1
志第十三歷七
Treatise Thirteen: Calendrics, Part Seven.
2
▲回回曆法一
▲ The Islamic (Huihui) Calendar Method, Part One
3
《回回曆法》，西域默狄納國王馬哈麻所作。 其地北極高二十四度半，經度偏西一百零七度，約在雲南之西八千餘昊。 其曆元用隋開皇己未，即其建國之年也。 洪武初，得其書於元都。 十五年秋，太祖謂西域推測天象最精，其五星緯度又中國所無。 命翰林李翀、吳伯宗同回回大師馬沙亦黑等譯其書。 其法不用閏月，以三百六十五日爲一歲。 歲十二宮，宮有閏日，凡百二十八年而宮閏三十一日。 以三百五十四日爲一週，週一十十月有閏日。 凡有閏閏凡百二十八年而而宮閏三十一日，以三百五十四日爲一週，周十二月，月有閏日。 凡三十年月閏十一日，歷千九百四十一年，宮月日辰再會。 此其立法之大概也。
The Islamic Calendar Treatise was composed by Muhammad, king of Medina in the Western Regions. Its latitude is 24.5°; longitude lies 107° west [of the Chinese standard], roughly 8,000+ li west of Yunnan. Its epoch is Sui Kaihuang jiwei (599 CE)—the year of the kingdom's founding. Early in the Hongwu era the treatise was recovered at the Yuan capital (Dadu). In autumn of Hongwu 15 the Founder observed that Western astronomical prediction is unrivaled in precision, and that planetary latitudes in their system were unknown in Chinese astronomy. He ordered Hanlin academicians Li Chong and Wu Bozong, together with the Muslim astronomer Jamal al-Din (Ma Sha Yih Hei) and others, to translate the work. The system has no intercalary months; the solar year is fixed at 365 days. The year comprises twelve zodiac signs; signs receive leap days—31 leap days in every 128-year cycle. The lunar cycle is 354 days; each twelve-month cycle admits intercalary days in its months. Over 128 years the solar cycle gains 31 leap days; the lunar cycle is 354 days in twelve months, with intercalary days inserted in the months. Eleven intercalary days are added to the months every 30 years; after 1,941 years the solar sign, lunar month, day, and hour realign. Such are the broad principles of the system's design.
4
按西域歷術見於史者，在唐有《九執曆》，元有札馬魯丁之《萬年曆》。 《九執因》最疏，《萬年曆》行之未久。 唯《回回曆》設科，隸欽天監，與《大統》參用二百七十餘年。 雖於交食之有無深淺，時有出入，然勝於《九執》、《萬年》遠矣。 但其書多脫誤。 盜蓋其人之隸籍臺官者，類以土盤布算，仍用其本國之書。 而明之習其術者，如唐順之、陳壤、袁黃輩之所論著又自成一家言。 以故翻譯之本不行於世，其殘缺宜也。 今爲博訪專門之裔，考究其原書，以補其脫落，正其訛舛，爲《回回曆尖》，著於篇。
Western calendrical systems recorded in history include the Tang Nine Offices Calendar and Jamal al-Din's Eternal Calendar of the Yuan. The Nine Offices Calendar was the crudest; the Eternal Calendar was employed only briefly. Only the Islamic calendar was instituted as a formal curriculum at the Imperial Observatory and used alongside the Datong calendar for more than 270 years. Predictions of eclipse magnitude and visibility still varied at times, but the system far surpassed the Nine Offices and Eternal calendars. Yet the transmitted text is full of lacunae and errors. Presumably because observatory staff of that tradition relied on sand-board calculation and continued to use texts in their native script. Meanwhile Ming scholars of the tradition—Tang Shunzhi, Chen Rang, Yuan Huang, and others—produced treatises advancing independent interpretations. Hence the translated edition never gained currency; its incompleteness is understandable. We have now sought out specialists in the tradition, collated the original sources, filled lacunae and corrected errors, and set down the Islamic Calendar Method in this chapter.
5
積年起西域阿喇必年，隋開皇己未。 下至洪武甲子，七百八十六年。
Accumulated years: from the Western Hijra era through Sui Kaihuang jiwei. Down to Hongwu jiazi (1384 CE)—786 years in total.
6
用數天周度三百六十。 每度六十分，每分六十秒，微纖以下俱准此。 宮十二。 每宮三十度。 目周分一千四百四十，時二十四，每時六十分。 刻九十六。 每刻十五分。 宮度起白羊，節氣首春分，命時起午正。 午初四刻屬前日。
Constants: celestial revolution = 360°. Each degree = 60 minutes; each minute = 60 seconds; smaller units follow the same ratio. Twelve zodiac signs. Each sign spans 30°. Diurnal arc = 1,440 parts; 24 hours per day, 60 parts per hour. Ninety-six quarter-hours (ke) per day. Each quarter-hour = 15 parts. Sign degrees begin at Aries; solar terms begin at the vernal equinox; clock hours begin at true noon. The first four ke after noon belong to the previous civil day.
7
七曜數日一，月二，火三，水四，木五，金六，土七。 以七曜紀不用甲子。
Seven luminaries numbering: Sun 1, Moon 2, Mars 3, Mercury 4, Jupiter 5, Venus 6, Saturn 7. Dates are reckoned by the seven-day planetary week, not the sexagenary cycle.
8
宮數白羊初，金牛一，陰陽二，世蟹三，獅子四，變女五，天秤六，天蠍七，人馬八，磨羯九，實寶瓶十，變魚十一。
Sign indices: Aries 0, Taurus 1, Gemini 2, Cancer 3, Leo 4, Virgo 5, Libra 6, Scorpio 7, Sagittarius 8, Capricorn 9, Aquarius 10, Pisces 11.
9
宮日白羊戌宮三十一日。 金牛酉宮三十一日。 陰陽申宮三十一日。 巨蟹未宮三十二日。 獅子午宮三十一日。 孌女巳宮三十一日。 天秤辰宮三十一日。 天蠍卯宮三十日。 人馬寅宮二十九日。 磨羯醜宮二十九日。 寶瓶子宮三十日。 變魚亥宮三十日。 已上十二宮，所謂不動之月，凡三百六十五日，乃歲周之日也。 若遇宮分有閏之年，於變魚宮加一日，凡三百六十六日。
Sign days—Aries (xu palace): 31 days. Taurus (you palace): 31 days. Gemini (shen palace): 31 days. Cancer (wei palace): 32 days. Leo (zi/wu palace): 31 days. Virgo (si palace): 31 days. Libra (chen palace): 31 days. Scorpio (mao palace): 30 days. Sagittarius (yin palace): 29 days. Capricorn (chou palace): 29 days. Aquarius (zi palace): 30 days. Pisces (hai palace): 30 days. The twelve signs above constitute the "fixed months" of the solar year, totaling 365 days—the days of the annual circuit. In a year with a solar leap day, one day is added in Pisces, for 366 days total.
10
月分大小單月大，變月小。 凡十二月，所謂動之月也。 月大三十日，月小二十九日，凡三百五十四日，乃十二月之日也。 遇月分有閏之處，於第十二月內增一日，凡三百五十五日。
Lunar month lengths: odd months long, even months short. These twelve constitute the "moving months" of the lunar year. Long months have 30 days and short months 29, totaling 354 days for the twelve lunar months. When a lunar leap day is required, one day is added within the twelfth month, yielding 355 days.
11
太陽五星最高行度隋己未測定。 太陽二宮二十九度二十一分。 土星八宮十四度四十八分。 木星六宮初度八分。 火星四宮十五度四分。 金星二宮十七度六分。 水星七宮六度十七分。
Solar and planetary apogee longitudes, fixed by observation in Sui jiwei (599 CE). Sun: sign 2, 29°21′. Saturn: sign 8, 14°48′. Jupiter: sign 6, 0°08′. Mars: sign 4, 15°04′. Venus: sign 2, 17°06′. Mercury: sign 7, 6°17′.
12
求宮分閏日無之餘日。 置西域歲前積年，減一，以一百五十九乘之，一百二十八年內，閏三十一日故以總數乘。 內加一十五，閏應。 以一百二十八屢減之，餘不滿之數，若在九十七已上，閏限。 其年宮分有閏日，已下無閏日。 於除得之數內加五，宮分立成起火三，故須加五。 滿七去之，餘即所求年白羊宮一日七曜。 有閏加一日，後同。
Procedure to determine whether the year has a solar leap day, and the weekday of the first day if there is none. Set the accumulated years before the current Western year, subtract 1, and multiply by 159 (the total leap days per 128-year solar cycle). Add 15 inside for the intercalary adjustment. Repeatedly subtract 128; if the remainder is 97 or greater, there is a solar leap day that year. If so, the year has a solar leap day; otherwise it does not. To the quotient add 5—the sign tables begin with Mars (no. 3), hence the increment. Cast out complete sevens; the remainder is the weekday of the first day of Aries for the sought year. If there is a leap day, add one day; subsequent procedures are the same.
13
求月分閏日朔之餘日。 置西域歲前積年，減一，以一百三十一年乘之，總數乘。 內加一百九十四，閏應。 以三十爲法屢減之，餘在十九已上，閏限。 其年月分有閏閏已下則無。 於除得之數，滿七去之，餘即所求年第一月一日七曜。
Procedure to find lunar leap days and the weekday of the first day of the first month. Set the accumulated years before the current Western year, subtract 1, and multiply by 131. Add 194 inside for the intercalary adjustment. Repeatedly subtract 30; if the remainder is 19 or greater, there is a lunar leap day that year. If so, the year has a lunar leap day; otherwise it does not. Cast out complete sevens from the quotient; the remainder is the weekday of the first day of the first month.
14
加次法置積日，全積並宮閏所得數。 減月閏內加三百三十一日，己未春正前日。 以三百五十四一年數除之，餘數內減去所加三百三十一，又減二十三，足成一年日數。 又減二十四，洪武甲子加次。 又減一，改應所損之一日。 爲實距年己未至今得數。 又法：以氣積宮閏並通閏爲氣積內減月閏，置十一，以距年乘之，外加十四，以三十除之，得月閏數。 以三百五十四除之，餘減洪武加次二十四，又減補日二十三，又減改應損日一，得數如前。 求通閏，置十一日，以距年乘之。 求宮閏前見。
Epoch correction: set accumulated days, combining the full accumulation with the solar leap-day count. Subtract the lunar leap count; add 331 days inside (the day before jiwei spring first). Divide by 354; from the remainder subtract the 331 added, then subtract 23 to complete one year's day count. Subtract 24 again for the Hongwu jiazi epoch correction. Subtract 1 again for the one-day revision adjustment. This yields the true interval in days from the jiwei epoch to the present. Alternate method: combine solar leap accumulation with general intercalation as the solar total; subtract lunar leap; set 11, multiply by years from epoch, add 14 outside, divide by 30 to obtain the lunar leap count. Divide by 354; from the remainder subtract the Hongwu correction (24), the supplement day (23), and the revision day (1)—as before. To find general intercalation: set 11 days and multiply by years from epoch. For solar leap days, see the procedure above.
15
▲太陽行度
▲ Solar Motion in Degrees
16
求最高總度置西域歲前積年，入總年零年月分日期立成內，各取前年前月前日最高行度並之。 如求十年，則取九年之類。 蓋立成中行度，俱本年本月日足數也。 如十年竟求十年，則逾數矣。 月日義同。 後仿此。
Procedure for total apogee longitude: set accumulated years before the current Western year; enter the lookup table for whole years with zero month and day; for each component take the prior year, prior month, and prior day apogee longitudes and sum them. To find year 10, take the entry for year 9, and so on. Table entries give complete values for the current year, month, and day. Using the year-10 entry when computing through year 10 would overshoot. The same rule applies to months and days. Subsequent procedures follow this pattern.
17
求最高行度置求到最高總度，加測定太陽最高行度，二宮二十九度二十一分。 即年求年白羊宮最高行度。 如求次宮，累加五秒零六微。 求次月，加四秒五十六微。
Procedure for apogee longitude: set the total apogee longitude obtained and add the fixed solar apogee at epoch—sign 2, 29°21′. This is the solar apogee longitude in Aries for the sought year. For the next sign, add 5′06″ cumulatively. For the next month, add 4′56″.
18
求中心行度日平行度。 置積年入總年零年月日立成內，各取日中心行度並之，取法同前。 內減一分四秒，即所求白羊宮第一日中心行度。 求各宮月日，按每日行度五十九分八秒累加之。 內減一分四秒，或云西域中國裏差，非是，蓋系己未年之末日度應也。
Procedure for mean longitude and daily mean motion. Set accumulated years; enter the lookup table for whole years with zero month and day; take each day's mean longitude and sum them—same retrieval method as above. Subtract 1′04″ inside; this is the mean longitude for the first day of Aries in the sought year. To find each sign's month-days: accumulate the daily motion of 59 parts 8 seconds day by day. Subtract 1 part 4 seconds within; some call this the longitudinal difference between the Western Regions and China, but that is wrong—it is the adjustment for the jìwèi epoch's year-end degree.
19
求自行度置其日中心行度，減其宮最高行度，即得。 即入盈縮歷度也。
To find the proper-motion degree: set that day's mean longitude and subtract that sign's apogee longitude. The result is the degree entered into the expansion–contraction sequence.
20
求加減差。 即盈縮差。 以自行宮度爲引數，入太陽加減立成內，照引數宮度取加減差。 是名未定差。 其度下小餘，用比例法，以本加減差，與後度加減差相減，餘數通爲秒，如一分通爲六十秒。 與引數小餘亦通秒相乘，得數爲纖，秒乘秒，得纖。 以六十收之，爲微、爲秒、爲分。 如數多，先以六十收之爲微，又以六十收之爲秒，又以六十收之爲分。 視前所得未定加減差數較，少於後數者後度加減差加之，多於後數者減之，是爲加減定差分。 如無小餘，竟用未定差爲定差。 後准此。
To find the equation correction. This is the expansion–contraction difference. Take the proper-motion sign-degree as the argument, enter the solar equation ready reckoner, and take the correction for that sign-degree. This is called the undetermined correction. For the fractional remainder below that degree, apply proportional interpolation: subtract this degree's correction from the next degree's correction, convert the remainder to seconds (one part equals sixty seconds), convert the argument's fractional remainder to seconds likewise and multiply; the product is in fibre units—seconds times seconds yield fibre. Collect by sixty into minute-fractions, seconds, and parts. If the amount is large, first collect by sixty into minute-fractions, then again into seconds, then again into parts. Compare the undetermined correction obtained above with the next value: if smaller, add the next degree's correction; if larger, subtract—this yields the fixed equation correction in parts. If there is no fractional remainder, use the undetermined correction directly as the fixed correction. Later cases follow the same rule.
21
求經度黃道度。 置其日中心行度，以加減定差分加減之，視定差引數自行宮度，在初宮至五宮爲減差，六宮至十一宮爲加差。 即得。
To find the ecliptic longitude. Set that day's mean longitude and add or subtract the fixed equation correction in parts; where the argument's proper-motion sign lies in signs 1–5 the correction is subtractive, in signs 6–11 additive. The result is the ecliptic longitude.
22
求七曜置積年入立成內，取總年零年月日下七曜數並之，累去七數，餘即所求白羊宮一日七曜。 如求次宮者，內加各宮七曜數。 如求逐日，累加一數，滿七去之。 求太陰、五星、羅計七曜並准此。
To find the planetary weekday: set the accumulated years, enter the ready reckoner, combine the weekday numbers under total years, zero years, month, and day, repeatedly discard sevens, and the remainder is the weekday for Aries day one. For the next sign, add that sign's weekday number within. For successive days, add one each day and discard sevens when full. The Moon, the five planets, and Rahu–Ketu weekdays all follow this rule.
23
▲太陰行度
▲ Lunar motion
24
求中心行度置積年入立成內，取總零年月日下中心行度並之，得數，內減一十四分，己未應轉。 即所求年白羊宮一日中心行度。 如求逐日，累加日行度。 十三度一零三五。
To find mean longitude: set the accumulated years, enter the ready reckoner, combine the mean longitudes under total, zero years, month, and day, and subtract 14 parts within for the jìwèi epoch adjustment. This is the mean longitude for Aries day one of the year sought. For successive days, accumulate the daily motion. Daily motion: 13°10′35″.
25
求加倍相離度月體在小輪行度，合朔後，與日相離。 置積年入立成內，取總年零年月日下加倍相離度並之，內減二十六分，即所求白羊宮一日度也。 如逐日，累加倍離日行度。 二十四度二二五三二二，半之，即小輪心離太陽數。
To find the doubled-separation degree—the Moon's motion on the epicycle, its distance from the Sun after conjunction. Set the accumulated years, enter the ready reckoner, combine the doubled-separation under total years, zero years, month, and day, subtract 26 parts within, and obtain Aries day one. For successive days, accumulate the doubled-separation daily motion. 24°22′53″22‴; halve this for the epicycle center's distance from the Sun.
26
求本輪行度即月轉度。 置積年入立成內，取總零年月日下本輪行度並之，內減一四分，即所求白羊宮一日度也。 如求各日，累加本輪日行度。 十三度三分五四。
To find deferent motion—that is the lunar rotation degree. Set the accumulated years, enter the ready reckoner, combine deferent motion under total, zero years, month, and day, subtract 14 parts within, and obtain Aries day one. For each day sought, accumulate the deferent daily motion. Daily motion: 13°3′54″.
27
求第一加減差又名倍離差。 以加倍相離宮度爲引數，入太陰第一加減立成內，取加減差。 未定差。 又與下差相減，餘乘引數小余，得數爲秒，分乘分以六十收之爲分，用加減未定差，後差多加少減，同太陽。 得第一分差。
To find the first equation correction, also called the doubled-separation correction. Take the doubled-separation sign-degree as the argument, enter the lunar first-equation ready reckoner, and take the correction. This is the undetermined correction. Subtract from the next correction, multiply the remainder by the argument's fractional remainder for seconds, collect parts-times-parts by sixty into parts, and apply to the undetermined correction—add when the next value is larger, subtract when smaller, as for the Sun. The result is the first correction in parts.
28
求本輪行度置其日本輪行度，以第一差分加減之。 視倍離度，前六宮加，後六宮減。
To find deferent motion: set that day's deferent motion and add or subtract the first correction in parts. By the doubled-separation degree: add in the first six signs, subtract in the last six.
29
求第二加減差以本輪行定度度爲引數，入太陰第二加減立成內，取未定差，依比例法，同前。 求得零數加減之爲第二加減差分。 視引數，六宮已前爲減差，後爲加差。
To find the second equation correction: take the deferent fixed motion as the argument, enter the lunar second-equation ready reckoner, take the undetermined correction, and interpolate proportionally as before. Obtain the fractional part and add or subtract it as the second correction in parts. By the argument: before sign 6 the correction is subtractive, after sign 6 additive.
30
求比數分以們離宮度，入第一加減立成內，取比數分。 如們離零分在三十分已上者，取下度比敷分。
To find the ratio parts: by the doubled-separation sign-degree, enter the first-equation ready reckoner and take the ratio parts. If the doubled-separation fractional part is 30 parts or more, take the next degree's ratio parts.
31
求遠近度以本輪行定宮度爲引敷，入陰第二加減立成內，取遠近度分。 其引數零分，亦依比例法取之。
To find the proximity correction: by the deferent fixed sign-degree as argument, enter the lunar second-equation ready reckoner and take the proximity parts. The argument's fractional part is also obtained by proportional interpolation.
32
求凡差定差置比敷分，以遠近度通分乘之，以六十約之爲分，即凡差。 以凡差加入第二加減差，即爲定差。
To find the general and fixed corrections: set the ratio parts, multiply by the proximity degree in common parts, and reduce by sixty to parts—this is the general correction. Add the general correction to the second correction to obtain the fixed correction.
33
求經度置其日太陰中心行度，以定差加減之，即太陰經度。 視本輪行定前減，以後加。
To find longitude: set that day's lunar mean longitude, add or subtract the fixed correction, and obtain the lunar longitude. By the deferent fixed motion: subtract before, add after.
34
▲太陰緯度
▲ Lunar latitude
35
求計都與月相離度入交定度。 置其日太限經度，內減其日計都行度，即計都與月相離度分。
To find the Rahu–Moon separation and nodal fixed degree. Set that day's lunar longitude and subtract that day's Rahu motion within to obtain the Rahu–Moon separation in parts.
36
求緯以計都與月相離宮度爲引數，入太陰緯度立成，上宮用右行順度，下宮用左行逆度。 取其度分，依比例法求得零分加減之，上六宮加，下六宮減。 得緯度分。 引數在六宮已前爲黃道北，六宮後爲黃道南。
To find latitude: take the Rahu–Moon separation sign-degree as the argument, enter the lunar-latitude ready reckoner—upper signs use right-motion direct degrees, lower signs left-motion retrograde degrees. Take the degree and parts, interpolate the fractional part proportionally, and add in the upper six signs or subtract in the lower six. The result is the latitude in parts. Arguments before sign 6 yield north latitude; after sign 6, south latitude.
37
求計羅行度置積年入總年零年月日立成內，取羅計中心行度並之，爲其年白羊宮一日行度。 求各宮一日，以各宮日行度加之，與十二宮相減，餘即所求宮一日計都行度。 如求計都逐日細行，以前後二段行度相減，餘以相距日數除之，爲日差。 又置前段計都行度，以日差累減之。 如求羅喉行度，置其日計都行度內。
To find Rahu's motion: set the accumulated years in the ready reckoner for total, zero years, month, and day, combine the Rahu mean motions, and obtain Aries day-one motion for that year. For each sign's first day: add that sign's daily motion and subtract from twelve signs; the remainder is Rahu's motion for that sign's day one. For Rahu's day-by-day fine motion: subtract the prior and following segment motions and divide the remainder by the intervening days to obtain the daily increment. Set the prior segment's Rahu motion and accumulate, subtracting the daily increment each day. For Ketu's motion: set it within that day's Rahu motion (opposite).
38
▲五星經度
▲ Longitude of the five planets
39
求最高總度數同太陽，依前太陽術求之。
To find apogee total degree: the reckoning matches the Sun; apply the solar procedure above.
40
求最高行度置所求本星最高總度，加測定本星最高行度，見前。 爲其年白羊最高行度。 求扣宮各日，加各宮日行度。
To find apogee motion: set the planet's apogee total degree, add the observed fixed apogee motion (see above), and obtain that year's Aries apogee motion. This is the apogee motion for Aries day one of the year sought. For each sign and day, add that sign's daily motion.
41
求日中心行度依太陽術求之。
To find mean longitude: follow the solar procedure.
42
求自行度置積年入立成總零年月日下，各取自行度並之，得其年白羊宮一日自行度。 土、木、金三星減一分，水星減三分，火星不減。 如求各宮各日，照本星自行度累加之。 水星如自行度遇三宮初度，作五日一段算，至九宮初度，作十日一段算緯度亦然。
To find proper motion: set accumulated years in the ready reckoner, combine each planet's proper motion under total, zero years, month, and day, and obtain Aries day-one proper motion for that year. Saturn, Jupiter, and Venus subtract 1 part; Mercury subtracts 3 parts; Mars subtracts none. For each sign and day, accumulate that planet's proper motion. For Mercury: from sign 3's initial degree compute in five-day segments; from sign 9's initial degree in ten-day segments—the same for latitude.
43
求中心行度中輪心度即入歷度五星本輪。 土、木、火三星，置太陽中心行度，減其星自行度，爲三星中心行度。 內又減最高行度，爲三星小輪心度。 金、水二星，其中心行度即太陽中心行度，內減其星最高行度，餘爲其星小輪心度。 不及減，加十二宮減之。
To find mean longitude and epicycle-center degree—that is the five planets' deferent entry into the sequence. Saturn, Jupiter, and Mars: set the Sun's mean longitude and subtract the planet's proper motion to obtain mean longitude. Subtract apogee motion within as well to obtain the epicycle-center degree. Venus and Mercury: mean longitude equals the Sun's mean longitude; subtract the planet's apogee within—the remainder is the epicycle-center degree. If subtraction fails, add twelve signs and subtract.
44
求第一加減差盈縮差。 以其星小輪心宮度爲引數，入本星第一加減立成，依比例法求之。 法同太陽、太陰。
To find the first equation correction—the expansion–contraction difference. Take the planet's epicycle-center sign-degree as the argument, enter its first-equation ready reckoner, and interpolate proportionally. The method matches the Sun and Moon.
45
求自行定度及小輪心定度視第一加減差引數，在初宮至五宮，用加減差，加自行度，減小輪心度，各爲定度。 在六宮至一宮，用加減差，減自行度，加小輪心度，各爲定度。
To find fixed proper motion and fixed epicycle center: where the first correction's argument lies in signs 1–5, add the correction to proper motion and subtract from the epicycle center—each yields a fixed degree. In signs 6–11, subtract the correction from proper motion and add to the epicycle center—each yields a fixed degree.
46
求第二加減差以其星自行定度，入本第二加減立成內，取其度分，用比例法加減之。 同前。
To find the second equation correction: by the planet's fixed proper motion, enter its second-equation ready reckoner, take degree and parts, and add or subtract proportionally. Same as before.
47
求比敷分如土、木、金、水星，以本星小輪心一宮度，入第一加減立成內，取比敷分，如引數小餘在三十分已上，取手行經敷分。 如火星，則必用比例法求之。
To find ratio parts: for Saturn, Jupiter, Venus, and Mercury, by the epicycle center's sign-degree enter the first-equation ready reckoner; if the argument's fractional part is 30 parts or more, take the next degree's ratio parts. For Mars, the proportional method must be used throughout.
48
求遠近度以自行定宮度，入第二加減立成內，取遠近度，依比例法求之。
To find proximity correction: by fixed proper-motion sign-degree, enter the second-equation ready reckoner, take the proximity value, and interpolate proportionally.
49
求凡差定差法同太陰。
To find general and fixed corrections: same method as the Moon.
50
求經度置小輪心定度，以定差加減之，視引數自行定度，在六宮已前加，已後減。 內加其星最高行度。
To find longitude: set the fixed epicycle-center degree, add or subtract the fixed correction—add before sign 6 by the fixed proper-motion argument, subtract after. Add the planet's apogee motion within.
51
求留段以其段小輪心，定宮諜爲引數，即立成內各星入歷定限。 入五星順退留立成內，於同宮近度，取本星度分，與前後行查減。 若取得在初宮至六宮，本行與後行相減。 六宮至初宮，本行與與前行相減。 又以引數宮度，減立成內同宮近度，兩減，餘通分相乘，用六度除之，立成內每隔六度。 六十分收之，順加逆減於前取度分，得數與其日自行定度同者，即本日留。 如自行定度多者已過留日，少者未到留日。 欲得細率，以所得數與其人日自行定度相減，餘以各星一日自行度約之，如土星一日自行五下七分有奇之類。 即得留日在本日前後數也。 土星留七日，其留日前三日，後三日，皆與留日數同。 木星留五日，其留日前二日，後二二與留日數同。 火、金、水三星不留，退而即退，但於行分極處留耳。
To find the station interval: take that segment's fixed epicycle-center sign-degree as the argument and enter each planet's sequence fixed limit in the ready reckoner. Enter the five-planet direct, retrograde, and station ready reckoner; at the nearest degree in the same sign take the planet's degree-parts and subtract from the prior and following motions. If the result falls in signs 1–6, subtract the following motion from this motion. From sign 6 back to sign 1, subtract the prior motion from this motion. Subtract the ready reckoner's nearest degree in the same sign from the argument sign-degree; convert both remainders to common parts, multiply, and divide by six (the table's six-degree interval). Collect by sixty parts; add for direct motion or subtract for retrograde from the prior degree-parts; when the result equals that day's fixed proper motion, that day is a station. If fixed proper motion is greater, the station day has passed; if less, it has not yet arrived. To obtain the fine ratio, subtract the value found from that day's proper-motion determined degree and divide the remainder by the planet's daily proper motion—for example, Saturn's daily proper motion of about 5°7′. The result is how many days before or after station fall on the current day. Saturn stations for seven days; the three days before and after the station day match the station count. Jupiter stations for five days; the two days before and after match the station count. Mars, Venus, and Mercury do not truly station; once retrograde begins they continue retrograde, pausing only at the extremity of their motion parts.
52
求細行分土、木、金、火四星，以前後兩段經度相減，以相距除之爲日行分。 水星以白羊宮初日經度，又與前一日經度相減，餘爲初日行分。 又置前後二段經度相減，餘以相距日除之，爲平行分。 與初日行分加減，倍之，以前段前一日與後段相距日數除之之爲日差。 以加減初日行分，初日行分少於平行分加，多減。 爲日行分。 五星各置前段經度，以逐日行分順加退減之，爲各星逐日經度。
Procedure to derive fine motion parts: for Saturn, Jupiter, Venus, and Mars, subtract prior and subsequent segment longitudes and divide by the interval to obtain daily motion parts. For Mercury, subtract the prior day's longitude from the Aries-first-day longitude; the remainder is the initial day's motion parts. Again subtract prior and subsequent segment longitudes and divide by the interval to obtain parallel motion parts. Add or subtract the initial day's motion parts, double the result, and divide by the days from the day before the prior segment to the subsequent interval to obtain the daily difference. Add or subtract from the initial day's motion parts: if they are less than the parallel motion parts, add; if greater, subtract. The result is the daily motion parts. For each planet, set the prior segment's longitude and add or subtract daily motion parts forward or backward to obtain day-by-day longitudes.
53
求伏見視各星自行定度，在伏見立成內限度已上者，即五星晨夕伏見也。
Procedure to derive concealment and visibility: if a planet's proper-motion determined degree lies above the limit in the concealment-and-visibility ready reckoner, that marks morning or evening concealment or visibility.
54
五星緯度求最高總行度、中心行度、自行度、小輪度，並依五星比經度術求之。
Procedure to derive the five planets' latitudes: obtain apogee total travel, center travel, proper-motion, and epicycle degrees by the five-planet ratio-longitude method.
55
求自行定度置自行宮度分，其宮以一十乘爲度。 如一宮，以十乘之得十度，此用約法折算，以造緯度立成。 其度以二十乘之爲分，滿六十約之爲度。 其分亦以二十乘之爲秒，滿六十約之屬分。 並之即得。
Procedure to derive the proper-motion determined degree: set proper-motion mansion degrees and minutes and multiply the mansion by ten for degrees. One mansion times ten yields ten degrees; this rounding shortcut builds the latitude ready reckoner. Multiply degrees by twenty for minutes and, when full, reduce by sixty to degrees. Multiply minutes by twenty for seconds and, when full, reduce by sixty to minutes. Combine them to obtain the value.
56
求小輪心定度置小輪心宮度分，其宮以五乘之爲度。 如一宮以五乘之，得五度。 其度以一十乘之爲分，滿六十約之迷度。 其分亦以一十乘之爲秒，滿六十約之爲分。 並之即得。
Procedure to derive the epicycle-center determined degree: set epicycle-center mansion degrees and minutes and multiply the mansion by five for degrees. One mansion times five yields five degrees. Multiply degrees by ten for minutes and, when full, reduce by sixty to degrees. Multiply minutes by ten for seconds and, when full, reduce by sixty to minutes. Combine them to obtain the value.
57
求緯度以小輪心定度及自行度，入本星緯度立成內兩取，一縱一橫。 得數與後行相減。 若遇交黃道者，與後行相併。 又以小輪心定立成上小輪心定相減，上橫行。 兩減餘相乘，以立成上小輪心度累加數除之。 如土星上橫行小輪心度每隔三度，火星每隔二度之類。 滿六十收之爲分，用加減兩取數，多於後行減，少加。 若遇交黃道者，即後行數多亦減。 寄左。 復以自行定度與立成上自行定度相減，首直行。 又以兩取數，與下行相減，若遇交黃道埏，與下行並。 兩減餘相乘，以立成上自行度累加數除之，如土星直行，自行度每隔十度，火星每隔四度之類。 收之爲分。 與前寄左數相加減，如兩取數多於下行者減，少加。 若遇交黃者，所得分多於寄左數，置所得分內，減寄左數，餘爲交過黃道南北分也。 即得黃道南北緯定分。
Procedure to derive latitude: with epicycle-center and proper-motion determined degrees, enter the planet's latitude ready reckoner and take one vertical and one horizontal value. Subtract the value taken from the lower row. If the planet crosses the ecliptic, add to the lower row instead. Again subtract the upper epicycle-center determined value on the ready reckoner from the epicycle-center determined value in the upper horizontal row. Multiply the two remainders and divide by the epicycle-center cumulative increment on the ready reckoner. For example, Saturn's upper horizontal row steps epicycle-center degree every three degrees and Mars every two. Reduce by sixty to minutes and add or subtract the two values: if greater than the lower row, subtract; if less, add. At ecliptic crossing, subtract even when the lower-row value is larger. Set aside on the left. Again subtract the ready-reckoner proper-motion determined degree in the first vertical row from the proper-motion determined degree. With the two values again, subtract from the lower row; at ecliptic crossing, add to the lower row. Multiply the two remainders and divide by the proper-motion cumulative increment—as when Saturn's vertical row steps every ten degrees and Mars every four. Reduce to minutes. Add or subtract the left-stored value: if the two values exceed the lower row, subtract; if less, add. At ecliptic crossing, if the new minutes exceed the left-stored amount, subtract the left-stored amount from the result; the remainder is the arc north or south of the ecliptic crossed. The result is the fixed ecliptic latitude north or south.
58
求緯度細行分置其星前段緯度，與後段緯度相減，餘以相距日除之，爲日差。 置前段緯度，以日差順加退減，即逐日緯度分。 按緯度前段少於後段者，以日差順加退減。 若前段多於後段者，宜以日差順減退加。 非可一例也。 若前後段南北不同者，置其星前後段緯度並之，以相距日除之，爲日差。 置前段緯度，以日差累減之，至不及減者，於日差內減之，餘以日差累加之，即得逐日緯度。
Procedure to derive fine latitude daily motion: subtract prior-segment from subsequent-segment latitude and divide by the interval for the daily difference. Set the prior-segment latitude and add or subtract the daily difference forward or backward for day-by-day latitude. When the prior latitude is less than the subsequent, add forward and subtract backward. When the prior exceeds the subsequent, subtract forward and add backward. The two cases cannot share one rule. When north and south differ between segments, add prior and subsequent latitudes and divide by the interval for the daily difference. Set the prior latitude and subtract the daily difference repeatedly; when subtraction fails, take the remainder from the daily difference and add forward to obtain day-by-day latitude.
59
推日食法日食諸數，如午前合朔，用前一日數推，午後合朔，用次日數推。
Procedure to derive solar eclipses: for eclipse parameters, use the prior day's values for pre-noon conjunction and the next day's for post-noon conjunction.
60
辨日食限視合朔太陰緯度，在黃道南四十五分已下，黃道北九十分已下，爲人食。 若合朔爲盡，則全見食。 若膈朔在日未出三時及日已入十五分，一時四分之一。 皆有帶食。 若合朔在夜刻者不算。
Discern eclipse limits: at conjunction, if lunar latitude is within 45′ south or 90′ north of the ecliptic, the eclipse is visible on Earth. If conjunction is total, the eclipse is seen in full. If the obscured new moon falls three watches before sunrise or a watch and a quarter (15 minutes) after sunset, there is a partial eclipse in every case. If conjunction falls in the night watches, do not compute it.
61
求食甚凡時即合朔。 置午正太陰行過太陽度，求法見後月食太陰逐時行過太陽分。 通秒，以二十四乘之爲實，置太陰日行度，減太陽日行度，通秒爲法，除之爲時。 時下零數以六下通之爲分，分下零數以六十通之爲秒，三十秒已上收爲一分，六十分收爲一時，共爲食甚凡時。
Derive the greatest-eclipse general time, which is conjunction. Set the Moon's longitude past the Sun at noon; see the lunar-eclipse procedure for the Moon's hourly motion past the Sun. Convert to seconds, multiply by 24 for the dividend, subtract solar from lunar daily motion (also in seconds) for the divisor, and divide to obtain hours. Convert the remainder below the hour to minutes and below minutes to seconds; carry at 30″ to the minute and 60′ to the hour—the sum is the greatest-eclipse general time.
62
求各朔太陽經度以食甚凡時通分，以太陽日行度通秒乘之，以二十四除之爲秒，滿六十約之爲秒分，用加減午正太陽度，午前合朔減之，午後加之。 得合朔時太陽經度。 即食甚日躔黃道度。
Derive solar longitude at each conjunction: convert greatest-eclipse general time to minutes, multiply by solar daily motion in seconds, divide by 24, reduce by 60, and add or subtract from noon solar longitude—subtract before noon, add after. The result is solar longitude at conjunction. This is the ecliptic degree on the day of greatest eclipse.
63
求加減分視合朔時太陽宮度，入晝夜加減立成內，取加減分，依比例法求之。
Derive add-subtract parts: at conjunction, enter the solar mansion in the latitude-longitude hourly add-subtract ready reckoner and obtain the parts by proportional method.
64
求子正至合朔時分秒置食甚凡時，以加減分分加減之，午前合朔減，午後加。 用加減十二時，午前合朔用減十二時，午後用加十二時。 即子正至合朔時分秒。 按命時起子正，乃變其術以劍《大統》，非其本法也。
Derive midnight-to-conjunction time: set greatest-eclipse general time and add or subtract the add-subtract minutes—subtract before noon, add after. Then add or subtract twelve hours—subtract twelve for pre-noon conjunction, add twelve for post-noon. The result is midnight-to-conjunction time in minutes and seconds. Note: reckoning from midnight standard alters the procedure to match the Datong calendar and departs from the original method.
65
求第一東西差經差。 視合朔時，太陽宮在立成經緯時加減立成右七宮取上行時，順行。 在左七宮取下行時，逆行。 以子正至合朔時，取經差，依比例法求之。 止用時下小餘求之。 下同。 第一東西差。
Derive the first east-west (longitude) difference. At conjunction, if the solar mansion lies in the right seven mansions of the hourly add-subtract ready reckoner, take the upper-row hourly value for direct motion. In the left seven mansions take the lower-row hourly value for retrograde motion. From midnight to conjunction, take the longitude difference and apply the proportional method. Use only the fractional remainder below the hour. The same applies below. This yields the first east-west difference.
66
求第二東西差視合朔時，太陽宮在立成內，同上。 取次宮子正至合朔時經差，依比例法求之，爲第二東西差。
Derive the second east-west difference: at conjunction, if the solar mansion lies in the ready reckoner, proceed as above. Take the next mansion's midnight-to-conjunction longitude difference and apply the proportional method to obtain the second east-west difference.
67
求第一南北差緯差。 以合朔時，太陽宮及子正，至合朔時入立成內，同上。 取緯差，依比例法求之，爲第一南北差。
Derive the first north-south (latitude) difference. At conjunction, with solar mansion and midnight-to-conjunction time, enter the ready reckoner as above. Take the latitude difference and apply the proportional method to obtain the first north-south difference.
68
求第二南北差以合朔太陽宮，取次宮子正至合朔時緯差，依比例法求之，爲第二南北差。
Derive the second north-south difference: with the conjunction solar mansion, take the next mansion's midnight-to-conjunction latitude difference by proportional method.
69
求第二時差以膈朔太陽宮及子正至膈朔時，入立成取時差，依比例法求之。
Derive the second time difference: with the obscured-new-moon solar mansion and midnight-to-obscured-new-moon time, enter the ready reckoner, take the time difference, and apply the proportional method.
70
第二時差公合朔太陽宮，取次宮子正至合朔時時差差，依比例法求之。
For the second time difference, with the conjunction solar mansion take the next mansion's midnight-to-conjunction time difference by proportional method.
71
求合朔時東西差以第一東西差與第二東西差相減，餘通秒，以乘合朔時太陽度分，亦通秒。 以三十度除之爲纖，以六十收之爲微、爲秒、爲分、經加減第一東西差，視第一東西差數少於第二差者加已，多者減之，下同。 爲合朔時東西差。
Derive east-west difference at conjunction: subtract the second from the first east-west difference, convert to seconds, and multiply by conjunction solar longitude in seconds. Divide by 30° for filaments, reduce by 60 to micro, seconds, and minutes, and add or subtract from the first east-west difference—add if the first is less than the second, subtract if greater; the same below. The result is the east-west difference at conjunction.
72
求合朔時南北差以第一南北差與第南北差相減，餘通秒，以乘太陽度分，以三十除之爲纖，依率收之爲微、秒、分，以加減第一南北差，爲各朔時南北差。
Derive north-south difference at conjunction: subtract the two north-south differences, convert the remainder to seconds and multiply by solar longitude, divide by 30, reduce by rate, and add or subtract from the first north-south difference.
73
求合朔時差以第一第二兩時差相減，乘太陽度分，以三十除之，依率帳之，用加減第一時差，爲合朔時差。
Derive time difference at conjunction: subtract the two time differences, multiply by solar longitude, divide by 30, reduce by rate, and add or subtract from the first time difference.
74
求合朔時本輪行度以本輪日行度一十三度四分通分，以乘食甚凡時，亦通分。 以二十四除之爲秒，依率收之爲分、爲度，以加減午正本輪行度，午前減，午後加。 爲合朔時行度。
Derive deferent motion at conjunction: set deferent daily motion at 13°4′ (unified) and multiply by greatest-eclipse general time, also unified. Divide by 24 for seconds, reduce by rate to minutes and degrees, and add or subtract from noon deferent motion—subtract before noon, add after. The result is deferent motion at conjunction.
75
求比敷分以本輪行度入立成，太陽、太陰時行影徑分立成。 取同宮近度太陰比敷分，依比例法求之。
Derive the proportional factor: enter deferent motion into the solar-lunar hourly motion and shadow-radius ready reckoner. Take the lunar proportional factor for the same mansion and nearby degree and apply the proportional method.
76
求東西定差置合朔時東西差通秒，以比敷分通秒乘之爲纖以六十收之爲微、爲秒、爲分，以加合朔東西差，有加、無減。 爲定差。
Derive the east-west fixed difference: convert conjunction east-west difference and proportional factor to seconds, multiply for filaments, reduce by 60, and add to the conjunction east-west difference—always add, never subtract. The result is the fixed difference.
77
求南北定差法同東西定差。
Derive the north-south fixed difference by the same method as the east-west fixed difference.
78
求食甚定時即食甚定分。 視其日合朔時，太陽度在立成經緯時加減立成左七宮，其時差，黑字減，白字加，在右七宮，白字減，黑字加，皆加減於子正至合朔時，得數命起正減之，得某時初正。 餘通爲秒，以一千乘之，以一百四下四除之，六十分爲一時，每日一千四百四十分，故以千乘之，又以一四四除之。 以六十約之，滿百爲刻，即食甚定時。
Derive greatest-eclipse fixed time, that is, the fixed minutes of greatest eclipse. Inspect conjunction solar longitude in the ready reckoner: in the left seven mansions subtract black entries and add white for the time difference; in the right seven mansions reverse the signs; add or subtract from midnight-to-conjunction time, name from midnight, and subtract to obtain the watch's initial standard. Convert the remainder to seconds, multiply by 1,000, and divide by 144—since 60 minutes make one hour and the day has 1,440 minutes. Reduce by 60; at 100 parts make one ke—the result is greatest-eclipse fixed time.
79
求食甚太陰經度於合朔太陽經度內，加減東西定差，即得食甚太陰經度。 其加減視食甚定時時差加減。
Derive greatest-eclipse lunar longitude: add or subtract the east-west fixed difference from conjunction solar longitude. Add or subtract according to the sign of the time difference at greatest-eclipse fixed time.
80
求合朔計都度置食甚凡時通分，以計都日行度三分一十一秒通秒乘之，以二十四除之爲微，滿六十收之爲秒、爲分，以加減其日午時計都行度，羅計逆行，午前合朔加，午後減。 爲合朔時計都度。
Derive conjunction Ketu longitude: convert greatest-eclipse general time to minutes, multiply by Ketu's daily motion of 3°11″, divide by 24, reduce by 60, and add or subtract from noon Ketu motion—Rahu and Ketu are retrograde: add before noon, subtract after. The result is Ketu's longitude at conjunction.
81
求合朔太陰緯度食甚時，太降經度內加減合朔時計都度，餘爲計都與月相離度，入太陰緯度立成取之。
Procedure to derive the Moon's latitude at conjunction: at greatest eclipse, add or subtract Ketu's longitude at conjunction within the lunar longitude; the remainder is Ketu's separation from the Moon; look it up in the lunar-latitude ready reckoner.
82
求食甚太陰緯度南北定差內。 加減合朔時太陰緯度，在黃道南加，北減。 得食甚緯度。
Procedure to derive the greatest-eclipse lunar latitude: apply the north–south fixed parallax. Add or subtract the lunar latitude at conjunction: add if south of the ecliptic, subtract if north. The result is the greatest-eclipse latitude.
83
求合朔時太陽自行度用太陽日行度五十九分八秒通秒，以乘食甚凡時，亦通分。 用二十四除之，得數爲微，滿六十收之爲秒、爲分，以加減其日午正自行度，午前合朔減，午後加。 得合朔自行度。
Procedure to derive the Sun's proper-motion degree at conjunction: convert the Sun's daily motion of 59 parts 8 seconds to seconds, multiply by the greatest-eclipse general time (parts likewise converted to seconds). Divide by 24; collect fine parts by sixties into seconds and parts; add or subtract from the day's noon proper-motion degree—subtract before noon, add after noon at conjunction. The result is the Sun's proper-motion degree at conjunction.
84
求太陽徑分以合朔太陽自行度爲引數，入立成影徑分立成內同宮近度，取太陽徑分，依比例法求之。
Procedure to derive the Sun's diameter parts: set the Sun's proper-motion degree at conjunction as the argument, enter the shadow-and-diameter ready reckoner at the same palace and nearest degree, and obtain the solar diameter parts by proportional interpolation.
85
求太陰徑分以合朔時本輪行度爲引數，入立成同上內取同宮近度太陰徑分，依比例法求之。
Procedure to derive the Moon's diameter parts: set the epicycle motion degree at conjunction as the argument, enter the same ready reckoner at the same palace and nearest degree, and obtain the lunar diameter parts by proportional interpolation.
86
求二半徑分並太陽、太陰雨徑分，半之。
Procedure to derive the two semi-diameter parts: add the solar and lunar diameter parts and halve the sum.
87
求太陽食限分置二半徑分，內減食甚太陰緯度，餘爲太陽食限。 如不及減者不食。 如太陰無緯度者，食既。 如太陰無緯度而日徑大於月徑者，食有金環。
Procedure to derive the solar eclipse limit: set the two semi-diameter parts and subtract the greatest-eclipse lunar latitude; the remainder is the solar eclipse limit. If subtraction fails, there is no eclipse. If the Moon has no latitude, the eclipse is total. If the Moon has no latitude and the solar diameter exceeds the lunar diameter, the eclipse is annular.
88
求太陽食甚定分以太陽食限分通秒，以一千乘之爲實，以太陽徑分通秒爲法除之，以百約之爲分，爲太陽食甚定分。
Procedure to derive the solar greatest-eclipse fixed parts: convert the solar eclipse limit to seconds, multiply by 1000 as the dividend, divide by the solar diameter in seconds, and scale by 100 to obtain the solar greatest-eclipse fixed parts.
89
求時差即定用分。 食甚太陰緯度通秒自乘，二半徑分亦通秒自乘，兩自乘數相減，餘以平方開之，以二十四乘之爲實，以其日太陰日行度內減太陽日行度通分爲法。 實如法而一，得數爲分，滿六十分爲一時，爲時差。
Procedure to derive the hour equation, namely the fixed usable parts. Square the greatest-eclipse lunar latitude in seconds, square the two semi-diameter parts in seconds, subtract the squares, take the square root of the remainder, multiply by 24 for the dividend, and set that day's lunar daily motion minus solar daily motion (in seconds) as the divisor. Divide the dividend by the divisor; the quotient is parts, and 60 parts make one hour—this is the hour equation.
90
求初虧置食甚定時，內減時差，餘時命起子正減之，得初正時。 餘分通秒，以一千乘之，以一百四十四除之，以六十約之，滿百爲刻，爲初虧時刻。
Procedure to derive first contact: set the greatest-eclipse fixed time, subtract the hour equation, count the remainder forward from midnight (zi), and obtain the initial proper hour. Convert the fractional remainder to seconds, multiply by 1000, divide by 144, reduce by 60, and when the quotient reaches 100 parts form one quarter-hour mark—this is the first-contact quarter-hour.
91
求復圓置食甚定時，內加時差，命起子正，如初虧法，得復圓時刻。
Procedure to derive last contact: set the greatest-eclipse fixed time, add the hour equation, and apply the same counting from midnight as for first contact to obtain the restoration quarter-hour.
92
求初虧食甚圓方位與《大統》法同。
Procedure to derive the azimuths of first contact, greatest eclipse, and last contact: same method as the Grand Concordance calendar.
93
推月食法月食諸數，午前望，用前一日推，午後望，用次一日推。
Procedure for lunar eclipse: for a pre-noon full moon reckon from the prior day; for a post-noon full moon reckon from the next day.
94
辨月食限視望日太陰經度與羅喉或計都度相離二十三度之內，太陽緯度在一度八分之下，爲有食。 又視合望在太陰未出二量，未入二時，其限有帶食。 其在二時已上者不算。
Discern the lunar-eclipse limit: if on the full-moon day the lunar longitude lies within 23° of Rahu or Ketu and the solar latitude is below 1°8′, an eclipse is possible. Also: if syzygy falls within two du before the Moon rises or within two hours before it sets, the eclipse may be visible only at the horizon. If the interval is two hours or more, do not reckon it.
95
求食甚凡時即經望。 置其日太陰經度內減六宮，如不及減，加十二宮減以減其日午正太陽度爲午前望。 如太陽度不及減，加入六宮減之，爲午後望。 置相減餘數相通秒，以二十四乘之爲實，置其日太陰經度，內減前一日太陰經度，若在午後望者，減後一日太陽度。 餘爲太陽日行度。 兩日行度相減，餘通秒爲法，除實得數爲時。 其時下餘數，以六十通之爲分、秒，即所求食甚凡時。
Procedure to derive the greatest-eclipse general time, namely the mean full moon. Set the day's lunar longitude and subtract six palaces; if subtraction fails, add twelve palaces and subtract the day's noon solar longitude—this identifies a pre-noon full moon. If the solar longitude cannot be subtracted, add six palaces and subtract again—this identifies a post-noon full moon. Convert the subtraction remainder to seconds and multiply by 24 for the dividend; set the day's lunar longitude and subtract the prior day's lunar longitude—for a post-noon full moon subtract the next day's solar longitude instead. The remainder is the solar daily motion. Subtract the two daily motions, convert the remainder to seconds as the divisor, and divide the dividend to obtain hours. For the fractional remainder under the hour, convert by 60 into parts and seconds—this is the greatest-eclipse general time sought.
96
求食甚月離黃道宮度置食甚凡時，與太陽日行度俱通秒相乘，以二十四除之，得數爲纖潢六十收之爲微、爲秒、爲分，以加減其日午正太陽度，午前望減，午後望加。 爲望時太陽度，加六宮，即得所求。
Procedure to derive the Moon's ecliptic palace at greatest eclipse: multiply the greatest-eclipse general time and solar daily motion (both in seconds), divide by 24, collect fine parts into micro-units, seconds, and parts, and add or subtract from the day's noon solar longitude—subtract before noon, add after noon at full moon. The result is the solar longitude at full moon; add six palaces to obtain the value sought.
97
求晝夜加減差以望時太陽宮度爲引數，入晝夜加減立成內，取加減分，依比例法求之。
Procedure to derive the day–night equation: set the solar palace at full moon as the argument, enter the day–night additive–subtractive ready reckoner, and obtain the correction by proportional interpolation.
98
求食甚定時置食甚凡時，以晝夜加減差法加減之。 午前望減，午後望加。 得數，用加減一十二時，如午後望加十二時，午前望與十二時相減。 命起子正，得初正時。 其小餘，如法收爲刻，法詳日食。 得定時。
Procedure to derive the greatest-eclipse fixed time: set the greatest-eclipse general time and apply the day–night equation. Subtract before noon at full moon; add after noon. Apply the result to twelve hours: for a post-noon full moon add 12 hours; for a pre-noon full moon subtract from 12 hours. Count from midnight (zi); obtain the initial proper hour. For the fractional remainder, collect quarter-hour marks by the same method as for solar eclipse. The result is the fixed time.
99
求望時計都度置食甚凡時，通秒爲實，以計都日行度三分一十一秒通秒乘之，以二十四除之，得數爲纖以六十收之爲微、爲秒、爲分，用加減其日午正計都行度，羅計逆行，午前望加，午後望減。 即得。
Procedure to derive Ketu's longitude at full moon: convert the greatest-eclipse general time to seconds as the dividend, multiply by Ketu's daily motion of 3 parts 11 seconds, divide by 24, collect fine parts into micro-units, seconds, and parts, and add or subtract from the day's noon Ketu longitude—Rahu and Ketu are retrograde: add before noon, subtract after noon at full moon. The result is obtained.
100
求望時太陰緯度置食甚月離黃道度，內減望時計都度，如不及減，加十二宮減。 餘爲計都與月相離度，入太陰緯度立成取之。
Procedure to derive the lunar latitude at full moon: set the Moon's ecliptic longitude at full moon and subtract Ketu's longitude at full moon; if subtraction fails, add twelve palaces and subtract again. The remainder is Ketu's separation from the Moon; look it up in the lunar-latitude ready reckoner.
101
求望時本輪行度即入遲疾歷。 置太陰本輪日行度，十三四分。 通分，以食甚凡時通秒乘之，以二十四除之爲微，以六十收之爲秒、爲分、爲度，用加減其日午正本輪行度，午前望減，午後加。 即得。
Procedure to derive the epicycle motion degree at full moon, namely entry into the slow–fast sequence. Set the Moon's epicycle daily motion at 13 parts 4. Convert to a common denominator, multiply by the greatest-eclipse general time in seconds, divide by 24 for fine parts, collect by sixties into seconds, parts, and degrees, and add or subtract from the day's noon epicycle motion—subtract before noon, add after noon at full moon. The result is obtained.
102
求太陰徑分以望時本輪行宮度，入影徑分立成求之。 法詳日食。
Procedure to derive the lunar diameter parts: set the epicycle palace degree at full moon and obtain the value from the shadow-and-diameter ready reckoner. The method is the same as for solar eclipse.
103
求太陰影徑分以望時本輪行宮度，放影徑分立成，取之。
Procedure to derive the lunar shadow diameter parts: set the epicycle palace degree at full moon and take the value from the shadow-diameter ready reckoner.
104
求望時太陽自行度以太陽日行度五十九分八秒與食甚凡時俱通秒相乘，以二十四除之，得數爲纖，滿六十收爲微、爲秒、爲分，以減其日午正太陽自行度。 法同日食求太陽經度。
Procedure to derive the Sun's proper-motion degree at full moon: convert the solar daily motion of 59 parts 8 seconds and the greatest-eclipse general time to seconds, multiply, divide by 24, collect fine parts into micro-units, seconds, and parts, and subtract from the day's noon solar proper-motion degree. The method matches solar-eclipse procedure for deriving solar longitude.
105
求影徑減差以其日太陽自行範度爲引數，入影徑立成內，於同宮近度取太陰影徑差分，依比例法求之。 法詳前。
Procedure to derive the shadow-diameter reduction: set that day's solar proper-motion palace as the argument, enter the shadow-diameter ready reckoner, take the lunar shadow-diameter difference at the same palace and nearest degree, and obtain it by proportional interpolation. The method is as given above.
106
求影徑定分置太陰影徑分，內減影徑減差分。
Procedure to derive the fixed shadow-diameter parts: set the lunar shadow-diameter parts and subtract the shadow-diameter reduction.
107
求二半徑分置太陰徑分，加影徑定分，半之。
Procedure to derive the two semi-diameter parts: add the lunar diameter parts and the fixed shadow-diameter parts, then halve the sum.
108
求太陰食限置二半徑分，內減望時太陰緯度。
Procedure to derive the lunar eclipse limit: set the two semi-diameter parts and subtract the lunar latitude at full moon.
109
求食甚定分置食限分，通秒，以一千乘之爲實，以太陰徑分秒爲法，除之，以百約之災分，爲食甚定分。
Procedure to derive the greatest-eclipse fixed parts: convert the eclipse limit to seconds, multiply by 1000 as the dividend, divide by the lunar diameter in seconds, scale by 100 to parts, and obtain the greatest-eclipse fixed parts.
110
求太陰逐時行過太陽分置太陰望時經度，減前一日太陰經度，又置望時太陽自行度，減前一日太陽自行度，以兩餘數相減，爲太陰晝夜行過太陽度。 通秒以二十四除之，滿六十收之，得逐時行過太陽分。
Procedure to derive the Moon's hourly motion relative to the Sun: subtract the prior day's lunar longitude from the full-moon lunar longitude, and the prior day's solar proper motion from the full-moon solar proper motion; subtract the two remainders to obtain the Moon's diurnal motion past the Sun. Convert to seconds, divide by 24, and collect by sixties to obtain the hourly motion past the Sun in parts.
111
求時差以太陰緯度分，通秒自乘，又以二半徑分通秒自乘，兩數相減，餘開平方爲實，以太陰行過太陽度通秒爲法除之，得數即時刻差。 即初虧至食甚定用分。
Procedure to derive the hour equation: square the lunar latitude in seconds and the two semi-diameter parts in seconds, subtract, take the square root as the dividend, divide by the Moon's hourly motion past the Sun in seconds, and obtain the hour-and-mark difference. This is the fixed usable interval from first contact to greatest eclipse.
112
求初虧復圓時刻以時差減食甚定時，得初虧時刻。 加食甚定時，得復圓時刻。 其命時收刻之法，並同日食。
Procedure to derive the quarter-hours of first contact and last contact: subtract the hour equation from the greatest-eclipse fixed time to obtain first contact. Add the hour equation to the greatest-eclipse fixed time to obtain last contact. The methods for counting hours and collecting quarter-hour marks are the same as for solar eclipse.
113
求食既至食甚時差置二半徑分，減太陰徑分，通秒自乘，又置太陰緯度亦通秒自乘，相減，平方開之爲實。 以太陰逐時行過太陽度通秒爲法除之，得數即時差。
Procedure to derive the hour equation from totality to greatest eclipse: subtract the lunar diameter from the two semi-diameter parts, square in seconds, square the lunar latitude in seconds, subtract, and take the square root as the dividend. Divide by the Moon's hourly motion past the Sun in seconds; the quotient is the hour equation.
114
求食既生光時刻以食既至食甚時差，減食甚定時，爲食既時刻。 加食甚定時，爲生光時刻。
Procedure to derive the quarter-hours of totality and emergence of light: subtract the totality-to-greatest-eclipse hour equation from the greatest-eclipse fixed time to obtain totality. Add the hour equation to the greatest-eclipse fixed time to obtain emergence of light.
115
求初虧食甚復圓方位與《大統》法同。
Procedure to derive the azimuths of first contact, greatest eclipse, and last contact: same method as the Grand Concordance calendar.
116
求日出入時以午正太陽經度爲引數，入西域晝夜時立成，取其度分，依比例法求之，爲未定分。 又引於數相對宮度內，取其度分，如初宮三度，向六宮三度取之。 亦依比例法求之，爲後未定分。 兩未定分相減，不及減，加三百六十度減。 餘通秒，用十五除之，六十收之爲分、爲時，得其日晝時分秒。 半之爲其日半晝時分秒。 以半晝時分秒減十二時，餘爲日出時分秒，加十二時爲日入時分秒。
Procedure to derive sunrise and sunset times: set noon solar longitude as the argument, enter the Western Regions day–night-time ready reckoner, and obtain degree parts by proportional interpolation as the first provisional value. Also take the opposing palace in the reckoner—for example, 3° in the first palace paired with 3° toward the sixth palace—and obtain degree parts the same way. Obtain a second provisional value by proportional interpolation. Subtract the two provisional values; if subtraction fails, add 360° and subtract again. Convert the remainder to seconds, divide by 15, collect by sixties into parts and hours, and obtain that day's daylight in hours, parts, and seconds. Halve to obtain half the daylight for that day in hours, parts, and seconds. Subtract half-daylight from 12 hours for sunrise; add 12 hours for sunset.
117
求日月出入帶食分秒視其日日出時分秒，較多於初虧時分秒，少於食甚定時及復時分秒者，即有帶食。 置其日日出時或日入時，與食甚定時分秒相減，餘爲帶食差。 置日月食甚定分，以帶食差通秒乘之，以時差通秒除之，得數爲帶食分。 於食甚定分內減帶食分，餘爲日月帶食所見之分。
Procedure to derive horizon-eclipse parts and seconds: if sunrise falls after first contact but before greatest eclipse and last contact, the eclipse is visible only at sunrise (horizon eclipse). Subtract the greatest-eclipse fixed time from that day's sunrise or sunset time; the remainder is the horizon-eclipse offset. Set the solar or lunar greatest-eclipse fixed parts, multiply by the horizon-eclipse offset in seconds, divide by the hour equation in seconds, and obtain the horizon-eclipse parts. Subtract the horizon-eclipse parts from the greatest-eclipse fixed parts; the remainder is the magnitude visible at the horizon.
118
求月食更點置二十四時，內減晝時，又減晨昏時，七十二分，即中歷之五刻弱也。 餘不夜時，通秒五約之爲更法。 寺分更法爲點法。 如食在子正以前者，置初虧食甚復圓等時，內減日入時，又減半晨昏時，三十六分。 餘通就，以更法減之爲更數。 不滿更法者，以點法減之爲點數。 食在子正已後者，置夜時半之，加初虧食甚復圓等時，以更法減之爲更數。 不滿更法者，以點法減之爲點數。 皆命起初更、初點。 更法減之，減一次爲一更，其減餘不滿法者，亦虛命爲一更。 點法仿此。
Procedure to derive lunar-eclipse watches and marks: set 24 hours, subtract daylight, then subtract morning-and-evening time of 72 parts—this is the Central Calendar's 5 quarter-hours less a fraction. For the remaining night hours, convert to seconds and divide by 5 to obtain the watch divisor. Divide the fractional parts by the watch divisor to obtain the mark divisor. If the eclipse is before midnight: set first contact, greatest eclipse, last contact, and the like, subtract sunset time, then subtract half the morning-and-evening interval of 36 parts. Convert the remainder to seconds and divide by the watch divisor to obtain the watch count. For the remainder less than one watch, divide by the mark divisor to obtain the mark count. If the eclipse is after midnight: halve the night hours, add first contact, greatest eclipse, last contact, and the like, and divide by the watch divisor for the watch count. For the remainder less than one watch, divide by the mark divisor to obtain the mark count. In every case count from the first watch and first mark. Subtract repeatedly by the watch divisor; each subtraction counts as one watch; if the final remainder is less than one divisor, still count it nominally as one watch. The mark divisor follows the same rule.
119
▲太陰五星凌犯
▲ Lunar and Five-Planets Encroachment
120
求太陰晝夜行度以本日經度與次日經度相減，餘即本日晝夜行度。
Procedure to derive the Moon's diurnal motion in degrees: subtract this day's longitude from the next day's; the remainder is the day's diurnal motion.
121
求太陰晨昏刻度置其日午太陰經度，內加立成太陰出入晨昏加減立成其日昏刻加差，即爲其日太陰昏刻經度。 置其次日午正太陰經度，減立成其日晨刻減差，即爲其日太陰晨刻經度。
Procedure to derive the Moon's morning-and-evening scale longitudes: set the day's noon lunar longitude, add that day's evening-quarter additive from the lunar ingress–egress morning–evening ready reckoner, and obtain the lunar longitude at evening quarter. Set the next day's noon lunar longitude, subtract that day's morning-quarter subtractive from the ready reckoner, and obtain the lunar longitude at morning quarter.
122
求月出入度置其日午正太陰經度，加立成內即前立成其日月入加差，即爲其日月入時太陰經度。 加立成內其日月出加差，即其日月出時太陰經度。
Procedure to derive the Moon's ingress and egress longitudes: set the day's noon lunar longitude, add that day's Sun–Moon ingress correction from the lunar ingress–egress ready reckoner, and obtain the lunar longitude at ingress. Add that day's Sun–Moon egress correction from the ready reckoner to obtain the lunar longitude at egress.
123
耱太陰所犯星座朔後視昏刻度至月入度，望後視月出度至晨刻度，入黃道南北各像星立成內，經緯度相近在一度已下者，取之。
Procedure to derive the star the Moon encroaches: after new moon, reckon from evening-scale longitude to ingress longitude; after full moon, from egress longitude to morning-scale longitude; look up north and south of the ecliptic in the star-image ready reckoners and select any star whose longitude and latitude lie within 1°.
124
求時刻置其日午正太陰經度，與取到各像星經度相減，通分，以二十四乘之，以太陰晝夜行度亦通分除之，得初正時。 其小餘，以六十通之爲分，以一竿千乘之，一百國十四除之，以百約之爲刻，即得所求時刻。
Procedure to derive the time: subtract the selected star's longitude from the day's noon lunar longitude, convert to a common denominator, multiply by 24, divide by the lunar diurnal motion (also converted), and obtain the initial proper hour. For the fractional remainder, convert by 60 into parts, multiply by 1000, divide by 144, scale by 100 into quarter-hours, and obtain the time sought.
125
求上下相離分置太陰緯度與年犯星緯度相減，餘爲上下相離分。 若月星同在南，月多爲下離。 同在北，月多爲上離，下爲下離。 若南北不同，月在北爲上離，南爲下離。
Procedure to derive the vertical separation: subtract the encroached star's latitude from the lunar latitude; the remainder is the vertical separation. If the Moon and star are both south of the ecliptic, the greater latitude is lower separation. If both are north, the greater latitude is upper separation and the lesser is lower separation. If they lie on opposite sides of the ecliptic, a northern Moon is upper separation and a southern Moon is lower separation.
126
求五星凌犯各星相離置其日五星經緯度，入黃道立成內，視各像內外星經緯度，在一度已下和取之。 其五星緯度與各星緯度相減，餘即上下相離分。
Procedure to derive each planet's separation from encroached stars: set the day's planetary longitude and latitude, enter the ecliptic ready reckoner, and select any star whose longitude and latitude lie within 1° of the planet's. Subtract each star's latitude from the planet's latitude; the remainder is the vertical separation.
127
求月犯五星，五星相犯視太陰經緯度，五星經緯度相近在一度已下者，取之。
Procedure to derive the Moon encroaching the five planets and mutual planetary encroachment: compare lunar and planetary longitudes and latitudes and record any pair within 1°.