1
志第十四歷八
Treatise Fourteen: Calendrics, Part Eight.
2
▲回回曆法二
▲ The Islamic (Huihui) Calendar Method, Part Two
3
日五星中心行度立成造法原本總年零年月分日期，及十二宮初日，凡五立成。 每立成內，首列本信立成處月日宮各紀數，次刑七曜，次刑日中心行度，及土、木、火、金、水各自行度日五星最高行度，交多不祿。 祿其造立成之法於左。
Method for constructing ready reckoners for the Sun's and five planets' mean longitudes: original tables for total years, zero years, months, days, and the twelve signs' ingress days—five ready reckoners in all. Within each ready reckoner, first list the tallies for month, day, and sign at that ready reckoner's principal entries; next list the seven luminaries; next list the Sun's center-travel degree and Saturn, Jupiter, Mars, Venus, and Mercury each self-travel degree and the five planets' greatest-high travel degrees; crossings mostly not recorded. Record its ready-reckoner construction method at left.
4
日中心行長日期立成一日行五十九分八秒，按日累加之，小月二十九日，得二十八度三十五分二秒，大月三十日，得二十九度三十四分一十秒。
Solar mean-motion daily ready reckoner: daily motion 59′08″; accumulate by day—a 29-day month yields 28°35′02″, a 30-day month 29°34′10″.
5
月分立成。 單月大，變月小，末置一閏日。 大月，二十妨度三十四分十秒。 小月二十八度三十五分二秒。 按月累加之，十二月計十一宮十八度五十五分九秒，閏日加五十九分八秒。
Monthly ready reckoner. Odd-numbered months are full (30 days), even months short (29 days), with one intercalary day at year's end. Large month, twenty-obstruct degrees 34 parts 10 seconds. Short month: 28°35′02″. Accumulate by month: twelve months total 11 signs 18°55′09″; an intercalary day adds 59′08″.
6
宮分初日立成。 於白羊宮初日起算，至金牛宮初日，凡三十一日，得一宮一度三十三分十八秒。 五十九分八秒之積。 視宮分日數多少，日數見前。 累加積之，至變魚宮初日，得十一宮一度十一秒。 自白羊至此凡三百三十五日之積。
Sign-ingress daily ready reckoner. From Aries ingress through Taurus ingress—31 days—yields 1 sign 1°33′18″. This is the accumulation of 59′08″ per day. Apply according to each sign's day-count (given above). Accumulate through the zodiac to Pisces (Transforming-Fish) ingress: 11 signs 1°11″. From Aries to Pisces: 335 days' total accumulation.
7
零年立成。 每年十一宮十八度五十五分九秒，三十年閏十一日，故二年、五年、七年、十年、十三年、十六年、十八年、二十一年、二十四年、二十六年、二十九年、皆閏日。 約法，每一年減十一度四分五十一秒，閏年減十度五分四十三秒，三十年爲一宮八度二十玢三十一秒。 每年三百五十四日，計一萬六百二十日，加閏十一日，共一萬六百三十一日。
Zero-year (within-cycle) ready reckoner. Annual increment: 11 signs 18°55′09″; eleven intercalary days in thirty years—years 2, 5, 7, 10, 13, 16, 18, 21, 24, 26, and 29 are leap years. Approximate method: each year subtract 11 degrees 4 parts 51 seconds; intercalary year subtract 10 degrees 5 parts 43 seconds; thirty years make one sign 8 degrees 20 jade 31 seconds. 354 days per year (10,620 in thirty years), plus eleven intercalary days—10,631 days total.
8
總年立成。 第一年爲三宮二十六度五分十九秒。 每三十年加一宮八度二十五分三十一秒，至一千四百罩十年，得五宮十五分三十三秒。
Total-years ready reckoner. Year 1: 3 signs 26°05′19″. Every thirty years add one sign 8 degrees 25 parts 31 seconds; reaching 1,400 cover ten years, obtain five signs 15 parts 33 seconds.
9
▲五星自行度立成造法
▲ Method for Constructing Five-Planet Proper-Motion Ready Reckoners
10
土星日期立成一日五十七分，按日遞加。 小月二十七度三十七分，大月二十八度三十四分。 其五日、十二日、二十日、二十八日增一分者，乃秒數所積也。
Saturn daily ready reckoner: 57′ per day, accumulated sequentially. Short month: 27°37′; full month: 28°34′. The extra 1′ on days 5, 12, 20, and 28 comes from accumulated seconds.
11
月分立成。 大月加二十八度三十四分，小月加二十七度三十革分。 按月累加，十二月計十一宮七度四分，閏日加五十七分。
Monthly ready reckoner. Large month add 28 degrees 34 parts; small month add 27 degrees 30 leather parts. Monthly accumulation: twelve months total 11 signs 7°04′; intercalary day adds 57′.
12
宮分初日立成。 金牛宮初日爲二十九度三十一分，自行三十一日之積。 餘四星准此。 視宮分日數累加之，至變魚宮祿日爲十宮十八度五十八分。
Sign-ingress daily ready reckoner. Taurus ingress: 29°31′—thirty-one days' proper-motion accumulation. The other four planets follow the same rule. According to each sign-portion's day-count accumulate; reaching Transforming-Fish sign record day is ten signs 18 degrees 58 parts.
13
零年立成。 每年十一宮七度四分，其閏日有無，視日中行度，零年有加本星一日行分，下四星准此。 至三十年，共一宮十二度一十六分。
Zero-year ready reckoner. Annual increment: 11 signs 7°04′; intercalary days follow the solar mean-motion table; zero-year entries add the planet's daily motion—the four planets below follow suit. Thirty years total: 1 sign 12°16′.
14
總年立成。 第一年十一宮二十九十度十八分，此隋己未測定根數，一雲即洪武甲子年數，加次在內。 下四星准此。 六百年四宮四度四十四分。 每三十年加一宮十二度二十七分，至一千四百四十年，計七宮十八度二十分。
Total-years ready reckoner. First year eleven signs twenty-ninety degrees 18 parts—this is the root number fixed at Sui jìwèi observation; one says it is the Hongwu jiazi year number, with the additive tier included. The four planets below follow the same pattern. Six hundred years: 4 signs 4°44′. Add 1 sign 12°27′ every thirty years; at year 1440, total 7 signs 18°20′.
15
木星日期立成一日五十四分，按日遞加。 小月二十六度十分，大月二十七度五分。 其四日、十一日、十七日、二十四日、三十日增一分者，秒數所積也。
Jupiter daily ready reckoner: 54′ per day, accumulated sequentially. Short month: 26°10′; full month: 27°05′. The extra 1′ on days 4, 11, 17, 24, and 30 comes from accumulated seconds.
16
月分立成按大、小月累加，十二月計十宮十九度二十九分，閏日加五十四分。
Monthly ready reckoner: accumulate by full and short months—twelve months total 10 signs 19°29′; intercalary day adds 54′.
17
宮分初日立成金牛宮初日二十七度五十九分，至變魚宮初日爲十宮二十九度二十六分。
Sign-ingress ready reckoner: Taurus ingress 27°59′; Pisces ingress 10 signs 29°26′.
18
零年立成每年十宮十九度，至三十年，計七宮二十四度三十九分。
Zero-year ready reckoner: 10 signs 19° per year; thirty years total 7 signs 24°39′.
19
總年立成第一年四宮二十五度十九分，六百年五宮八度二十七分。 每三十年加七宮二四三九，至千四百四十年，計八呂八度五十分。
Total-years ready reckoner: year 1, 4 signs 25°19′; six hundred years, 5 signs 8°27′. Every thirty years add seven signs two-four-three-nine; reaching 1,400 forty years, totaling eight Lü eight degrees 50 parts.
20
火星日期立成一日二十八分，按日遞加。 小月十三度二十三分，大月十三度五十一分。 其二日、五日、九日、十二日、十五日、十八日、二十二日、二十五日、二十八日各減一分。
Mars daily ready reckoner: 28′ per day, accumulated sequentially. Short month: 13°23′; full month: 13°51′. Subtract 1′ on days 2, 5, 9, 12, 15, 18, 22, 25, and 28.
21
月分立成按大小月累加，十二月計五宮十三度二十四分，閏日加二十八分。
Monthly ready reckoner: twelve months total 5 signs 13°24′; intercalary day adds 28′.
22
宮分初日立成金牛宮初日十四度九分，至變魚宮初日五宮十八度二十九分。
Sign-ingress ready reckoner: Taurus ingress 14°09′; Pisces ingress 5 signs 18°29′.
23
零年立成每年五宮十三度二十四分，至三十年，計七宮十七度一分。
Zero-year ready reckoner: 5 signs 13°24′ per year; thirty years total 7 signs 17°01′.
24
總年立成第一年八宮三十四度六分，六百年四宮四工三十三分。 每三十年加七宮度一分，至一千四百四十年，計一度一十一分。
Total-year ready reckoner: first year eight signs 34 degrees 6 parts; six hundred years four signs four-work 33 parts. Every thirty years add seven signs degree 1 part; reaching 1,400 forty years, totaling 1 degree 11 parts.
25
金星日期立成一日三十七分，按日遞加。 小月十七度五十三分，大月十八度三十分。
Venus daily ready reckoner: 37′ per day, accumulated sequentially. Short month: 17°53′; full month: 18°30′.
26
月分立成按大小月累加，十二月計七宮八度十五分，閏日加三十七分。
Monthly ready reckoner: twelve months total 7 signs 8°15′; intercalary day adds 37′.
27
宮分初日立成金牛宮初日十九度七分，至變魚宮初日七宮十五度二分。
Sign-ingress ready reckoner: Taurus ingress 19°07′; Pisces ingress 7 signs 15°02′.
28
零年立成每年七宮八度十五分，至三十年，計二宮十四度十五分。
Zero-year ready reckoner: 7 signs 8°15′ per year; thirty years total 2 signs 14°15′.
29
總年立成第一年一宮十五度二十九分，六百年三宮零三十四分。 每三十年加二宮十四度十五分，一百四十年，計九度五十一分。
Total-years ready reckoner: year 1, 1 sign 15°29′; six hundred years, 3 signs 0°34′. Every thirty years add two signs 14 degrees 15 parts; one hundred forty years, totaling nine degrees 51 parts.
30
水星日期立成一日三度六分，按日遞加。 小月三宮初度六分，大月三宮三度十二分。 其二日、四日、七日、九日、十二日、十四日、十七日、十九日、二十二日、二十四日、二十七日、二十九日各增一分。
Mercury daily ready reckoner: 3°06′ per day, accumulated sequentially. Short month: 3 signs 0°06′; full month: 3 signs 3°12′. Add 1′ on days 2, 4, 7, 9, 12, 14, 17, 19, 22, 24, 27, and 29.
31
月分立成按大小月累加，十二月計初宮十九度四十七分，閏月加三度六分。
Month-portion ready reckoner: according to large and small months accumulate; twelve months total initial sign 19 degrees 47 parts; intercalary month adds three degrees 6 parts.
32
宮分初日立成金牛宮初日三宮六度十九分，至變魚宮初日十宮二十度四十五分。
Sign-ingress ready reckoner: Taurus ingress 3 signs 6°19′; Pisces ingress 10 signs 20°45′.
33
零年立成每年初宮十九度四十七分，至三十年，計八宮二十七度四十四分。
Zero-year ready reckoner: 0 signs 19°47′ per year; thirty years total 8 signs 27°44′.
34
總年立成第一年二宮二十五度三十四分，六百年一宮十度九分。 每三十年加八宮二十七度四十四分，至一千四百四十年，計十一宮六度三十五分。
Total-years ready reckoner: year 1, 2 signs 25°34′; six hundred years, 1 sign 10°09′. Add 8 signs 27°44′ every thirty years; at year 1440, total 11 signs 6°35′.
35
▲日五星最高行度立成造法日五星同用。
▲ Method for Constructing Sun-and-Planet Apogee Ready Reckoners (shared by all six bodies)
36
最高行日分立成。 一日一十微，按日遞加。 其四日、十一日、十八日、二十五日。 各減一微，大月四秒五十六微，小月四秒四十六微。
Apogee daily ready reckoner. Daily motion: 10 micro-units; accumulate sequentially. On the 4th, 11th, 18th, and 25th days subtract 1 micro-unit each; full month 4″56‴, short month 4″46‴.
37
月分立成按大小月累加，十二月計五十八秒一十微。 有閏日加十微。
Monthly ready reckoner: twelve months total 58″10‴. Intercalary day adds 10 micro-units.
38
宮分初日立成金牛宮初日五秒六微，至變魚宮初日五十五秒五微。
Sign-ingress ready reckoner: Taurus ingress 5″06‴; Pisces ingress 55″05‴.
39
零年言成每年五十八秒，去二十微。 按年遞加，三年積六十微加一秒，三十年計二十九分十秒。
Zero-year speak-complete: each year fifty-eight seconds, remove twenty micro. Accumulate annually; 60 micro-units in three years make 1″; thirty years total 29′10″.
40
總年立成一年初宮十度四十分二十八秒，洪武甲子加次。 六百年五十八分十三秒。 每三十年加二十九分七秒，一千四百四十年，計十二度三十六分五十五秒。
Total-years ready reckoner: year 1, 0 signs 10°40′28″, with Hongwu jiazi additive tier. Six hundred years: 58′13″. Add 29′07″ every thirty years; at year 1440, total 12°36′55″.
41
▲太陰經度立成造法
▲ Method for Constructing Lunar Ecliptic-Longitude Ready Reckoners
42
日期立成。 中心行度，一日十三度十一分，按日累加。 大月一宮五度十七分，小月初宮二十二度工分。 內二日、四日、六日、九日、十一、十四、十六、十八、二十一、二十三、二十六、二十八、三十日。 各減一分，共減十三分。 加倍相離度，一日二十四度二十三分，按日遞加。 大月初宮十一度二十七分，小月十一宮十七度四分。 內五日、十四日、二十三日，各減一分。 七輪行度，一日十三度四分，按日遞加。 大月一宮一度五十七分，小月十八度五十三分。 其中逢五，皆減一分。 羅計中心行度，一日三分，按日遞加。 大月一度三十五分，小月一度三十二分。 內三日、九日、十五、二十、二十六日，各增一分。
Daily ready reckoner. Mean longitude: 13°11′ per day, accumulated sequentially. Large month one sign five degrees seventeen parts; small month initial sign twenty-two degrees work parts. On days 2, 4, 6, 9, 11, 14, 16, 18, 21, 23, 26, 28, and 30 subtract 1′ each—thirteen parts total. Doubled-separation longitude: 24°23′ per day, accumulated sequentially. Full month: 0 signs 11°27′; short month: 11 signs 17°04′. On days 5, 14, and 23, subtract 1′ each. Seven-wheel longitude: 13°04′ per day, accumulated sequentially. Full month: 1 sign 1°57′; short month: 18°53′. Whenever the day-number is divisible by five, subtract 1′. Rahu–Ketu mean longitude: 3′ per day, accumulated sequentially. Daily motion: large month 1°35′, small month 1°32′. On days 3, 9, 15, 20, and 26 within the month, add 1 part each.
43
月分立成。 中心行度，大月一宮五度十七分，小月二十二度七分，按月加之，十二月計十一宮十四度二十七分。 內三月、七月、十一月，各增一分。 有閏日，加十三度十一分。 加倍相離度，大月十一度二十七分，小月一宮十牙度四分，十二月計十一宮二十一度三分。 內二、六、十月，各減一分。 有閏日，加二十四度二十三分。 本輪行度，大月一宮一度五十七分，小月十八度五十三分，十二月計十宮五度零分。 有閏日，加十三行度四分。 羅計行度，大月一度三十五分，小月一度三十二分，十二月十八度四十五分。 內三、七、十一月，各增一分。 有閏日，加三分。
Monthly-division ready reckoner. Mean motion: large month 1 sign 5°17′, small month 22°07′; accumulate by month; twelve-month total 11 signs 14°27′. In months 3, 7, and 11, add 1 part each. With an intercalary day, add 13°11′. Doubled-separation degrees: large month eleven degrees twenty-seven parts, small month one palace ten [ya] degrees four parts; twelve months total eleven palaces twenty-one degrees three parts. In months 2, 6, and 10, subtract 1 part each. With an intercalary day, add 24°23′. Deferent rotation: large month 1 sign 1°57′, small month 18°53′; twelve-month total 10 signs 5°00′. With an intercalary day, add 13°04′ deferent motion. Rahu motion: large month 1°35′, small month 1°32′; twelve-month total 18°45′. In months 3, 7, and 11, add 1 part each. With an intercalary day, add 3 parts.
44
協零年立成。 中心行度，每年十一宮十四度二十七分，三十年一宮八度十五分。 三十年閏十一日，與太陽零年同。 下准此。 銳離度，每年十一宮二十一度三分，三十年十一宮二十九度四十分。 閏日，加二十四度二十三分。 本輪行度，每年十宮五度，三十年九宮二十三度四十七分。 閏日，加十三度四分。 羅計行度，每年十八度四十五分，三十年六宮二十二度五十八分。 閏日，加三分。
Phased zero-year ready reckoner. Mean motion: per year 11 signs 14°27′; per 30 years 1 sign 8°15′. Eleven intercalary days every 30 years—the same rule as the solar zero-year table. The entries below follow the same rule. Doubled-separation [the cited text for the cited text]: per year 11 signs 21°03′; in 30 years 11 signs 29°40′. Intercalary day: add 24°23′. Deferent rotation: per year 10 signs 5°; in 30 years 9 signs 23°47′. Intercalary day: add 13°04′. Rahu motion: per year 18°45′; in 30 years 6 signs 22°58′. Intercalary day: add 3 parts.
45
總年立成。 中心行度，第一年四宮二十八度四十九分，六百年六宮八度四十二分，每三十加一宮八度十五分，一千四百四十年，五宮二十九度四十七分。 倍離度，第一年，一宮二十五度二十八分，六百年一宮十八度三十三分。 每三十年加十一宮二直九度四十分，一千四百四十年，一宮九度二十一分。 本輪行度，第一年，四宮十二度二分，六百年八宮八度分。 每三十年加二十三度六分，六百年十一宮十度三十四分。 每三十年加六宮二十二度五十八分，一千寂百四十年，八宮十五度五十分。
Total-year ready reckoner. Mean motion: year 1, 4 signs 28°49′; at 600 years, 6 signs 8°42′; every 30 years add 1 sign 8°15′; at 1,440 years, 5 signs 29°47′. Doubled-separation: year 1, 1 sign 25°28′; at 600 years, 1 sign 18°33′. Every thirty years add eleven palaces two [zhi] nine degrees forty parts; one thousand four hundred forty years one palace nine degrees twenty-one parts. Deferent rotation: year 1, 4 signs 12°02′; at 600 years, 8 signs 8°00′ [fraction omitted in text]. Every 30 years add 23°06′; at 600 years, 11 signs 10°34′. Every thirty years add six palaces twenty-two degrees fifty-eight parts; one thousand [ji] hundred forty years eight palaces fifteen degrees fifty parts.
46
▲總零年宮月日七曜立成造法
▲ Construction of the total-, zero-year, sign-, month-, day-, and seven-luminaries ready reckoners
47
總年立成，第一年起金六，六百年起日一，每三十年加寺數。 零年立成，起水四。 宮分立成，金牛宮起火三。 月分立成，起月二。 日期立成，起日一。 求法：有閏日，滿歲七曜。 不滿歲，用月七曜。 並之，得逐月末日七曜。
Total-year ready reckoner: first year starts Metal six; six hundred years starts Sun one; every thirty years add [si] numbers. Zero-year table starts at Water 4. Sign-division table: Taurus starts at Fire 3. Month-division table starts at Moon 2. Date-division table starts at Sun 1. Procedure: with an intercalary day, use the yearly weekday number. If the year is incomplete, use the monthly weekday number. Combine them to obtain the weekday for each month's final day.
48
太陽加減立成自行宮度爲引數。 原本宮縱列首行，度橫列上行，每三宮順布三十度，內列加減差，又列加減分。 其加減分，乃本度加減差與次度加減差之較也。 今去之，止列加減差數，將引數宮列上橫行，度列首直行，用順逆查之，得數無異，而簡捷過之。 月、五星加減立成，准此。
Solar equation ready reckoner: proper-motion sign-degree is the argument. Originally signs ran down the first column and degrees across the top row; every three signs, 30° were set in sequence, with equation corrections listed within, and correction differences beside them. The correction-difference column records the interval between this degree's equation correction and the next degree's correction. That column is now omitted; only equation corrections remain. Place the argument sign on the top row and degree on the first column, and look up directly or in reverse—the result is unchanged, but the table is simpler. Lunar and five-planet equation ready reckoners follow the same rule.