加二，晨見先差退位，後差從下加三，退位，後差退位；金星定合，二因之，夕見先差伏倍用，後差從下加三，晨疾伏先差從下加二，後差二因，夕退伏、晨退見六因，先後退位；水星夕見後差從下加三，先差二因，晨疾先差從下加三，後差倍用，定合乃用加減次定度為定度，置定度及分，以加天正冬至加時黃道日度及分，命從鬥宿初度起算，至不滿宿，算外，即得其變加時宿度，其火星前、後退及前遲變皆為次定星，又置之，以留退定差度及分，增者增之，減者減之，得為前、後退定度，前遲，置前留定差，以三除之，乃用增減前遲定度也。又火星留差，以一百二十四半減前遲次定度，又以二百四十六度少加後退定度，若在一百八十二度六十二分以下為入在增；以上者，以減去一百八十二度六十二分為入在減。置入在增、減度及分，如在七十二度以下者為上限；以上者，返減一百八十二度六十二分，餘為下限。各置所入上、下限增減度及分，在上限四因之，在下限倍，身外加三，皆以一百約之為度及分，若在後留者，三因之為定差度及分。又，《儀天》有火星退定差度及分，以二百四十一度少加前退後次定度，又以一百一十九度減退次定度及分，餘，在一百八十二度六十二分以下者為入在增；以上者，減去一百八十二度六十二分，餘為入在減。又置入上、下限度分，若在七十二度以下者為上限，如在七十二度以上者為減一百八十二度六十二分，餘為下限。又置上下限增減度分，在上為度，不滿為分，即各得退定差度及分，其定差，如在後退者，倍之為定差。又有火星留定日，各置前、後留常中日，前留以前遲變入限增減定度及分，增者增之，減者減之，各以前、後留定差度及分，增者加之，減者損之，即得前、後留定日，其增減差通入曆用之。又有火星前、後退定度，各置前、後變次定度及分，以前、後退定差度及分，如在增者加之，在減者損之，即得定度及分；置定度及分，以加天正冬至黃道日度及分，命從鬥宿初度去之，至不滿宿，算外，即得退行所在宿度及分也，其增減定度，三除乃用之。

Adjust the mean degrees and parts with the applicable excess-and-shortage fixed difference, then add the yang portion and subtract the yin portion of the yin-yang fixed parts. For Venus and Mercury evening-visibility and morning-rapid variations, reverse the procedure to obtain the fixed star; then find the lodge degrees by adding to the mean conjunction. For Mars front-slow and rear-retreat difference degrees, add 236 degrees to the front-slow fixed star and 257 degrees to the rear-retreat fixed star; if at or below half the full circuit the result is yang degrees; If above, subtract half the full circuit; the remainder gives yin degrees; If front-slow yin-yang degrees exceed 110 degrees, subtract half the full circuit in reverse and multiply the remainder by five; if rear-retreat entering yin-yang degrees are at or below 74 degrees, likewise multiply by five; reduce by one hundred to degrees and parts, subtract yang and add yin to the fixed star, and obtain the front-slow and rear-retreat fixed star; Find the lodge degrees by adding to the mean conjunction. Under Qianyuan, set the planet's variation mid-star and adjust by the entering-epact front and rear degrees, adding before and subtracting after; where yin-yang degrees also apply, subtract yin and add yang to obtain the fixed star; Add the winter solstice ecliptic solar degree, count outward from the Dipper lodge, and obtain the lodge sequence entered by that variation. If the planet is in a stationary variation, do not seek a separate fixed star; use only the previous variation's fixed star as the stationary variation fixed star. For Mars stationary difference, subtract 119 degrees from the front-slow fixed star and 134 degrees from the rear-retreat fixed star; if at or below 182 degrees and a half the value is front, otherwise subtract to obtain rear; set the front and rear degrees—if at or below 73 degrees it is front, otherwise reverse-subtract 183 degrees and a half for the rear degrees; double each, divide by one hundred for degrees, and name the result the stationary-difference degrees and parts. For front-retreat fixed-star degrees, subtract 123 degrees from the front-retreat fixed star and 131 degrees from the rear-retreat fixed star; if at or below 182 degrees and a half the value is front, otherwise subtract to obtain rear; if the front and rear degrees are at or below 73 degrees treat them as front, otherwise reverse-subtract 182 degrees and a half for rear; double each, divide by one hundred for degrees, and obtain the front and rear retreat difference degrees and parts; Apply front subtraction and rear addition to the segment fixed star to obtain the fixed star. Furthermore, the five planets use yin-yang degrees at the following points: Jupiter, Mars, and Saturn at morning visibility, later rapid motion, and evening conjunction; Venus at evening visibility and retrograde, evening conjunction, morning visibility, later rapid motion, and mean conjunction all use sun's-march and yin-yang degrees; Mercury uses them at all segments. Under Yitian, set each planet's variation mid-degrees and parts and adjust by the increment-and-decrement fixed degrees and parts for the entered variation limit, adding increases and subtracting decreases. For Venus at fixed conjunction and evening visibility, reverse the evening fixed degrees and parts—subtract increases and add decreases—to obtain the fixed day and secondary fixed day; after all adjustments, where the sun's-march front-and-rear fixed difference applies, subtract first and add afterward with its degrees and parts to obtain the fixed degrees and parts. For the sun's-march difference: at Jupiter fixed conjunction, multiply by five, halve, and drop one place; at morning visibility, first double and drop one place, afterward multiply by five, halve, and drop one place; For afterward fixed rapid motion, apply five to the first difference, halve, and drop one place; for the fixed difference, double and drop one place; For Mars fixed conjunction, divide by two outside the body; at morning visibility, multiply the first difference by seven and drop one place; for the afterward difference, divide by two outside the body, then multiply by seven and drop one place; For Saturn fixed conjunction, drop one place and add two from below; at morning visibility, drop the first difference one place; for the afterward difference, add three from below, drop one place, and drop the afterward difference one place; For Venus fixed conjunction, double the factor; at evening visibility, double the first difference at occultation; for the afterward difference, add three from below; at morning rapid occultation, add two from below to the first difference and double the afterward difference; at evening retrograde occultation and morning retrograde visibility, multiply by six and drop one place front and rear; For Mercury evening visibility, add three from below to the afterward difference and double the first difference; at morning rapid motion, add three from below to the first difference and double the afterward difference; at fixed conjunction, apply addition and subtraction to the secondary fixed degrees to obtain the fixed degrees; set the fixed degrees and parts, add the winter solstice added-time ecliptic solar degrees and parts, count from the Dipper lodge initial degree until the remainder falls within a lodge, and obtain the variation's added-time lodge degrees; Mars front and rear retreat and front-slow variations all yield secondary fixed stars—set them again, adjust by the stationary-retreat fixed difference degrees and parts, adding increases and subtracting decreases, to obtain front and rear retreat fixed degrees; for front-slow motion, set the front stationary fixed difference, divide by three, and use it to adjust the front-slow fixed degrees. For Mars stationary difference, subtract 124 and a half from the front-slow secondary fixed degrees and add 246 degrees and a little less to the rear-retreat fixed degrees; if at or below 182 degrees 62 parts the value is entering in increase; If above, subtract 182 degrees 62 parts to obtain entering in decrease. Set the entering increase-and-decrease degrees and parts; if at or below 72 degrees it is the upper limit; If above, reverse-subtract 182 degrees 62 parts; the remainder is the lower limit. Set the increment-and-decrement degrees and parts for the entered upper or lower limit; at the upper limit multiply by four, at the lower limit double, add three outside the body, and reduce by one hundred to degrees and parts; if in rear stationary motion, triple to obtain the fixed-difference degrees and parts. Furthermore, the Yitian calendar has Mars retreat fixed-difference degrees and parts: add 241 degrees and a little less to the front-retreat rear secondary fixed degrees and subtract 119 degrees from the retreat secondary fixed degrees and parts; if the remainder is at or below 182 degrees 62 parts it is entering in increase; If above, subtract 182 degrees 62 parts; the remainder is entering in decrease. Set the entered upper and lower limit degrees and parts; if at or below 72 degrees it is the upper limit; if above 72 degrees, subtract 182 degrees 62 parts and treat the remainder as the lower limit. Set the upper and lower limit increment-and-decrement degrees and parts; take the whole as degrees and the remainder as parts to obtain the retreat fixed-difference degrees and parts; if the fixed difference is in rear retreat, double it. There is also the Mars stationary fixed day: set the front and rear stationary regular mid-days; for front stationary, adjust by the front-slow variation entered limit increment-and-decrement fixed degrees and parts, adding increases and subtracting decreases; then apply the front and rear stationary difference degrees and parts, adding increases and reducing decreases, to obtain the front and rear stationary fixed days; the increment-and-decrement difference is applied together through entering the epact. There are also Mars front and rear retreat fixed degrees: set the front and rear variation secondary fixed degrees and parts and, by the front and rear retreat fixed-difference degrees and parts, add if in increase and reduce if in decrease to obtain the fixed degrees and parts; Set the fixed degrees and parts, add the winter solstice ecliptic solar degrees and parts, count from the Dipper lodge initial degree until the remainder falls within a lodge, and obtain the lodge degrees and parts of retrograde motion; divide the increment-and-decrement fixed degrees by three before use.

48

日率度率：以本段定積減後段定積，為泛日率；以本段定星減後段定星，為定度率。又置後段甲子，以前段甲子減之，餘為距後實日率。《乾元》以前段定積減後段定積為日率，以其段定星減後段定星為度率。《儀天》各置其段定日定度，以前段定日定度減之，餘者為其段日率、度率。其退行段，置前段定度減之，餘為退行度率。

Day rate and degree rate: subtract the later segment's fixed accumulation from this segment's fixed accumulation to obtain the general day rate; Subtract the later segment's fixed star from this segment's fixed star to obtain the fixed degree rate. Set the later segment's jiazi day, subtract the earlier segment's jiazi day, and the remainder gives the actual day rate to the later segment. Under Qianyuan, subtract the later segment's fixed accumulation from the earlier segment's fixed accumulation for the day rate, and subtract the later segment's fixed star from that segment's fixed star for the degree rate. Under Yitian, set each segment's fixed day and fixed degree, subtract the earlier segment's fixed day and fixed degree, and the remainder gives that segment's day rate and degree rate. For a retrograde segment, set the earlier segment's fixed degree and subtract it; the remainder is the retrograde degree rate.

49

平行分：《儀天》謂之求每日平行度及分。

Parallel motion parts: the Yitian calendar terms this procedure "finding the daily parallel degrees and parts."

50

以距後日率除度率，為平行分。《乾元》以日率除度率為行分。《儀天》各置其段度率及分，以其段日率除之，即得其星平行分。

Divide the degree rate by the distance-to-later day rate to obtain the parallel motion parts. Under Qianyuan, divide the degree rate by the day rate to obtain the motion parts. Under Yitian, set each segment's degree rate and parts, divide by that segment's day rate, and obtain the planet's parallel motion parts.

51

初末行分：《儀天》謂之求每段初末日度及分。

First and last motion parts: the Yitian calendar terms this "finding the opening and closing solar degrees and parts of each segment."

52

置其段平行分，與後段平行分相減，為合差；半之，加減平行分，為初、末行分；後多者減平行分為初，加平行分為末；後少者加平行分為初，減平行分為末。《乾元》法同。《儀天》各以其段平行分與後段平行分相減，餘為會差，半會差，以加減其段平行分，余同《應天》。又五星前留一段及後退段，皆加為初、減為末；後留一段及前退段，皆以半總差減為初、加為末。其總差消息前後段初、末分，令衰殺等以用總差，即得前後段初、末行分相應也。

Set the segment's parallel parts and subtract the later segment's parallel parts to obtain the aggregate difference; Halve it, then add to or subtract from the parallel parts to obtain the first and last motion parts; If the later segment is greater, subtract the parallel parts for the first and add them for the last; If the later segment is lesser, add the parallel parts for the first and subtract them for the last. The Qianyuan calendar uses the same method. Under Yitian, subtract each segment's parallel parts from the later segment's parallel parts; the remainder is the conjunction difference; halve the conjunction difference and add to or subtract from the segment's parallel parts; the rest is the same as under Yingtian. For the five planets, in a pre-stationary segment and a rear retrograde segment, always add for the first and subtract for the last; In a post-stationary segment and a front retrograde segment, subtract half the total difference for the first and add it for the last. Let the total difference wax and wane the first and last parts of the front and rear segments so that their deceleration and acceleration are equal when the total difference is applied, and the first and last motion parts of the consecutive segments will correspond.

53

求日差：以距後日除合差為日差。《乾元》以日率除合差為日差。《儀天》置其段總差，以減其日率，一百除之，即為每日差行之分。

To find the daily difference: divide the aggregate difference by the distance-to-later days to obtain the daily difference. Under Qianyuan, divide the aggregate difference by the day rate to obtain the daily difference. Under Yitian, set each segment's total difference, subtract the day rate, divide by 100, and obtain the daily differential motion parts.

54

求每日行分：以日差後多者益、後少者損初日行分，為每日行分。《乾元》、《儀天》法同。

To find the daily motion parts: if the later segment is greater, add the daily difference to the first day's motion parts; if lesser, subtract; this gives the daily motion parts. The Qianyuan and Yitian calendars use the same method.

55

求每日星所在：以每日行分順加逆減其星，命如前，即得所求。其木火土水前、後遲段平行分倍之，前為初，後為末分，各以距後日除，為日差；前遲日損、後遲日益，為每日行分。《乾元》以日差累損益初日行分，累加其段宿次，即得每日星行宿次及分。《儀天》求每日差行度及分，各置其段總差，以減其日率一日以餘之，即為每日差行之分。以每日差分累損益初日行分，為每日行度及分。初日行分多於末日行分，累損初日行分；少於末日行分，累益初日行分。將其每日行度及分累加其星初日所在宿次，各得每日所在宿次及分。如是退行段，將每日行分累減其初日宿次及分，即得退行所在宿度及分。又《儀天》有直求其日星所在宿次，置其所求日，減一，以乘每日差分，所得為積差，以積差加減初日行分，初日多於末日減之；末日多於初日加之，即得其日行分；以初日行分並之，乃半之，為平行分；置平行分，以求日數乘之，為積度及分；以其積度及分加其星初日宿度，命去之，即其星其日所在宿次及分、如是退行段，以其積度及分減其星初日宿度，餘，為其星所在宿度及分。

To find each day's star position: successively add or subtract the daily motion parts from the star (adding in direct motion, subtracting in retrograde), count as before, and obtain the result sought. For Jupiter, Mars, Saturn, and Mercury in front and rear slow segments, double the parallel motion parts—the front for the first and the rear for the last; divide each by the distance-to-later days to obtain the daily difference; In front slow segments decrease daily; in rear slow segments increase daily—this yields the daily motion parts. Under Qianyuan, cumulatively add or subtract the daily difference from the first day's motion parts, and cumulatively add the segment's lodge positions to obtain each day's star lodge position and parts. Under Yitian, to find the daily differential motion degrees and parts: set each segment's total difference, subtract the day rate minus one day from the remainder, and obtain the daily differential motion parts. Cumulatively add or subtract the daily difference from the first day's motion parts to obtain each day's motion degrees and parts. If the first day's motion parts exceed the last day's, cumulatively decrease the first day's motion parts; If they are less than the last day's, cumulatively increase the first day's motion parts. Cumulatively add the daily motion degrees and parts to the star's lodge position on the first day to obtain each day's lodge position and parts. For retrograde segments, cumulatively subtract the daily motion parts from the first day's lodge position and parts to obtain the retrograde lodge degrees and parts. Yitian also has a direct method to find a given day's star lodge: set the day sought, subtract one, multiply by the daily difference to obtain the accumulated difference; adjust the first day's motion parts—subtract if the first day exceeds the last; add if the last exceeds the first—to obtain that day's motion parts; add to the first day's motion parts and halve to obtain the parallel motion parts; set the parallel motion parts, multiply by the number of days sought, and obtain the accumulated degrees and parts; add the accumulated degrees and parts to the star's first-day lodge degree and count off the lodges to obtain that day's lodge position and parts; for retrograde segments, subtract the accumulated degrees and parts from the star's first-day lodge degree, and the remainder gives the star's lodge degrees and parts.

56

漏刻，《周禮》，挈壺氏主挈壺水以為漏，以水火守之，分以日夜，所以視漏刻之盈縮，辨昏旦之短長。自秦、漢至五代，典其事者，雖立法不同，而皆本于《周禮》。惟後漢、隋、五代著于史志，其法甚詳，而曆載既久，傳用漸差。國朝復挈壺之職，專司辰刻，署置於文德殿門內之東偏，設鼓樓、鐘樓於殿庭之左右。其制有銅壺、水稱、渴烏、漏箭、時牌、契之屬：壺以貯水，烏以引注，稱以平其漏，箭以識其刻，牌以告時於晝，牌有七，自卯至酉用之，制以牙，刻字填金。契以發鼓於夜，契有二：一曰放鼓。二曰止鼓制以木，刻字於上。

Clepsydra marks: in the Zhou Rites, the Clepsydra-Plunger office drew water from vessels to measure time, guarding it with fire and water to divide day from night, thereby observing variations in the clepsydra marks and distinguishing the length of dusk and dawn. From the Qin and Han through the Five Dynasties, officials charged with this duty legislated differently, yet all derived their methods from the Zhou Rites. Only the Eastern Han, Sui, and Five Dynasties preserved detailed methods in the historical annals, but as calendar eras lengthened, transmitted practice gradually drifted. The dynasty restored the clepsydra-plunger office to oversee the hours and marks, stationed within the east side of the Wende Palace gate, with drum and bell towers set to the left and right of the palace courtyard. The apparatus included bronze vessels, water scales, thirst-bird siphons, clepsydra arrows, time placards, and tally slips: vessels store water, birds draw and pour it, scales level the flow, arrows mark the divisions, and placards announce the hours by day—seven placards, from mao to you, carved in ivory with gilt-filled characters. Tally slips trigger the drums at night; there are two kinds: one called drum-release. The second is called drum-stop; it is made of wood with characters carved upon it.

57

常以卯正後一刻為禁門開鑰之節，盈八刻後以為辰時，每時皆然，以至於酉。每一時，直官進牌奏時正，雞人引唱，擊鼓一十五聲，惟午正擊鼓一百五十聲。

The eighth mark after mao proper serves as the signal to open the forbidden gates and turn the keys; eight marks later marks the chen hour, and each hour follows this pattern through you. At each hour the duty officer presents the placard announcing the correct time, the cockherd leads the chant, and drums sound fifteen times—except at noon proper, when drums sound one hundred fifty times.

58

至昏夜雞唱，放鼓契出，發鼓、擊鐘一百聲，然後下漏。每夜分為五更，更分為五點，更以擊鼓為節，點以擊鐘為節。每更初皆雞唱，轉點即移水稱，以至五更二點，止鼓契出，凡放鼓契出，禁門外擊鼓，然後衙鼓作，止鼓契出亦然，而更鼓止焉。

At dusk, when cocks crow, the drum-release tally is issued, drums and bells sound one hundred times, and then the clepsydra is drained. Each night is divided into five watches, and each watch into five points; watches are marked by drumbeats and points by bell strokes. At the opening of each watch cocks crow; at each point the water scale is shifted, until the second point of the fifth watch, when the drum-stop tally appears; whenever the drum-release tally is issued, drums beat outside the forbidden gates before the office drums begin, and the same when the drum-stop tally appears—then the watch drums cease.

59

五點擊鐘一百聲。雞唱、擊鼓，是謂攢點，至八刻後為卯時正，四時皆用此法。禁鐘又別有更點在長春殿門之外，玉清昭應宮、景靈宮、會靈觀、祥源觀及宗廟陵寢亦皆置焉，而更以鼓為節，點以鉦為節。大中祥符三年，春官正韓顯符上《銅渾儀法要》，其中有二十四氣晝夜進退、日出沒刻數立成之法，合於宋朝曆象，今取其氣節之初，載之于左：

At five points the bell sounds one hundred times. Cockcrow and drumbeats mark the accumulated points; eight marks later is mao proper, and all four seasons use this method. Separate forbidden bells and watch points stand outside the Changchun Hall gate; the Yujing Zhaoying Palace, Jingling Palace, Huiling Abbey, Xiangyuan Abbey, and the imperial temples and mausoleums also have them, with watches marked by drums and points by gongs. In the third year of Dazhong Xiangfu, Spring Office Director Han Xianfu submitted the Essentials of the Copper Armillary Sphere, which included ready-made tables for the day-and-night advance and retreat of the twenty-four qi and for sunrise and sunset marks, conforming to Song calendrical astronomy; the beginnings of the qi nodes are excerpted at left:

60

殿前報時雞唱，唐朝舊有詞，朱梁以來，因而廢棄，止唱和音。景德四年，司天監請復用舊詞，遂詔兩制詳定，付之習唱。每大禮、禦殿、登樓、入閣、內宴、晝改時、夜改更則用之，常時改刻、改點則不用。

The palace timekeeping cockcrow once had Tang-era lyrics, but since the Later Liang these were abandoned and only melodic tones were sung. In the fourth year of Jingde, the Astronomical Bureau petitioned to restore the old lyrics; an edict then directed the two academies to review and fix them and deliver them for training in performance. They are used at great ceremonies, imperial hall audiences, tower ascents, gate entries, inner banquets, and when the day watch or night watch is changed; they are not used for ordinary mark or point adjustments.

61

五更五點後發鼓曰：

After five watches and five points the drum-release chant says:

62

朝光發，萬戶開，群臣謁。平旦寅，朝辨色，泰時昕。日出卯，瑞露晞，祥光繞。食時辰，登六樂，薦八珍。禺中巳，少陽時，大繩紀。日南午，天下明，萬物睹。日昳未，飛夕陽，清晚氣。晡時申，聽朝暇，湛凝神。日入酉，群動息，嚴扃守。

Dawn light issues, myriad gates open, and ministers attend audience. At level dawn in yin hour, the court distinguishes colors as the great morning brightens. The sun rises at mao hour; auspicious dew dries and auspicious light encircles. At mealtime in chen hour, six kinds of music rise and eight delicacies are offered. At yuzhong in si hour, in growing yang the great cord is tied. The sun stands south at wu hour; all under heaven is bright and myriad things are seen. The sun slants at wei hour; the setting sun flies and the evening air clears. At afternoon meal in shen hour, the court recesses and the spirit focuses deeply. The sun sets at you hour; all activity ceases and the gates are strictly guarded.

63

初夜發鼓曰：

The initial night drum-release chant says:

64

日欲暮，魚鑰下，龍韜布。甲夜己，設鉤陳，備蘭錡。乙夜庚，杓位易，太階平。丙夜辛，清鶴唳，夢良臣。丁夜壬，丹禁靜，漏更深。戊夜癸，曉奏聞，求衣始。

The sun nears dusk; fish keys descend and dragon strategies deploy. In the first night watch at ji hour, the garrison formation is set and ritual axes prepared. In the second night watch at geng hour, the Dipper's position shifts and the grand steps level. In the third night watch at xin hour, clear cranes call and worthy ministers appear in dreams. In the fourth night watch at ren hour, the crimson precinct grows quiet and the clepsydra runs deeper. In the fifth night watch at gui hour, dawn petitions are heard and the sovereign seeks his garments.

65

端拱中，翰林天文鄭昭晏上言：「唐貞觀二年三月朔，日有食之，前志不書分數、宿度、分野、虧初復末時刻。臣以《乾元曆》法推之，得其歲戊子，其朔戊申，日所食五分，一分在未出時前，四分出後，其時出在寅六刻，虧在三刻，食甚在八刻，復在卯四刻，當降婁九度。」又言：「按曆書雲，凡欲取驗將來，必在考之既往。謹按《春秋》交食及漢氏以來五星守犯，以新曆及唐《麟德》、《開元》二曆覆驗三十事，以究其疏密。」

In the Duanguan era, Hanlin Astronomical Official Zheng Zhaoyan submitted a memorial: "On the new moon of the third month of the second year of Tang Zhenguan, there was a solar eclipse; the earlier annals did not record the fraction obscured, the lodge degree, the regional field, or the marks of first contact, maximum, and re-emergence. I calculated by the Qianyuan calendar method and obtained the year wuzi and new moon wushen; the sun was obscured five-tenths—one tenth before sunrise and four tenths after; sunrise fell at the sixth mark of yin hour, first contact at the third mark, maximum at the eighth, re-emergence at the fourth mark of mao hour, at descending Lou nine degrees." He further said: "Calendar texts state that to verify what is to come, one must examine what has passed. I respectfully reviewed Spring and Autumn eclipses and Han-dynasty records of the five planets' occupations and transgressions, re-verifying thirty cases against the new calendar and the Tang Lindé and Kaiyuan calendars to assess their accuracy."

66

日食：

Solar eclipses:

67

五星守犯：

Five-planet occupations and transgressions:

68

後漢永元五年七月壬午，歲星犯軒轅大星。《麟德》星五度。《開元》張五度。《乾元》張八度。

In the fifth year of Later Han Yongyuan, on rewu day of the seventh month, Jupiter transgressed the great star of Xuanyuan. Under Lindé: five degrees into the Star lodge. Under Kaiyuan: five degrees into Extended Net. Under Qianyuan: eight degrees into Extended Net.

69

元初三年七月甲寅，歲星入輿鬼。《麟德》井二十九度。《開元》鬼一度。《乾元》柳五度。

In the third year of Yuanchu, on jiayin day of the seventh month, Jupiter entered Carriage Ghost. Under Lindé: twenty-nine degrees into the Well lodge. Under Kaiyuan: one degree into Ghost. Under Qianyuan: five degrees into Willow.

70

後魏大延二年八月丁亥，歲星入鬼。《麟德》井二十八度。《開元》鬼二度。《乾元》柳三度。

In the second year of Later Wei Dayan, on dinghai day of the eighth month, Jupiter entered Ghost. Under Lindé: twenty-eight degrees into the Well lodge. Under Kaiyuan: two degrees into Ghost. Under Qianyuan: three degrees into Willow.

71

正始二年六月己未，歲星犯昴。《麟德》昴二度。《開元》昴三度。《乾元》昴四度。

In the second year of Zhengshi, on jiwei day of the sixth month, Jupiter transgressed Hairy Head. Under Lindé: two degrees into Hairy Head. Under Kaiyuan: three degrees into Hairy Head. Under Qianyuan: four degrees into Hairy Head.

72

宋大明三年五月戊辰，歲星犯東井鉞。《麟德》參四度。《開元》參六度。《乾元》井初度。

In the third year of Song Daming, on wuchen day of the fifth month, Jupiter transgressed the axe of Eastern Well. Under Lindé: four degrees into Three Stars. Under Kaiyuan: six degrees into Three Stars. Under Qianyuan: the initial degree of the Well lodge.

73

後漢永和四年七月壬午，熒惑入南斗，犯第三星。《麟德》箕七度。《開元》鬥一度。《乾元》鬥十二度。

In the fourth year of Later Han Yonghe, on rewu day of the seventh month, Mars entered Southern Dipper and transgressed the third star. Under Lindé: seven degrees into Winnowing Basket. Under Kaiyuan: one degree into the Dipper lodge. Under Qianyuan: twelve degrees into the Dipper lodge.

74

魏嘉平三年十月癸未，熒惑犯亢南星。《麟德》角六度。《開元》亢五度。《乾元》亢三度。

In the third year of Wei Jiaping, on guiwei day of the tenth month, Mars transgressed the southern star of Gullet. Under Lindé: six degrees into the Horn lodge. Under Kaiyuan: five degrees into the Gullet lodge. Under Qianyuan: three degrees into the Gullet lodge.

75

晉永和七年五月乙未，熒惑犯軒轅大星。《麟德》星七度。《開元》張二度。《乾元》張二度。

In the seventh year of Jin Yonghe, on yiwei day of the fifth month, Mars transgressed the great star of Xuanyuan. Under Lindé: seven degrees into the Star lodge. Under Kaiyuan: two degrees into Extended Net. Under Qianyuan: two degrees into Extended Net.

76

後魏太常二年五月癸巳，熒惑犯右執法。《麟德》翼六度。《開元》翼十二度。《乾元》翼十三度。

In the second year of Later Wei Taichang, on guisi day of the fifth month, Mars transgressed the Right Enforcer. Under Lindé: six degrees into the Wings lodge. Under Kaiyuan: twelve degrees into the Wings lodge. Under Qianyuan: thirteen degrees into the Wings lodge.

77

陳天嘉四年八月甲午，熒惑犯軒轅大星。《麟德》張二度。《開元》張五度。《乾元》張四度。

In the fourth year of Chen Tianjia, on jiawu day of the eighth month, Mars transgressed the great star of Xuanyuan. Under Lindé: two degrees into Extended Net. Under Kaiyuan: five degrees into Extended Net. Under Qianyuan: four degrees into Extended Net.

78

後漢延光三年九月壬寅，鎮星犯左執法。《麟德》翼十九度。《開元》軫二度。《乾元》翼五度。

In the third year of Later Han Yanguang, on renyin day of the ninth month, Saturn transgressed the Left Enforcer. Under Lindé: nineteen degrees into the Wings lodge. Under Kaiyuan: two degrees into the Chariot Shaft lodge. Under Qianyuan: five degrees into the Wings lodge.

79

晉永和十年正月癸酉，鎮星掩鉞星。《麟德》參六度。《開元》參七度。《乾元》井三度。

In the tenth year of Jin Yonghe, on guiyou day of the first month, Saturn occulted the Axe Star. Under Lindé: six degrees into Three Stars. Under Kaiyuan: seven degrees into Three Stars. Under Qianyuan: three degrees into the Well lodge.

80

後魏神瑞二年三月己卯，鎮星再犯輿鬼積屍。《麟德》井二十八度。《開元》井三十度。《乾元》柳初度。

In the second year of Later Wei Shenrui, on jimao day of the third month, Saturn again transgressed the Accumulated Corpses in Carriage Ghost. Under Lindé: twenty-eight degrees into the Well lodge. Under Kaiyuan: thirty degrees into the Well lodge. Under Qianyuan: the initial degree of the Willow lodge.

81

齊永明九年七月庚戌，鎮星逆在泣星東北。《麟德》危二度。《開元》虛九度。《乾元》危四度。

In the ninth year of Qi Yongming, on gengxu day of the seventh month, Saturn was in retrograde northeast of the Weeping Star. Under Lindé: two degrees into the Rooftop lodge. Under Kaiyuan: nine degrees into the Emptiness lodge. Under Qianyuan: four degrees into the Rooftop lodge.

82

陳永定三年六月庚子，鎮星入參。《麟德》參七度。《開元》參八度。《乾元》井二度。

In the third year of Chen Yongding, on gengzi day of the sixth month, Saturn entered Three Stars. Under Lindé: seven degrees into Three Stars. Under Kaiyuan: eight degrees into Three Stars. Under Qianyuan: two degrees into the Well lodge.

83

後漢永初四年六月癸酉，太白入鬼。《麟德》參五度。《開元》井三十度。《乾元》鬼初度。

In the fourth year of Later Han Yongchu, on guiyou day of the sixth month, Venus entered Ghost. Under Lindé: five degrees into Three Stars. Under Kaiyuan: thirty degrees into the Well lodge. Under Qianyuan: the initial degree of the Ghost lodge.

84

延光三年二月辛未，太白入昴。《麟德》晨伏。《開元》昴六度。《乾元》昴一度。

In the third year of Yanguang, on xinwei day of the second month, Venus entered Hairy Head. Under Lindé: morning culmination. Under Kaiyuan: six degrees into Hairy Head. Under Qianyuan: one degree into Hairy Head.

85

魏黃初三年閏六月丁丑，太白晨伏。《麟德》丁亥晨伏，後十日。《開元》同，丁丑晨伏。《乾元》十月置閏，七月丁丑晨伏。

In the third year of Wei Huangchu, on dingchou day of the intercalary sixth month, Venus reached morning culmination. Under Lindé: morning culmination on dinghai day, ten days later. Under Kaiyuan: the same—morning culmination on dingchou day. Under Qianyuan: intercalation placed in the tenth month; morning culmination on dingchou day of the seventh month.

86

晉咸康七年四月己丑，太白入輿鬼。《麟德》柳三度。《開元》鬼一度。《乾元》柳一度。

In the seventh year of Jin Xiankang, on jichou day of the fourth month, Venus entered Carriage Ghost. Under Lindé: three degrees into the Willow lodge. Under Kaiyuan: one degree into Ghost. Under Qianyuan: one degree into the Willow lodge.

87

晉永和十一年九月己未，太白犯天江。《麟德》尾四度。《開元》尾九度。《乾元》尾十二度。

In the eleventh year of Jin Yonghe, on jiwei day of the ninth month, Venus transgressed the Celestial River. Under Lindé: four degrees into the Tail lodge. Under Kaiyuan: nine degrees into the Tail lodge. Under Qianyuan: twelve degrees into the Tail lodge.

88

漢太始二年七月辛亥，辰星夕見。《麟德》伏末見。《開元》夕見軫九度。《乾元》夕見軫九度。

In the second year of Han Taishi, on xinhai day of the seventh month, Mercury was seen in the evening. Under Lindé: not seen at the end of its period of invisibility. Under Kaiyuan: seen in the evening at nine degrees into the Chariot Shaft lodge. Under Qianyuan: seen in the evening at nine degrees into the Chariot Shaft lodge.

89

後漢元初五年五月庚午，辰星犯輿鬼。《麟德》井二十七度。《開元》井二十八度。《乾元》井二十九度。

In the fifth year of Later Han Yuanchu, on gengwu day of the fifth month, Mercury transgressed Carriage Ghost. Under Lindé: twenty-seven degrees into the Well lodge. Under Kaiyuan: twenty-eight degrees into the Well lodge. Under Qianyuan: twenty-nine degrees into the Well lodge.

90

漢安二年五月丁亥，辰星犯輿鬼。《麟德》夕見井二十二度。《開元》夕見鬼二度。《乾元》夕見鬼一度。

In the second year of Han An, on dinghai day of the fifth month, Mercury transgressed Carriage Ghost. Under Lindé: seen in the evening at twenty-two degrees into the Well lodge. Under Kaiyuan: seen in the evening at two degrees into Ghost. Under Qianyuan: seen in the evening at one degree into Ghost.

91

晉隆安三年五月辛未，辰星犯軒轅大星。《麟德》夕見星五度。《開元》夕見星三度。《乾元》夕見星五度。

In the third year of Jin Longan, on xinwei day of the fifth month, Mercury transgressed the great star of Xuanyuan. Under Lindé: seen in the evening at five degrees into the Star lodge. Under Kaiyuan: seen in the evening at three degrees into the Star lodge. Under Qianyuan: seen in the evening at five degrees into the Star lodge.

92

後魏太和十五年六月丙子，辰星隨太白於西方。《麟德》張二度。《開元》星五度。《乾元》張初度。

In the fifteenth year of Later Wei Taihe, on bingzi day of the sixth month, Mercury followed Venus in the west. Under Lindé: two degrees into Extended Net. Under Kaiyuan: five degrees into the Star lodge. Under Qianyuan: the initial degree of Extended Net.

93

端拱二年四月己未，翰林祗候張玭夜直禁中，太宗手詔曰：「覽《乾元曆》細行，此夕熒惑當退軫宿乃順行，今止到角宿即順行，得非曆差否？」奏曰：「今夕一鼓，占熒惑在軫末、角初，順行也。據曆法，今月甲寅至軫十六度，乙卯順行，驗天差二度。臣占熒惑明潤軌道，兼前歲逆出太微垣，按曆法差疾者八日，此皆上天祐德之應，非曆法之可測也。」至道元年，昭晏又上言：「承詔考驗司天監丞王睿雍熙四年所上曆，以十八事按驗，所得者六，所失者十二。」太宗嘉之，謂宰相曰：「昭晏曆術用功，考驗否臧，昭然無隱。」由是賜昭晏金紫，令兼知曆算。二年，屯田員外郎呂奉天上言：

In the second year of Duanguan, on jiwei day of the fourth month, Hanlin Attendant Zhang Bi was on night duty in the palace when Emperor Taizong sent a handwritten edict: "Having reviewed the fine motions in the Qianyuan calendar, Mars tonight should withdraw from the Chariot Shaft lodge and then resume direct motion; yet it has resumed direct motion upon reaching only the Horn lodge—is this not a calendar error?" He memorialized in reply: "At the first drum tonight, Mars was observed at the end of Chariot Shaft and the beginning of Horn, in direct motion. According to the calendar method, on jiayin day of this month Mars should reach sixteen degrees into Chariot Shaft and resume direct motion on yimao day; compared with heaven, it differed by two degrees. Your servant observed that Mars was bright and its course smooth; moreover, the year before it had emerged in retrograde from the Supreme Palace Enclosure, and by the calendar method it was eight days too fast—all of this, I submit, is a response of Heaven blessing virtue, not something calendrical methods can predict." In the first year of Zhidao, Zhaoyan submitted another memorial: "Having received an edict to examine the calendar submitted in the fourth year of Yongxi by Vice Director Wang Rui of the Astronomical Bureau, I tested it against eighteen cases: six matched and twelve failed." Emperor Taizong commended this and told the chief ministers: "Zhaoyan has applied himself diligently to calendrical science; in testing success and failure, his findings are clear and without concealment." Thereupon he bestowed on Zhaoyan the gold-and-purple insignia and ordered him to serve concurrently as overseer of calendar calculation. In the second year, Outer Gentleman of the Field Administration Lu Fengtian submitted a memorial:

94

「按經史年曆，自漢、魏以降，雖有編聯，周、秦以前，多無甲子。太史公司馬遷雖言歲次，詳求朔閏，則與經傳都不符合，乃言周武王元年歲在乙酉。唐兵部尚書王起撰《五點陣圖》，言周桓王十年，歲在甲子，四月八日佛生，常星不見；又言孔子生於周靈王庚戌之歲，卒于周悼王四十一年壬戌之歲，皆非是也。馬遷乃古之良史，王起又近世名儒，後人因循，莫敢改易。臣竊以史氏凡編一年，則有一十二月，月有晦朔、氣閏，則須與歲次合同，苟不合同，何名歲次？本朝文教聿興，禮樂鹹備，惟此一事，久未刊詳。臣探索百家，用心十載，乃知唐堯即位之年，歲在丙子，迄太平興國元年，亦在丙子，凡三千三百一年矣。虞、夏之間，未有甲子可證，成湯既沒，太甲元年始有二月乙丑朔旦冬至，伊尹祀于先王，至武王伐商之年正月辛卯朔，二十有八日戊午，二月五日甲子昧爽。又康王十二年六月戊辰朔，三日庚午朏，王命作冊畢。自堯即位年，距春秋魯隱西元年，凡一千六百七年；從隱西元年，距今至道二年，凡一千七百一十五年；從太甲元年，距今至道二年，凡二千七百三十二年；從魯莊公七年四月辛卯夜常星不見，距今至道二年，凡一千六百八十一年，從周靈王二十年孔子生，其年九月庚戌、十月庚辰兩朔頻食，距今至道二年，凡一千五百四十五年；從魯哀公十六年四月乙丑孔子卒，距今至道二年，凡一千四百七十二年。以上並據經傳正文，用古曆推校，無不符合，乃知《史記》及《五點陣圖》所編之年，殊為闊略。諸如此事，觸類甚多，若盡披陳，恐煩聖覽。臣耽研既久，引證尤明，起商王小甲七年二月甲申朔旦冬至，自此之後，每七十六年一得朔旦冬至，此乃古曆一蔀；每蔀積月九百四十、積日二萬七千七百五十九，率以為常，直至《春秋》魯僖公五年正月辛亥朔旦冬至，了無差爽。用此為法，以推經傳，縱小有增減，抑又經傳之誤，皆可以發明也。古曆到齊、梁以來，或差一日，更有近曆校課，亦得符合。伏望聖慈，許臣撰集，不出百日，其書必成。儻有可觀，願藏秘府。」

"Examining the year-calendars in the classics and histories, from Han and Wei downward there are compilations and linkages, but before Zhou and Qin most lack sexagenary designations. Grand Historian Sima Qian did speak of year sequences, but when one seeks the new moons and intercalations in detail, they match neither classics nor commentaries; he placed the first year of King Wu of Zhou in yiyou. Wang Qi, Director of the Tang Ministry of War, composed the Five-Point Array Diagram, placing Duke Huan of Zhou's tenth year in jiazi and the Buddha's birth on the eighth day of the fourth month, when ordinary stars were not seen; he also placed Confucius's birth in the gengxu year of King Ling of Zhou and his death in the forty-first year of King Dao of Zhou, the renxu year—all of which is incorrect. Qian was a fine historian of antiquity and Wang Qi a renowned scholar of recent times; later men followed their lead and none dared to change it. Your servant holds that whenever historians compile a year, it must contain twelve months, and the months must have dark moons, new moons, qi nodes, and intercalations that agree with the year sequence—if they do not agree, how can it be called a year sequence? In our dynasty culture and teaching flourish and rites and music are fully in place, yet this one matter has long gone uncorrected in detail. Your servant has explored the hundred schools and applied himself for ten years, and thereby learned that the year in which Tang Yao acceded to the throne was bingzi, and that down to the first year of Taiping Xingguo is also bingzi—a span of three thousand three hundred and one years. Between Yu and Xia there is no sexagenary proof; after Cheng Tang's death, the first year of Taijia first yields the second month, yichou new moon at dawn on the winter solstice, when Yi Yin sacrificed to the former kings; down to the year King Wu attacked Shang, the first month had xinmao new moon, and on the twenty-eighth day, wuwu, and on the fifth day of the second month, jiazi, before dawn. Also in the twelfth year of King Kang, the sixth month had wuchen new moon; on the third day, gengwu, at the crescent moon, the king ordered the composition of the Document of Bi. From the year of Yao's accession to the first year of Duke Yin of Lu in the Spring and Autumn Annals is one thousand six hundred and seven years; from Duke Yin's first year to the present, the second year of Zhidao, is one thousand seven hundred and fifteen years; from the first year of Taijia to the present, the second year of Zhidao, is two thousand seven hundred and thirty-two years; from the night in the fourth month, xinmao, of the seventh year of Duke Zhuang of Lu, when ordinary stars were not seen, to the present, the second year of Zhidao, is one thousand six hundred and eighty-one years; from the twentieth year of King Ling of Zhou, when Confucius was born—in that year the new moons of the ninth month, gengxu, and the tenth month, gengchen, suffered frequent eclipses—to the present, the second year of Zhidao, is one thousand five hundred and forty-five years; From the death of Confucius on yichou in the fourth month of the sixteenth year of Duke Ai of Lu to the present, the second year of Zhidao, is one thousand four hundred and seventy-two years. All of the above is based on the main text of the classics and commentaries and verified by the ancient calendar, with nothing inconsistent—whereby one knows that the years compiled in the Records of the Grand Historian and Wang Qi's Diagram of the Five Night Watches are exceedingly broad and rough. Matters such as these, by analogy, are very numerous; if I were to lay them all bare, I fear I would weary Your Majesty's reading. Your servant has long devoted himself to this research, and the citations are especially clear: starting from the winter solstice at dawn on the new moon, jiashen, of the second month of the seventh year of King Xiao Jia of Shang, thereafter a new-moon dawn winter solstice occurs every seventy-six years—this is one obscuration of the ancient calendar; each obscuration accumulates nine hundred and forty months and twenty-seven thousand seven hundred and fifty-nine days, taken as the constant rule, down to the winter solstice at dawn on the new moon, xinhai, of the first month of the fifth year of Duke Xi of Lu in the Spring and Autumn Annals, without the slightest discrepancy. Using this as the method to extrapolate the classics and commentaries, even where there are slight additions or subtractions—or indeed errors in the classics and commentaries—all can be clarified. From Qi and Liang down, the ancient calendar sometimes differed by a day; tested against more recent calendars, it also accords. I humbly hope for Your Majesty's gracious permission to compile and gather this material; within a hundred days the work will surely be finished. If it proves at all worth reading, I ask that it be preserved in the imperial archives."

95

詔許之。書終不就。

An edict approved the request. The book was never completed.

96

又司天冬官正楊文鎰上言：「新曆甲子，請以百二十年。」事下有司，以其無所依據，議寢不行。太宗曰：「支幹相承，雖止於六十，儻再周甲子，成上壽之數，使期頤之人得見所生之年，不亦善乎？」遂詔新曆甲子所紀百二十歲。

The Director of the Astronomy Bureau, Yang Wenyi, also memorialized, saying, "For the sexagenary cycle of the new calendar, I ask that one hundred and twenty years be used." The matter was referred to the responsible offices; because it had no basis, discussion suspended it and it was not implemented. Taizong said, "The stems and branches succeed one another, stopping at sixty cycles; yet if the sexagenary cycle were run through twice, reaching the number of great longevity, so that men of a hundred years could see the year of their birth—would that not be well?" Accordingly an edict ordered that the sexagenary cycle recorded in the new calendar should span one hundred and twenty years.

97

國初，有司上言：「國家受周禪，周木德，木生火，則本朝運膺火德，色當尚赤。臘以戌日。」詔從之。

At the founding of the state, the responsible officials memorialized: "Our realm received the abdication of Zhou; Zhou held the virtue of Wood, and Wood generates Fire; therefore our dynasty's destiny receives the virtue of Fire, and the color it should honor is red. The La sacrifice should be on a xu day." An edict approved this.

98

雍熙元年四月，布衣趙垂慶上書言：「本朝當越五代而上承唐統為金德，若梁繼唐，傳後唐，至本朝亦合為金德。矧自國初符瑞色白者不可勝紀，皆金德之應也。望改正朔，易車旗服色，以承天統。」事下尚書省集議，常侍徐鉉與百官奏議曰：「五運相承，國家大事，著於前載，具有明文。頃以唐末喪亂，朱梁篡弑，莊宗早編屬籍，親雪國仇，中興唐祚，重新土運，以梁室比羿、浞、王莽，不為正統。自後數姓相傳，晉以金，漢以水，周以木，天造有宋，運膺火德。況國初祀赤帝為感生帝，於今二十五年，豈可輕議改易？」又云：「梁至周不合迭居五運，欲國家繼唐統為金德，且五運迭遷，親承歷數，質文相次，間不容髮，豈可越數姓之上，繼百年之運？此不可之甚也。按《唐書》天寶九載，崔昌獻議自魏、晉至周、隋，皆不得為正統，欲唐遠繼漢統，立周、漢子孫為王者後，備三恪之禮。是時，朝議是非相半，集賢院學士衛包上言符同，李林甫遂行其事。至十二載，林甫卒，復以魏、周、隋之後為三恪，崔昌、衛包由是遠貶，此又前載之甚明也。伏請祗守舊章，以承天祐。」從之。

In the fourth month of the first year of Yongxi, the commoner Zhao Chuiqing submitted a memorial stating, "Our dynasty ought to skip over the Five Dynasties and inherit the Tang line as the virtue of Metal; if Liang succeeded Tang and passed the throne to Later Tang, down to our dynasty it would likewise accord with Metal. Moreover, from the founding of the state, white auspicious omens too numerous to record have all been responses of the virtue of Metal. I hope the calendar will be corrected, and the colors of chariots, banners, and robes changed, to inherit Heaven's mandate." The matter was sent to the Ministry of Personnel for collective deliberation; Regular Attendant Xu Xuan and the officials submitted a deliberation: "The succession of the Five Phases is a great matter of state, recorded in earlier histories with clear textual authority. Recently, because of the disorders at the end of Tang, when Zhu's Liang usurped and slew the throne, Emperor Zhuangzong had long been enrolled in the imperial genealogy; he personally avenged the state's wrong and restored the house of Tang, reestablishing the cycle of Earth—placing the house of Liang alongside Yi, Zhuo, and Wang Mang and denying it orthodox standing. Thereafter several houses succeeded one another: Jin with Metal, Han with Water, Zhou with Wood—and Heaven produced Song, whose destiny receives the virtue of Fire. Moreover, from the founding we have sacrificed to the Red Emperor as the Spirit-of-Birth Emperor—for twenty-five years now—how can this be lightly debated and changed?" They also said, "From Liang through Zhou does not fit sequential occupation of the Five Phases. To wish our state to inherit the Tang line as Metal, moreover—the Phases shift in turn, each directly receiving the heavenly sequence, substance and pattern succeeding without the interval of a hair—how could one leap over several houses and inherit a cycle of a hundred years? This is utterly inadmissible. According to the Book of Tang, in the ninth year of Tianbao Cui Chang offered a deliberation that from Wei and Jin through Zhou and Sui none might count as orthodox lineage; he wished Tang to inherit the Han line from afar, installing descendants of Zhou and Han as successors of kings, fulfilling the rites of the Three Respectful Lines. At that time court opinion was evenly divided; Academy Scholar Wei Bao memorialized in agreement, and Li Linfu thereupon carried it out. By the twelfth year Linfu died, and Wei, Zhou, and Sui successors were again made the Three Respectful Lines; Cui Chang and Wei Bao were thereby banished far away—another precedent made very clear in earlier records. I humbly ask that the old statutes be reverently upheld, to inherit Heaven's protection." The proposal was accepted.

99

大中祥符三年，開封府功曹參軍張君房上言：「自唐室下衰，土德隤圮，朱氏強稱金統，而莊宗旋復舊邦，則朱梁氏不入正統明矣。晉氏又復稱金，蓋謂乘于唐氏，殊不知李?建國於江南耳。漢家二主，共止三年，紹晉而興，是為水德。洎廣順革命，二主九年，終於顯德。以上三朝七主，共止二十四年，行運之間，陰隱而難賾。伏自太祖承周木德而王，當於火行，上系于商，開國在宋，自是三朝迄今以為然矣。愚臣詳而辨之，若可疑者。太祖禪周之歲，歲在庚申。夫庚者，金也，申亦金位，納音是木，蓋周氏稱木，為二金所勝之象也。太宗登極之後，詔開金明池于金方之上，此誰啟之？乃天之靈符也。陛下履極當強圉之歲，握符在作噩之春，適宋道之隆興，得金天之正氣。臣試以瑞應言之，則當年丹徒貢白鹿，姑蘇進白龜，條支之雀來，潁川之雉至。臣又聞當封禪之時，魯郊貢白兔，鄆上得金龜，皆金符之至驗也。願以臣章下三事大臣，參定其事。」疏奏，不報。

In the third year of Dazhong Xiangfu, Merit Officer Zhang Junfang of Kaifeng Prefecture memorialized: "From the decline of the house of Tang, when the virtue of Earth crumbled, the house of Zhu forcibly claimed the Metal succession; yet Emperor Zhuangzong soon restored the old realm—thus the house of Zhu's Liang clearly did not enter the orthodox succession. Jin again claimed Metal, supposing it succeeded the house of Tang—utterly unaware that Li Bian had established a state only in the south of the Yangtze. Han had two rulers totaling only three years, rising as successor to Jin—this was the virtue of Water. From the Guangshun revolution, two rulers over nine years, ending in Xiande. The above three dynasties and seven rulers together lasted only twenty-four years; among their shifting cycles, the phases lie hidden and are hard to discern. From the time when Taizu received Zhou's virtue of Wood and became king—fittingly belonging to the phase of Fire, connected above to Shang, founding the state in Song—the three dynasties down to the present have taken this as settled. This foolish servant has examined and distinguished the matter, and there are points that seem doubtful. The year Taizu received Zhou's abdication was gengshen. Geng is Metal, and shen is also a Metal position; the received tone is Wood—Zhou claimed Wood, and this is the image of two Metals overcoming it. After Taizong ascended the throne, he ordered Jinming Pool opened in the direction of Metal—who initiated this? It was Heaven's spiritual token. Your Majesty took the throne in the year qiangyu, grasping the mandate in the spring of zuoe—just as Song's Way rose gloriously, receiving the correct qi of Metal and Heaven. If I speak of auspicious responses, that year Dantu presented a white deer, Gusu offered a white turtle, a sparrow from Tiaozhi came, and pheasants from Yingchuan arrived. I also heard that at the time of the feng and shan rites, the Lu outskirts presented a white rabbit, and on Yun a golden turtle was found—all supreme verification of Metal tokens. I ask that my memorial be sent down to the grand ministers of the Three Bureaus to jointly determine the matter." The memorial was submitted and received no reply.

100

天禧四年，光祿寺丞謝絳上書曰：

In the fourth year of Tianxi, Assistant Director Xie Jiang of the Court of Imperial Sacrifices submitted a memorial, saying:

101

臣按古志，凡帝王之興，必推五行之盛德，所以配天地而符陰陽也。故神農氏以火德，聖祖以土德，夏以木德，商以金德，周以火德。自漢之興，王火德者，以謂承堯之後。且漢，堯之裔也。五帝之大，莫大於堯，漢能因之，是不墜其緒而善繼其盛德也。國家膺開光之慶，執敦厚之德，宜以土瑞而王天下，然其推終始傳，承周之木德而火當其次。且朱梁不預正統者，謂莊宗復興於後。自石晉、漢氏以及于周，則李昪建國于江左而唐祚未絕，是三代者亦不得正其統矣。昔者，秦祚促而德暴，不入正統，考諸五代之際，亦是類矣。國家誠能下黜五代，紹唐之土德，以繼聖祖，亦猶漢之黜秦，興周之火德以繼堯者也。

According to the ancient records, whenever an emperor rises he must claim the flourishing virtue of one of the Five Phases, thereby matching Heaven and Earth and tallying with Yin and Yang. Thus the clan of Shennong took Fire; the Sacred Ancestor took Earth; Xia took Wood; Shang took Metal; Zhou took Fire. From the rise of Han, those who claimed the virtue of Fire did so on the ground that they succeeded Yao. Moreover, Han was descended from Yao. Among the Five Emperors, none was greater than Yao; Han was able to follow him—thus not dropping his thread and well inheriting his flourishing virtue. Our state received the blessing of founding splendor and holds sincere, steadfast virtue; it ought to take the earthly auspice and rule all under Heaven—yet by reckoning end and beginning, having inherited Zhou's virtue of Wood, Fire comes next. Moreover, Zhu's Liang is excluded from the orthodox succession because Zhuangzong restored the realm afterward. From Later Jin and Han down to Zhou, Li Bian established a state on the left bank of the river while the fortune of Tang had not expired—thus these three dynasties also cannot be made orthodox. Formerly Qin's reign was brief and its virtue harsh, and it did not enter the orthodox succession; considering the period of the Five Dynasties, the cases are of the same kind. If our state could truly set aside the Five Dynasties, renew Tang's virtue of Earth, and thereby succeed the Sacred Ancestor, it would be as when Han dismissed Qin and raised Zhou's virtue of Fire to succeed Yao.

102

夫五行定位，土德居中，國家飛運于宋，作京於汴，誠萬國之中區矣。《傳》曰：「土為群物主，故曰後土。」《洪範》曰「土爰稼穡，稼穡作甘。」方今四海給足，嘉生蕃衍，邇年京師甘露下，泰山醴泉湧，作甘之兆，斯亦見矣。矧靈木異卉，資生於土，千品萬類，不可勝道，非土德之驗乎？

The Five Phases have fixed positions, with Earth at the center; our state's ascending fortune is in Song, with the capital at Bian—truly the central region of the myriad states. The Commentary says, "Earth is lord of the myriad things, and is therefore called Queen Earth." The Great Plan says, "Earth is the field of sowing and harvest; sowing and harvest yield sweetness." At present the four seas are abundantly supplied and life flourishes; in recent years sweet dew has fallen on the capital and sweet springs have welled at Mount Tai—the omen of yielding sweetness is already manifest. Moreover, sacred trees and exotic plants live by relying on Earth—thousands upon myriad kinds too numerous to tell—are these not verification of the virtue of Earth?

103

臣又聞之，太祖生於洛邑，而胞絡惟黃；鴻圖既建，五緯聚於奎躔，而鎮星是主。及陛下升中之次，日抱黃珥；朝祀于太清宮，有星曰含輿，其色黃而潤澤。斯皆凝命有表，微德攸屬，天意人事響效之大者，則土德之符在矣。是故天心之在茲，陛下拒而罔受；民意之若是，陛下謙而弗答。氣壅未宣，河決遂潰，豈不神哉！然則天淵之勃流，水德之浸患，考六府之厭鎮，驗五行之勝剋，亦宜興土之運，禦時之災。伏望順考符應，詳習法度，惟陛下時而行之。

I have also heard that Taizu was born in the capital at Luoyi, his afterbirth entirely yellow; once the grand design was established, the five cords gathered at the Kui asterism, with the Station Star as master. and when Your Majesty performed the feng ascent, the sun wore yellow halos; while performing morning rites at the Palace of Grand Purity, there was a star called Harmon's Cart, yellow and lustrous. These are all great manifestations of a concentrated mandate, to which subtle virtue belongs—great echoes between Heaven's intent and human affairs—whereby the tokens of Earth are present. Therefore, though Heaven's heart is here, Your Majesty has refused and not accepted; and though the people's intent is thus, Your Majesty has been modest and not answered. Qi blocked and unreleased, the river's dike then burst—is this not divine! Then—the surging flood of the heavenly abyss, the inundating calamity of Water's virtue—examining how the six treasuries are to be suppressed and secured and verifying how the Five Phases overcome and restrain, one ought also to raise the cycle of Earth to ward off the calamity of the age. I humbly hope Your Majesty will accord with these tokens and responses, examine the laws in detail, and act at the proper time.

104

大理寺丞董行父又上言曰：「在昔泰皇以萬物生於東，至仁體乎木，故德始於木。木以生火，神農受之為火德；火以生土，黃帝受之為土德；土以生金，少昊受之為金德；金以生水，顓頊受之為水德；水以生木，高辛受之為木德；木以生火，唐堯受之為火德；火以生土，虞舜傳之為土德。土以生金，夏為金德；金以生水，商為水德；水以生木，周為木德；木以生火，漢應圖讖為火德；火以生土，唐受曆運為土德。陛下紹天之統，受天之命，固當上繼唐祚，以金為德，顯黃帝之嫡緒，彰聖祖之丕烈。臣又按聖祖先降于癸酉，太祖受禪于庚申，陛下即位于丁酉，天書下降于戊申。庚，金也，申、酉皆金也，天之體也。陛下紹唐、漢之運，繼黃帝之後，三世變道，應天之統，正金之德，斯又順也。」詔兩制詳議。既而獻議曰：「竊詳謝絳所述，以聖祖得瑞，宜承土德，且引漢承堯緒為火德之比，雖班彪敘漢祖之興有五，其一曰帝堯之苗裔。及序承正統，乃越秦而繼周，非用堯之行。今國家或用土德，即當越唐上，承于隋，彌以非順，失其五德傳襲之序。又據董行父請越五代紹唐為金德，若其度越累世，上承百代之統，則晉、漢洎周，鹹帝中夏，太祖實受終於周室而陟於元後，豈可弗遵傳繼之序，續於遐邈之統？三聖臨禦六十餘載，登封告成，昭姓紀號，率循火行之運，以輝炎靈之曜。茲事體大，非容輕議，矧雍熙中徐鉉等議之詳矣。其謝絳、董行父等所請，難以施行。」詔可。

Assistant Director Dong Xingfu of the Grand Court of Justice also memorialized, saying, "Of old the Supreme Sovereign, because the myriad things are born in the east and utmost benevolence is embodied in Wood, therefore virtue began with Wood. Wood generates Fire, and Shennong received it as the virtue of Fire; Fire generates Earth, and the Yellow Emperor received it as the virtue of Earth; Earth generates Metal, and Shaohao received it as the virtue of Metal; Metal generates Water, and Zhuanxu received it as the virtue of Water; Water generates Wood, and Gaoxin received it as the virtue of Wood; Wood generates Fire, and Tang Yao received it as the virtue of Fire; Fire generates Earth, and Yu Shun handed it down as the virtue of Earth. Earth generates Metal; Xia took Metal; Metal generates Water; Shang took Water; Water generates Wood; Zhou took Wood; Wood generates Fire; Han answered the charts and prophecies as the virtue of Fire; Fire generates Earth; Tang received the calendar fortune as the virtue of Earth. Your Majesty continues Heaven's lineage and receives Heaven's mandate; you ought indeed to succeed the fortune of Tang above, taking Metal as your virtue, displaying the direct line of the Yellow Emperor and manifesting the great glory of the Sacred Ancestor. I also note that the Sacred Ancestor first descended on guiyou; Taizu received the abdication on gengshen; Your Majesty acceded to the throne on dingyou; and the heavenly writ descended on wushen. Geng is Metal; shen and you are both Metal—the body of Heaven. Your Majesty continues the cycles of Tang and Han and succeeds the Yellow Emperor; through three generations the Way has shifted—responding to Heaven's lineage and correcting to the virtue of Metal—this too is accord." An edict ordered detailed deliberation by the Two Departments. Soon afterward they submitted a deliberation: "We have carefully examined what Xie Jiang stated—that because the Sacred Ancestor obtained an auspicious token he ought to inherit the virtue of Earth, and that he cited Han's succession to Yao's line as a parallel to Fire—although Ban Biao, narrating the rise of Han's founder, listed five grounds, one being that he was a descendant of Emperor Yao. Yet when arranging the orthodox succession, he skipped Qin and succeeded Zhou—not applying Yao's line. If our state were now to adopt the virtue of Earth, it would have to leap over Tang above and inherit from Sui—still less accordant, losing the order of the Five Phases' transmission. Again, according to Dong Xingfu's request to skip the Five Dynasties and renew Tang as Metal—if one were to pass over accumulated generations and inherit the lineage of a hundred ages above, then Jin, Han, and Zhou all held emperors over the Central Plains; Taizu truly received the end from the house of Zhou and ascended as supreme sovereign—how could one fail to follow the order of transmission and continue a lineage far removed in time? Three sages have ruled for sixty-odd years; the feng ascent has been announced complete, surname and reign title proclaimed—all following the cycle of the phase of Fire, to shine with the brilliance of the flame spirit. This matter is weighty and not to be debated lightly—moreover Xu Xuan and others deliberated it in full during Yongxi. What Xie Jiang, Dong Xingfu, and the others requested cannot readily be implemented." An edict approved this.

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