1
志二十三
Treatise 23
2
時憲四
Shixian Calendar 4
3
康熙甲子元法中
Kangxi jiazi calendrical system — middle section
4
日躔用數
Constants for solar motion
5
康熙二十三年甲子天正冬至為法元。 癸亥年十一月冬至。
The calendrical epoch is the winter solstice of the first month in jiazi year, Kangxi 23. That is the winter solstice of the eleventh month in the guihai year.
6
周天三百六十度。 平分之為半周，四分之為象限，十二分之為宮，每度六十分，秒微纖以下皆以六十遞析。 周天入算，化作一百二十九萬六千秒。
A full circuit of heaven comprises 360 degrees. Half the circuit gives a semicircle, a quarter gives a quadrant, and a twelfth gives a mansion; each degree contains sixty minutes, and seconds and still finer units below are all broken down by successive factors of sixty. In calculation the full circuit is expressed as 1,296,000 seconds.
7
周日一萬分。 時則二十四，刻則九十六，刻下分則一千四百四十，秒則八萬六千四百。
The day-circle is divided into 10,000 parts. There are 24 hours, 96 quarters, 1,440 fractional parts beneath a quarter, and 86,400 seconds.
8
周歲三百六十五日二四二一八七五。
The tropical year is 365.2421875 days.
9
紀法六十。
The sexagenary cycle divisor is 60.
10
宿法二十八。
The lunar-mansion divisor is 28.
11
太陽每日平行三千五百四十八秒，小餘三三０五一六九。
The sun's mean daily motion is 3,548 seconds, fractional remainder 0.3305169.
12
最卑歲行六十一秒，小餘一六六六六。
The apogee advances 61 seconds per year, fractional remainder 0.16666.
13
最卑日行十分秒之一又六七四六九。
The apogee's daily motion is 0.0001 second plus 0.67469.
14
本天半徑一千萬。
The deferent radius is 10,000,000.
15
本輪半徑二十六萬八千八百一十二。
The deferent epicycle radius is 268,812.
16
均輪半徑八萬九千六百零四。
The equating epicycle radius is 89,604.
17
宿度見天文志。
Lodge longitudes appear in the Treatise on Astronomy.
18
歲差五十一秒。
Annual precession is 51 seconds.
19
各省及蒙古北極高度、東西偏度、見天文志。
North polar altitudes and longitudinal offsets for each province and Mongolia are listed in the Treatise on Astronomy.
20
黃赤大距，二十三度二十九分三十秒。
Obliquity of the ecliptic: 23°29′30″.
21
最卑應，七度十分十一秒十微。
Apogee epoch: 7°10′11.10″.
22
氣應，七日六五六三七四九二六。
Solar-term epoch offset: 7.656374926 days.
23
宿應，五日六五六三七四九二六。
Lodge epoch offset: 5.656374926 days.
24
日干，甲、乙、丙、丁、戊、己、庚、辛、壬、癸。
The ten day-stems: jia, yi, bing, ding, wu, ji, geng, xin, ren, and gui.
25
支，子、丑、寅、卯、辰、巳、午、未、申、酉、戌、亥。
The twelve branches: zi, chou, yin, mao, chen, si, wu, wei, shen, you, xu, and hai.
26
宿名，角、亢、氐、房、心、尾、箕、斗、牛、女、虛、危、室、壁、奎、婁、胃、昂、畢、參、觜、井、鬼、柳、星、張、翼、軫。
The twenty-eight lodges: Jiao, Kang, Di, Fang, Xin, Wei, Ji, Dou, Niu, Nü, Xu, Wei, Shi, Bi, Kui, Lou, Wei, Mao, Bi, Shen, Zi, Jing, Gui, Liu, Xing, Zhang, Yi, and Zhen.
27
時名，從十二支各分初、正。 起子正，盡夜子初。
Hour names: each of the twelve branches is split into early and exact halves. They run from exact zi through early night zi.
28
推日躔法求天正冬至，置周歲，以距元年數減一得積年乘之，得中積分，加氣應得通積分，上考往古，則減氣應得通積分。 其日滿紀法去之，餘為天正冬至日分。 上考往古，則以所餘轉與紀法相減，餘為天正冬至日分。 自初日起甲子，其小餘以刻下分通之，如法收為時刻。 周日一萬分為一率，小餘為二率，刻下分為三率，求得四率為時分。 滿六十分收為一時，十五分收為一刻。 初時起子正，中積分加宿應，滿宿法去之，為天正冬至值宿日分，初日起角宿。
To find the winter solstice of the civil year: take the tropical year constant, multiply by the number of years elapsed since the epoch (minus one) to obtain the mean accumulated interval, then add the solar-term epoch offset for total accumulated parts; when projecting backward into antiquity, subtract the solar-term epoch instead. Remove full sexagenary cycles from the day count; the remainder is the day-fraction for the civil-year winter solstice. For backward projection, subtract the remainder from the cycle divisor instead; what is left is the winter-solstice day-fraction. Count the day-stems from jiazi on day one; convert the fractional remainder via sub-quarter parts and reduce to hours and quarters. Use the 10,000-part day-circle as the first ratio, the fractional remainder as the second, and sub-quarter parts as the third; the fourth ratio gives the hour-fraction. Combine every 60 parts into one hour and every 15 parts into one quarter. Hours start at exact zi; add the lodge epoch to the mean accumulated interval and remove full mansion cycles to obtain the lodge day-fraction for the winter solstice, counting mansions from Jiao on the first day.
29
求平行，以周日為一率，太陽每日平行為二率，天正冬至小餘與周日相減餘為三率，求得四率為年根秒數。 又置太陽每日平行，以本日距冬至次日數乘之，得數為秒。 與年根相併，以宮度分收之，得平行。
For mean motion: use the day-circle as the first ratio, the sun's daily mean motion as the second, and the winter-solstice fractional remainder minus the day-circle as the third; the fourth ratio is the year's-root in seconds. Multiply the sun's daily mean motion by the number of days from the current date to the day after the winter solstice to obtain seconds. Add this to the year's root and reduce to mansion, degree, and minute to obtain the mean longitude.
30
求實行，置最卑歲行，以積年乘之。 又置最卑日行，以距冬至次日數乘之。 兩數相併，加最卑應，上考則減最卑應。 以減平行為引數。 用平三角形，以本輪半徑三分之二為對正角之邊，以引數為一角，求得對角之邊倍之。 又求得對又一角之邊，與本天半徑相加減。 引數三宮至八宮則相加，九宮至二宮則相減。 復用平三角形，以加倍之數為小邊，加減本天半徑之數為大邊，正角在兩邊之中，求得對小邊之角為均數。 置平行以均數加減之，引數初宮至五宮為加，六宮至十一宮為減。 得實行。 求宿度，以積年乘歲差，得數加甲子法元黃道宿度，為本年宿鈐，以減實行，餘為日躔宿度。 若實行不及減宿鈐，退一宿減之。
For true motion: multiply the apogee's annual rate by the accumulated years. Multiply the apogee's daily rate by the days elapsed since the day after the winter solstice. Sum the two results and add the apogee epoch (subtract it when projecting backward). Subtract this from the mean longitude to obtain the equation argument. By plane triangle: let two-thirds of the deferent epicycle radius be the side opposite the right angle, take the equation argument as one angle, and double the side opposite that angle. Find the side opposite the other angle and add or subtract it from the deferent radius. If the argument falls in mansions 3–8, add; if in mansions 9–2, subtract. Apply another plane triangle with the doubled value as the short side and the deferent radius after addition or subtraction as the long side; the right angle lies between them, and the angle opposite the short side is the equation of center. Add or subtract the equation from the mean longitude: add for mansions 1–5, subtract for mansions 6–11. This yields the true longitude. For lodge longitude: multiply years elapsed by annual precession, add to the jiazi-epoch ecliptic lodge position to obtain this year's lodge pivot, subtract from the true longitude; the remainder is the sun's lodge position. If the true longitude is less than the lodge pivot, step back one lodge before subtracting.
31
求紀日值宿，置距冬至次日數，加冬至，日滿紀法去之。 初日起甲子，加冬至值宿，日滿宿法去之。 初日起角宿，得紀日值宿。
For the sexagenary day and its lodge: take days since the day after the winter solstice, add the solstice day-count, and remove full cycles of 60. Count day-stems from jiazi on day one, add the solstice lodge, and remove full cycles of 28 mansions. Starting mansions from Jiao on day one gives the cycle-day and its lodge.
32
求節氣時刻，日躔初宮丑，星紀。 初度為冬至，十五度為小寒。 一宮子，元枵。 初度為大寒，十五度為立春。 二宮亥，娵訾。 初度為雨水，十五度為驚蟄。 三宮戌，降婁。 初度為春分，十五度為清明。 四宮酉，大梁。 初度為穀雨，十五度為立夏。 五宮申，實沈。 初度為小滿，十五度為芒種。 六宮未，鶉首。 初度為夏至，十五度為小暑。 七宮午，鶉火。 初度為大暑，十五度為立秋。 八宮巳，鶉尾。 初度為處暑，十五度為白露。 九宮辰，壽星。 初度為秋分，十五度為寒露。 十宮卯，大火。 初度為霜降，十五度為立冬。 十一宮寅，析木。 初度為小雪，十五度為大雪。 皆以子正日躔未交節氣宮度者，為交節氣本日； 已過節氣宮度者，為交節氣次日。 乃以本日實行與次日實行相減為一率，每日刻下分為二率，本日子正實行與節氣宮度相減為三率，求得四率為距子正後之分數，乃以時刻收之，即得節氣初正時刻。 如實行適與節氣宮度相符而無餘分，即為子正初刻。 求各省節氣時刻，皆以京師為主，視偏度加減之。 每偏一度，加減時之四分。 偏東則加，偏西則減。 推節氣用時法，以交節氣本日均數變時為均數時差，反其加減。 又以半徑為一率，黃赤大距餘弦為二率，本節氣黃道度正切為三率，求得四率為赤道正切。 檢表得度，與黃道相減，餘變時為升度時差。 二分後為加，二至後為減。 皆加減節氣時刻，為節氣用時。 求距緯度，以本天半徑為一率，黃赤大距度之正弦為二率，實行距春秋分前後度之正弦為三率，實行初宮初度至二宮末度，與三宮相減，餘為春分前； 三宮初度至五宮末度，則減去三宮，為春分後。 六宮初度至八宮末度，與九宮相減，餘為秋分前； 九宮初度至十一宮末度，則減去九宮，為秋分後。 求得四率為正弦，檢表得距緯度。 實行三宮至八宮，其緯在赤道北； 九宮至二宮，其緯在赤道南。
To find solar-term times: the sun's first mansion is Chou (Xingji). Degree 0 is the winter solstice; degree 15 is Lesser Cold. Mansion 1 is Zi (Yuanxiao). Degree 0 is Greater Cold; degree 15 is Beginning of Spring. Mansion 2 is Hai (Zouzi). Degree 0 is Rain Water; degree 15 is Awakening of Insects. Mansion 3 is Xu (Jianglou). Degree 0 is the spring equinox; degree 15 is Clear and Bright. Mansion 4 is You (Daliang). Degree 0 is Grain Rain; degree 15 is Beginning of Summer. Mansion 5 is Shen (Shichen). Degree 0 is Lesser Fullness; degree 15 is Grain in Ear. Mansion 6 is Wei (Chunshou). Degree 0 is the summer solstice; degree 15 is Lesser Heat. Mansion 7 is Wu (Chunhuo). Degree 0 is Greater Heat; degree 15 is Beginning of Autumn. Mansion 8 is Si (Chunwei). Degree 0 is End of Heat; degree 15 is White Dew. Mansion 9 is Chen (Shouxing). Degree 0 is the autumn equinox; degree 15 is Cold Dew. Mansion 10 is Mao (Dahu). Degree 0 is Frost's Descent; degree 15 is Beginning of Winter. Mansion 11 is Yin (Ximu). Degree 0 is Lesser Snow; degree 15 is Greater Snow. At exact zi, if the sun has not yet reached the term's mansion degree, that day is the term-entry day; if it has already passed that degree, term entry falls on the following day. Take the difference between today's and tomorrow's true longitudes as the first ratio, the daily sub-quarter parts as the second, and the gap between today's exact-zi true longitude and the term's mansion degree as the third; the fourth ratio is the interval after exact zi, which reduces to hours and quarters to give the term's early and exact times. If the true longitude exactly matches the term's mansion degree with no remainder, the term falls at exact zi in the first quarter. Provincial solar-term times are adjusted from the capital according to each place's longitudinal offset. Each degree of offset shifts the time by one quarter-hour. Add for places east of the capital, subtract for places west. For apparent solar-term time: convert the term-entry day's equation into a time correction and apply it with reversed sign. Next use the radius as the first ratio, the cosine of obliquity as the second, and the tangent of the term's ecliptic longitude as the third; the fourth ratio is the equatorial tangent. Look up the degree in the table, subtract from the ecliptic longitude, and convert the difference to the ascension time correction. Add after the equinoxes, subtract after the solstices. Apply both corrections to the mean term time to obtain apparent time. For ecliptic latitude: use the deferent radius as the first ratio, the sine of obliquity as the second, and the sine of the sun's distance from the equinoxes as the third; from mansion 1 degree 0 through the end of mansion 2, subtract mansion 3 — the remainder is before the spring equinox; From mansion 3 degree 0 through the end of mansion 5, subtract mansion 3 for the interval after the spring equinox. From mansion 6 degree 0 through the end of mansion 8, subtract mansion 9 for before the autumn equinox; From mansion 9 degree 0 through the end of mansion 11, subtract mansion 9 for after the autumn equinox. The fourth ratio is a sine; look it up in the table to obtain the latitude. For true motion in mansions 3–8, latitude is north of the equator; for mansions 9–2, latitude is south of the equator.
33
求日出入晝夜時刻，以本天半徑為一率，北極高度之正切為二率，本日距緯度之正切為三率，求得四率為正弦，檢表得日出入在卯酉前後赤道度。 變時，一度變時之四分，凡言變時皆仿此。 為距卯酉分。 以加減卯酉時，即得日出入時刻。 春分前、秋分後，以加卯正為日出，減酉正為日入。 春分後、秋分前，以減卯正為日出，加酉正為日入。 又倍距卯酉分，以加減半晝分，得晝夜時刻。 春分後以加得晝刻，以減得夜刻，秋分後反是。
For sunrise, sunset, and day-night length: use the deferent radius as the first ratio, the tangent of north polar altitude as the second, and the tangent of the day's ecliptic latitude as the third; the fourth ratio is a sine looked up in the table as the equatorial arc from Mao or You at rising and setting. Convert degrees to time at one quarter-hour per degree; all time conversions follow this rule. This gives the interval from Mao or You. Add or subtract from the Mao and You times to obtain sunrise and sunset. Before the spring equinox and after the autumn equinox, add to exact Mao for sunrise and subtract from exact You for sunset. After the spring equinox and before the autumn equinox, subtract from exact Mao for sunrise and add to exact You for sunset. Double the Mao-You interval and add or subtract it from the half-daylight value to obtain day and night lengths. After the spring equinox, add to get daylight quarters and subtract for night quarters; after the autumn equinox, reverse the operations.
34
月離用數
Constants for lunar motion
35
太陰每日平行四萬七千四百三十五秒，小餘０二一一七七。
The moon's mean daily motion is 47,435 seconds, fractional remainder 0.021177.
36
太陰每時四刻。 平行一千九百七十六秒，小餘四五九二一五七。
The moon advances four quarters per hour. Hourly mean motion is 1,976 seconds, fractional remainder 0.4592157.
37
月孛即最高，每日行四百０一秒，小餘０七七四七七。
The lunar apogee moves 401 seconds per day, fractional remainder 0.077477.
38
正交每日平行一百九十秒，小餘六四。
The ascending node moves 190 seconds per day, fractional remainder 0.64.
39
本輪半徑五十八萬。
Deferent epicycle radius: 580,000.
40
均輪半徑二十九萬。
Equating epicycle radius: 290,000.
41
負圈半徑七十九萬七千。
Auxiliary circle radius: 797,000.
42
次輪半徑二十一萬七千。
Secondary epicycle radius: 217,000.
43
次均輪半徑一十一萬七千五百。
Secondary equating epicycle radius: 117,500.
44
朔、望黃白大距四度五十八分三十秒。
Ecliptic-lunar latitude at syzygy: 4°58′30″.
45
兩弦黃白大距五度一十七分三十秒。
Ecliptic-lunar latitude at quadrature: 5°17′30″.
46
黃白大距中數五度０八分。
Mean ecliptic-lunar latitude: 5°08′.
47
黃白大距半較九分三十秒。
Half-variation of ecliptic-lunar latitude: 9′30″.
48
太陰平行應一宮０八度四十分五十七秒十六微。
Lunar mean-motion epoch: 1 mansion 8°40′57.16″.
49
月孛應三宮０四度四十九分五十四秒０九微。
Lunar apogee epoch: 3 mansions 4°49′54.09″.
50
正交應六宮二十七度十三分三十七秒四十八微。
Node epoch: 6 mansions 27°13′37.48″.
51
推月離法求天正冬至，同日躔。
To find the civil-year winter solstice for lunar motion, use the same method as for solar motion.
52
求太陰平行，置中積分，加氣應詳日躔。 小餘，不用日，下同。 減天正冬至小餘，得積日。 上考則減氣應小餘，加天正冬至小餘。 與太陰每日平行相乘，滿周天秒數去之，餘數收為宮度分。 以加太陰平行應，得太陰年根。 上考則減，又置太陰每日平行，以距天正冬至次日數乘之，得數為秒。 以宮度分收之，與年根相併，滿十二宮去之。 為太陰平行。
For lunar mean motion: take the mean accumulated interval and add the solar-term epoch offset, as described under solar motion. Use only the fractional remainder, not the day count; the same applies below. Subtract the winter-solstice fractional remainder to obtain accumulated days. When projecting backward, subtract the solar-term fractional remainder and add the winter-solstice fractional remainder. Multiply by the moon's daily mean motion, remove full circuits of heaven in seconds, and reduce the remainder to mansions, degrees, and minutes. Add to the lunar mean-motion epoch to obtain the year's root. Subtract when projecting backward; multiply the moon's daily mean motion by days since the day after the winter solstice to obtain seconds. Reduce to mansions, degrees, and minutes, add to the year's root, and remove full cycles of twelve mansions. This is the lunar mean longitude.
53
求月孛行，以積日見前條，下同。 與月孛每日行相乘，滿周天秒數去之，餘數收為宮度分。 以加月孛應，得月孛年根。 上考則減。 又置月孛每日行以距天正冬至次日數乘之，得數為秒，以宮度分收之，與年根相併，滿十二宮去之。 為月孛行。
For lunar apogee motion, use accumulated days as above; the same applies below. Multiply by the apogee's daily rate, remove full circuits in seconds, and reduce the remainder to mansions, degrees, and minutes. Add to the apogee epoch to obtain the apogee year's root. Subtract when projecting backward. Multiply the apogee's daily rate by days since the day after the winter solstice, reduce to mansions, degrees, and minutes, add to the year's root, and remove full twelve-mansion cycles. This is the lunar apogee longitude.
54
求正交平行，以積日與正交每日平行相乘，滿周天秒數去之，餘數收為宮度分，以減正交應，正交應不足減者，加十二宮減之。 得正交年根。 上考則加。 又置正交每日平行，以距天正冬至次日數乘之，得數為秒，以宮度分收之，以減年根，年根不足減者，加十二宮減之。 為正交平行。
For the node's mean motion: multiply accumulated days by the node's daily rate, reduce to mansions, degrees, and minutes, and subtract from the node epoch; if the epoch is too small, add twelve mansions before subtracting. This gives the node's year's root. Add when projecting backward. Multiply the node's daily rate by days since the day after the winter solstice, reduce to mansions, degrees, and minutes, and subtract from the year's root; if needed, add twelve mansions before subtracting. This is the node's mean longitude.
55
求用時太陰平行，以本日太陽均數變時，詳日躔。 得均數時差。 均數加者，時差為減； 均數減者，時差為加。 又以本日太陽黃、赤經度詳日躔。 相減餘數變時，得升度時差。 二分後為加，二至後為減。 乃以兩時差相加減，為時差總。 兩時差加減同號者，則相加為總，加者仍為加，減者仍為減。 加減異號者，則相減為總，加數大者為加，減數大者為減。 化秒，與太陰每時平行相乘為實，以一度化秒為法除之，得數為秒，以度分收之，得時差行。 以加減太陰平行，時差總為加者則減，減者則加。 為用時太陰平行。
For the moon's mean longitude at apparent time: convert the sun's equation to a time correction, as under solar motion. This yields the equation time correction. If the equation is added, the time correction is subtracted; if the equation is subtracted, the time correction is added. Also obtain the sun's ecliptic and equatorial longitudes for the day, as under solar motion. Subtract them and convert the difference to the ascension time correction. Add after the equinoxes, subtract after the solstices. Combine the two time corrections to obtain the total time difference. If both corrections share the same sign, add them; addition stays addition and subtraction stays subtraction. If the signs differ, subtract the smaller from the larger and take the sign of the larger value. Convert the total to seconds, multiply by the moon's hourly mean motion, divide by the seconds in one degree, and reduce the result to degrees and minutes as the time-difference displacement. Apply it to lunar mean motion with reversed sign: subtract if the total correction adds, add if it subtracts. This is the lunar mean longitude at apparent time.
56
求初實行，置用時太陰平行，減去月孛行，得引數。 用平三角形，以本輪半徑之半為對正角之邊，以引數為一角，求得對角之邊三因之。 又求得對又一角之邊，與本天半徑相加減。 引數九宮至二宮相加，三宮至八宮相減。 復用平三角形，以三因數為小邊，加減本天半徑數為大邊，正角在兩邊之中，求得對小邊之角為初均數，★求得對正角之邊。 即次輪最近點距地心之線。 乃置用時太陰平行，以初均數加減之，引數初宮至五宮為減，六宮以後為加。 為初實行。
For the first true longitude: subtract the apogee longitude from the apparent-time mean longitude to obtain the equation argument. By plane triangle: use half the deferent epicycle radius as the side opposite the right angle, take the equation argument as one angle, and triple the opposite side. Find the side opposite the other angle and add or subtract it from the deferent radius. Add for argument mansions 9–2, subtract for mansions 3–8. Apply another plane triangle with the tripled value as the short side and the adjusted deferent radius as the long side; the angle opposite the short side is the first equation, and the side opposite the right angle is also found. This is the line from Earth's center to the secondary epicycle's nearest point. Add or subtract the first equation from the apparent-time mean longitude: subtract for mansions 1–5, add from mansion 6 onward. This is the first true longitude.
57
求白道實行，置初實行，減本日太陽實行得次引。 即距日度。 用平三角形，以次輪最近點距地心線為一邊，倍次引之通弦本天半徑為一率，次引之正弦為二率，次輪半徑為三率，求得四率倍之即通弦。 為一邊； 以初均數與引數減半周之度引數不及半周，則與半周相減，如過半周，則減去半周。 相加，又以次引距象限度次引不及象限，則與象限相減； 如過象限及過三象限，則減去象限及三象限，用其餘； 如過二象限，則減去二象限，餘數仍與象限相減，為次引距象限度。 加減之，初均數減者，次引過象限或過三象限則相加，不過象限或過二象限則相減。 初均數加者反是。 為所夾之角，若相加過半周，則與全周相減，用其餘為所夾之角。 若相加適足半周或相減無餘，則無二均數。 若次引為初度，或適足半周，亦無二均數。 求得對通弦之角為二均數，如無初均數，以次輪心距地心為一邊，次輪半徑為一邊； 次引倍數為所夾之角，次引過半周者，與全周相減，用其餘； 在最高為所夾之內角，在最卑為所夾之外角，求得對次輪半徑之角為二均數。 隨定其加減號。 以初均數與均輪心距最卑之度相加，為加減泛限。 泛限適足九十度，則二均加減與初均同。 如泛限不足九十度，則與九十度相減，餘數倍之，為加減定限。 初均減者，以次引倍度； 初均加者，以次引倍度減全周之餘數，皆與定限較。 如泛限過九十度者，減去九十度，餘數倍之，為加減定限。 初均加者，以次引倍度； 初均減者，以次引倍度減全周之餘數，皆與定限較。 並以大於定限，則二均之加減與初均同； 小於定限者反是。 ★求得對角之邊，為次均輪心距地心線。 又以此線及次引，用平三角形，以次均輪心距地為一邊，次均輪半徑為一邊，次引倍度為所夾之角，次引過半周者，與全周相減，用其餘。 求得對次均輪半徑之角為三均數，隨定其加減號。 次引倍度不及半周為加，過半周為減。 乃以二均數與三均數相加減，為二三均數。 兩均數同號則相加，異號則相減。 以加減初實行，兩均數同為加者仍為加，同為減者仍為減。 一為加一為減者，加數大為加，減數大為減。 為白道實行。
For true lunar latitude motion: subtract the sun's true longitude from the first true longitude to obtain the secondary argument. This is the elongation from the sun. By plane triangle: one side is the line to the secondary epicycle's nearest point; use double the secondary argument's versed sine with deferent radius as the first ratio, the sine of the secondary argument as the second, and the secondary epicycle radius as the third; double the fourth ratio to obtain the chord. As one side; Combine the first equation with the argument reduced by a semicircle: if the argument is less than 180°, subtract it from 180°; if greater, subtract 180°. Add these, then take the secondary argument's distance from a quadrant: if less than 90°, subtract from 90°; if beyond 90° or 270°, subtract 90° or 270° and use the remainder; if beyond 180°, subtract 180° and again subtract from 90° to obtain the secondary argument's quadrant distance. Combine them: if the first equation subtracts, add when the secondary argument exceeds 90° or 270°, subtract otherwise. Reverse the rule when the first equation adds. This is the included angle; if the sum exceeds 180°, subtract from 360°. If the sum is exactly 180° or subtraction leaves zero, there is no second equation. There is also no second equation if the secondary argument is at 0° or exactly 180°. The angle opposite the chord is the second equation; without a first equation, use the line to the secondary epicycle center and the secondary epicycle radius as the two sides; take double the secondary argument as the included angle, reducing by 360° if it exceeds 180°; use the interior angle at apogee and the exterior at perigee; the angle opposite the secondary epicycle radius is the second equation. Determine the sign of addition or subtraction accordingly. Add the first equation to the arc from the equating epicycle center to perigee to obtain the general comparison limit. If the general limit is exactly 90°, the second equation has the same sign as the first. If less than 90°, subtract from 90° and double the remainder for the fixed comparison limit. If the first equation subtracts, compare the doubled secondary argument; if the first equation adds, compare the remainder of double the secondary argument minus 360° against the fixed limit. If the general limit exceeds 90°, subtract 90° and double the remainder for the fixed limit. If the first equation adds, compare the doubled secondary argument; if the first equation subtracts, compare the remainder of double the secondary argument minus 360° against the fixed limit. If the value exceeds the fixed limit, the second equation matches the first in sign; if less, the sign is reversed. The side opposite the angle is the line from Earth's center to the secondary equating epicycle center. With this line and the secondary argument, form a plane triangle using the line to the secondary equating epicycle center and its radius as sides and double the secondary argument as the included angle, reducing by 360° if needed. The angle opposite the secondary equating epicycle radius is the third equation; determine its sign accordingly. Add if double the secondary argument is under 180°, subtract if over. Combine the second and third equations to obtain the combined second-third correction. If both equations share a sign, add them; if their signs differ, subtract. Apply the result to the first true longitude: if both equations add, add; if both subtract, subtract. If one equation adds and the other subtracts, take the sign of the larger term. This is the true lunar-path longitude.
58
求黃道實行，用弧三角形，以黃白大距中數為一邊，大距半較為一邊，次引倍度為所夾之角，次引過半周與全周相減，用其餘。 求得對角之邊為黃白大距，並求得對半較之角為交均。 以交均加減正交平行，次引倍度不及半周為減，過半周為加。 得正交實行。 又加減六宮為中交實行，置白道實行，減正交實行，得距交實行。 以本天半徑為一率，黃白大距之餘弦為二率，距交實行之正切為三率，求得四率為黃道之正切。 檢表得度分，與距交實行相減，餘為升度差，以加減白道實行，距交實行不過象限，或過二象限為減，過象限及過三象限為加。 為黃道實行。
For true ecliptic longitude, solve a spherical triangle with the mean lunar latitude, the latitude half-difference, and double the secondary argument as the included angle, reducing by 360° if needed. The side opposite the angle gives the lunar latitude; the angle opposite the half-difference is the node equation. Apply the node equation to the node's mean longitude: subtract if double the secondary argument is under 180°, add if over. This gives the true node longitude. Add or subtract six mansions for the middle-node longitude; subtract the node longitude from the lunar-path true longitude to obtain the arc from the node. Use the deferent radius as the first ratio, the cosine of lunar latitude as the second, and the tangent of the arc from the node as the third; the fourth ratio is the ecliptic tangent. Look up degrees and minutes in the table and subtract from the arc from the node to obtain the ascension difference, then apply it to the lunar-path longitude: subtract if the arc is within one or two quadrants of the node, add if within one or three. This is the true ecliptic longitude.
59
求黃道緯度，以本天半徑為一率，黃白大距之正弦為二率，距交實行之正弦為三率，求得四率為正弦。 檢表得黃道緯度，距交實行初宮至五宮為黃道北，六宮至十一宮為黃道南。
For ecliptic latitude, use the deferent radius as the first ratio, the sine of lunar latitude as the second, and the sine of the arc from the node as the third; the fourth ratio is the sine. Look up the ecliptic latitude in the table; the latitude is north from mansions 0–5 of the arc from the node and south from mansions 6–11.
60
求四種宿度，依日躔求宿度法，求得本年黃道宿鈐。 以黃道實行、月孛行及正交、中交實行各度分視其足減宿鈐內某宿則減之，餘為四種宿度。
For the four lodge longitudes, use the solar-motion lodge method to obtain this year's ecliptic lodge pivot. For each of the ecliptic, apogee, node, and middle-node longitudes, subtract the appropriate lodge from the pivot wherever possible; the remainders are the four lodge positions.
61
求紀日值宿，同日躔。
For the lodge of the sexagenary day, use the same method as for solar motion.
62
求交宮時刻，以太陰本日實行與次日實行相減未過宮為本日，已過宮為次日。 餘為一率，刻下分為二率，太陰本日實行不用宮。 與三十度相減餘為三率，求得四率為距子正分數。 如法收之，得交宮時刻。
For the time the moon crosses a node mansion, compare today's and tomorrow's true longitudes: if the crossing has not yet occurred, use today; if it has, use tomorrow. Take the remainder as the first ratio and the fractional minutes as the second; use today's true longitude without its mansion. Subtract from 30° for the third ratio; the fourth ratio is the time from midnight. Reduce by the standard method to obtain the node-crossing time.
63
求太陰出入時刻，以本日太陽黃道經度求其相當赤道經度。 又用弧三角形，以太陰距黃極為一邊，黃極距北極為一邊，即黃赤大距。 太陰距冬至黃道經度為所夾之外角，過半周者與全周相減，用其餘。 求得對邊為太陰距北極度。 與九十度相減，得赤道緯度。 不及九十度者，與九十度相減，餘為北緯。 過九十度者，減去九十度，餘為南緯。 又求得近北極之角，為太陰距冬至赤道經度。 乃以本天半徑為一率，北極高度之正切為二率，太陰赤道緯度之正切為三率，求得四率為正弦。 檢表得太陰出入在卯酉前後赤道度，太陰在赤道北，出在卯正前，入在酉正後； 太陰在赤道南，出在卯正後，入在酉正前。 以加減前減後加。 太陰距太陽赤道度，太陰赤道經度內減去太陽赤道經度即得。 得數變時。 自卯正酉正後計之，出地自卯正後，入地自酉正後。 得何時刻，再加本時太陰行度之時刻，約一小時行三十分，變為時之二分。 即得太陰出入時刻。
For moonrise and moonset, convert the sun's ecliptic longitude for the day to the corresponding equatorial longitude. Next solve a spherical triangle with the moon's distance from the ecliptic pole and the obliquity as sides. Take the moon's ecliptic longitude from the winter solstice as the exterior angle, reducing by 360° if it exceeds 180°. The opposite side gives the moon's polar distance. Subtract from 90° to obtain equatorial declination. If under 90°, subtract from 90°; the remainder is north declination. If over 90°, subtract 90°; the remainder is south declination. The angle at the north pole gives the moon's equatorial longitude from the winter solstice. Use the deferent radius as the first ratio, the tangent of the observer's latitude as the second, and the tangent of the moon's declination as the third; the fourth ratio is the sine. Look up the equatorial degrees before and after Mao and You for moonrise and moonset; north of the equator, the moon rises before Mao and sets after You; south of the equator, it rises after Mao and sets before You. Apply the correction: subtract before the cardinal point, add after. The moon's elongation in equatorial longitude is the moon's equatorial longitude minus the sun's. Convert the result to time. Measure from after Mao for rising and from after You for setting. Find the base time, then add the time for the moon's motion in the current hour — about 30′ of motion per hour, converted to two fractional hours. This gives moonrise and moonset.
64
求合朔弦望，太陰實行與太陽實行同宮同度為合朔限，距三宮為上弦限，距六宮為望限，距九宮為下弦限，皆以太陰未及限度為本日，已過限度為次日。 乃以太陰、太陽本日實行與次日實行各相減，兩減餘數相較為一率，刻下分為二率，本日太陽實行加限度上弦加三宮，望加六宮，下弦加九宮。 減本日太陰實行，餘為三率，求得四率為距子正之分。 如法收之，得合朔弦望時刻。
For new moon and the quarters and full moon: conjunction is when the moon and sun share mansion and degree; limits fall three, six, and nine mansions apart for first quarter, full moon, and last quarter — use today if the moon has not reached the limit, tomorrow if it has. Subtract today's and tomorrow's true longitudes for moon and sun; take the ratio of the remainders as the first ratio and the fractional minutes as the second; add the limit to today's solar longitude — three mansions for first quarter, six for full moon, nine for last quarter. Subtract today's lunar longitude for the third ratio; the fourth ratio is the time from midnight. Reduce by the standard method to obtain the times of new moon and the quarters and full moon.
65
求正升斜升橫升，合朔日，太陰實行自子宮十五度至酉宮十五度為正升，自酉宮十五度至未宮初度為斜升，自未宮初度至寅宮十五度為橫升，自寅宮十五度至子宮十五度為斜升。
For direct, oblique, and horizontal ascension on new-moon day: direct from Zi 15° to You 15°, oblique from You 15° to Wei 0°, horizontal from Wei 0° to Yin 15°, oblique from Yin 15° back to Zi 15°.
66
求月大小，以前朔後朔相較，日干同者前月大，不同者前月小。
For month length, compare the preceding and following new moons: if the day stem matches, the preceding month is long (30 days); if not, short (29 days).
67
求閏月，以前後兩年有冬至之月為準。 中積十三月者，以無中氣之月，從前月置閏。 一歲中兩無中氣者，置在前無中氣之月為閏。
For the intercalary month, use the months containing winter solstice in the surrounding years as the standard. When thirteen mean months accumulate, intercalate after the month without a mid-term qi, following the preceding month. If two months in one year lack mid-term qi, intercalate after the first such month.
68
土星用數
Constants for Saturn
69
每日平行一百二十秒，小餘六０二二五五一。
Mean motion 120 seconds per day, fractional remainder 0.6022551.
70
最高日行十分秒之二又一九五八０三。
Apogee motion is 0.0002 second plus 0.195803 per day.
71
正交日行十分秒之一又一四六七二八。
Node motion is 0.0001 second plus 0.146728 per day.
72
本輪半徑八十六萬五千五百八十七。
Deferent radius 865,587.
73
均輪半徑二十九萬六千四百一十三。
Equating epicycle radius 296,413.
74
次輪半徑一百零四萬二千六百。
Secondary epicycle radius 1,042,600.
75
本道與黃道交角二度三十一分。
Orbital inclination to the ecliptic: 2°31′.
76
土星平行應七宮二十三度十九分四十四秒五十五微。
Saturn mean-motion epoch: 7 mansions 23°19′44.55″.
77
最高應十一宮二十八度二十六分六秒五微。
Apogee epoch: 11 mansions 28°26′6.05″.
78
正交應六宮二十一度二十分五十七秒二十四微。
Node epoch: 6 mansions 21°20′57.24″.
79
木星用數
Constants for Jupiter
80
每日平行二百九十九秒，小餘二八五二九六八。
Mean motion 299 seconds per day, fractional remainder 0.2852968.
81
最高日行十分秒之一又五八四三三。
Apogee motion is 0.0001 second plus 0.58433 per day.
82
正交日行百分秒之三又七二三五五七。
Node motion is 0.0003 second plus 0.723557 per day.
83
本輪半徑七十萬五千三百二十。
Deferent radius 705,320.
84
均輪半徑二十四萬七千九百八十。
Equating epicycle radius 247,980.
85
次輪半徑一百九十二萬九千四百八十。
Secondary epicycle radius 1,929,480.
86
本道與黃道交角一度十九分四十秒。
Orbital inclination to the ecliptic: 1°19′40″.
87
木星平行應八宮九度十三分十三秒十一微。
Jupiter mean-motion epoch: 8 mansions 9°13′13.11″.
88
最高應九宮九度五十一分五十九秒二十七微。
Apogee epoch: 9 mansions 9°51′59.27″.
89
正交應六宮七度二十一分四十九秒三十五微。
Node epoch: 6 mansions 7°21′49.35″.
90
火星用數
Constants for Mars
91
每日平行一千八百八十六秒，小餘六七００三五八。
Mean motion 1,886 seconds per day, fractional remainder 0.6700358.
92
最高日行十分秒之一又八三四三九九。
Apogee motion is 0.0001 second plus 0.834399 per day.
93
正交日行十分秒之一又四四九七二三。
Node motion is 0.0001 second plus 0.449723 per day.
94
本輪半徑一百四十八萬四千。
Deferent radius 1,484,000.
95
均輪半徑三十七萬一千。
Equating epicycle radius 371,000.
96
最小次輪半徑六百三十萬二千七百五十。
Minimum secondary epicycle radius 6,302,750.
97
本天高卑大差二十五萬八千五百。
Deferent eccentricity range: 258,500.
98
太陽高卑大差二十三萬五千。
Solar eccentricity range: 235,000.
99
本道與黃道交角一度五十分。
Orbital inclination to the ecliptic: 1°50′.
100
火星平行應二宮十三度三十九分五十二秒十五微。
Mars mean-motion epoch: 2 mansions 13°39′52.15″.
101
最高應八宮初度三十三分十一秒五十四微。
Apogee epoch: 8 mansions 0°33′11.54″.
102
正交應四宮十七度五十一分五十四秒七微，餘見日躔。
Node epoch: 4 mansions 17°51′54.07″; see solar motion for the remainder.
103
推土、木、火星法
Methods for Saturn, Jupiter, and Mars
104
求天正冬至，同日躔。
For the civil-year winter solstice, use the same method as for solar motion.
105
求三星平行，以積日詳月離。 與本星每日平行相乘，滿周天秒數去之，餘收為宮度分，為積日平行。 以加本星平行應，得本星年根。 上考則減。 又置本星每日平行，以所求距天正冬至次日數乘之，得數與年根相併，得本星平行。
For the three planets' mean longitudes, use accumulated days as in the lunar-motion section. Multiply by the planet's daily mean rate, reduce modulo a full circuit in seconds, and collect as mansions, degrees, and minutes to obtain the mean motion from accumulated days. Add to the planet's mean-motion epoch to obtain its year's root. Subtract when projecting backward. Multiply the planet's daily mean rate by days since the day after the winter solstice, add to the year's root, and obtain the planet's mean longitude.
106
求三星最高行，以積日與本星最高日行相乘，得數以加本星最高應，得最高年根。 上考則減。 又置本星最高日行，以所求距天正冬至次日數乘之，得數與年根相併，得本星最高行。
For apogee longitude: multiply accumulated days by the planet's apogee daily rate, add to the apogee epoch, and obtain the apogee year's root. Subtract when projecting backward. Multiply the planet's apogee daily rate by days since the day after the winter solstice, add to the year's root, and obtain the apogee longitude.
107
求三星正交行，以積日與本星正交日行相乘，得數以加本星正交應，得正交年根。 上考則減。 又置本星正交日行，以所求距天正冬至次日數乘之，得數與年根相併，得本星正交行。
For node longitude: multiply accumulated days by the planet's node daily rate, add to the node epoch, and obtain the node's year's root. Subtract when projecting backward. Multiply the planet's node daily rate by days since the day after the winter solstice, add to the year's root, and obtain the node longitude.
108
求三星初實行，置本星平行，減最高行，得引數。 用平三角形，以均輪半徑減本輪半徑為對正角之邊，以引數為一角，求得對引數角之邊及對又一角之邊。 又用平三角形，以對引數角之邊與均輪通弦相加求通弦法，詳月離。 為小邊，以對又一角之邊與本天半徑相加減引數三宮至八宮相減，九宮至二宮相加。 為大邊，正角在兩邊之中，求得對小邊之角為初均數。 並求得對正角之邊為次輪心距地心線，以初均數加減本星平行，引數初宮至五宮減，六宮至十一宮加。 得本星初實行。
For the first true longitude: subtract apogee longitude from mean longitude to obtain the equation argument. Form a plane triangle with the deferent minus equating epicycle radius as the side opposite the right angle and the equation argument as one angle; solve for the remaining sides. Next add the side opposite the equation-argument angle to the equating epicycle chord by the chord method, as in the lunar-motion section. This is the small side; add or subtract the other side and the deferent radius — subtract for argument mansions 3–8, add for mansions 9–2. This is the large side; with the right angle between them, the angle opposite the small side is the first equation. Also obtain the side opposite the right angle as the secondary epicycle center's geocentric distance; apply the first equation to the planet's mean longitude — subtract for argument mansions 0–5, add for mansions 6–11. This gives the planet's first true longitude.
109
求三星本道實行，置本日太陽實行減本星初實行，得次引。 即距日度。 用平三角形，以次輪心距地心線為一邊，次輪半徑為一邊，惟火星次輪半徑時時不同，求法詳後。 次引為所夾之外角，過半周者與全周相減，用其餘。 求得對次輪半徑之角為次均數，並求得對次引角之邊為星距地心線。 乃以次均數加減初實行，加減與初均相反。 得本星本道實行。 求火星次輪實半徑，以火星本輪全徑命為二千萬為一率，本天高卑大差為二率，均輪心距最卑之正矢為三率，引數與半周相減，即均輪心距最卑度。 求得四率為本天高卑差。 又以太陽本輪全徑命為二千萬為一率，太陽高卑大差為二率，本日太陽引數之正矢為三率，引數過半周者與全周相減，用其餘。 求得四率為太陽高卑差。 乃置火星最小次輪半徑，以兩高卑差加之，得火星次輪實半徑。
For true longitude in the planet's own path: subtract the planet's first true longitude from the sun's true longitude that day to obtain the secondary argument. This is the elongation from the sun. Form a plane triangle with the secondary epicycle center's geocentric distance as one side and the secondary epicycle radius as the other — Mars's secondary epicycle radius alone varies; see below. Take the secondary argument as the exterior angle; if it exceeds a semicircle, subtract from a full circuit and use the remainder. The angle opposite the secondary epicycle radius is the second equation; the side opposite the secondary-argument angle is the star's geocentric distance. Apply the second equation to the first true longitude with the sign opposite to the first equation. This gives the planet's true longitude in its own path. For Mars's true secondary epicycle semidiameter: use Mars's primary epicycle diameter (20,000,000) as the first ratio, the primary-heaven great high-low difference as the second, the versed sine of the equating epicycle center's perigee distance as the third, and the equation argument subtracted from a semicircle as the perigee distance in degrees. The fourth ratio is the primary-heaven high-low correction. Next use the sun's primary epicycle diameter (20,000,000) as the first ratio, the solar great high-low difference as the second, and the versed sine of that day's solar equation argument as the third; if the argument exceeds a semicircle, subtract from a full circuit and use the remainder. The fourth ratio is the solar high-low correction. Add both high-low corrections to Mars's minimum secondary epicycle semidiameter to obtain its true secondary epicycle semidiameter.
110
求三星黃道實行，置本星初實行，減本星正交行，得距交實行。 次輪心距正交。 乃以本天半徑為一率，本道與黃道交角之餘弦為二率，距交實行之正切為三率，求得四率為正切。 檢表得黃道度，與距交實行相減，得升度差，以加減本道實行，距交實行不過象限及過二象限為減，過象限及過三象限為加。 得本星黃道實行。
For ecliptic true longitude: subtract the planet's node longitude from its first true longitude to obtain the arc from the node. This is the secondary epicycle center's arc from the node. Use primary heaven radius as the first ratio, the cosine of the planet-path/ecliptic angle as the second, and the tangent of the arc from the node as the third; the fourth ratio is the tangent. Look up ecliptic degrees from the table, subtract from the arc from the node to get the ascension difference, and apply it to true longitude in the planet's path — subtract when the node arc is within a quadrant or passes two quadrants; add when it passes one or three quadrants. This gives the planet's ecliptic true longitude.
111
求三星視緯，以本天半徑為一率，本道與黃道交角之正弦為二率，距交實行之正弦為三率，求得四率為正弦，檢表得初緯。 又以本天半徑為一率，初緯之正弦為二率，次輪心距地心線為三率，求得四率為星距黃道線。 乃以星距地心線為一率，星距黃道線為二率，本天半徑為三率，求得四率為正弦。 檢表得本星視緯，隨定其南北。 距交實行初宮至五宮為黃道北，六宮至十一宮為黃道南。
For apparent latitude: use primary heaven radius as the first ratio, the sine of the planet-path/ecliptic angle as the second, and the sine of the arc from the node as the third; look up the resulting sine for the preliminary latitude. Next use primary heaven radius as the first ratio, the sine of the preliminary latitude as the second, and the secondary epicycle center's geocentric distance as the third; the fourth ratio is the star's distance from the ecliptic. Then use the star's geocentric distance as the first ratio, its distance from the ecliptic as the second, and primary heaven radius as the third; the fourth ratio is the sine. Look up the planet's apparent latitude and assign north or south accordingly. Node arcs in mansions 0–5 place the planet north of the ecliptic; mansions 6–11, south.
112
求黃道宿度及紀日，同日躔。
For ecliptic lodge degree and sexagenary day assignment, use the solar-motion method.
113
求交宮時刻，同月離。
For the moment of node mansion crossing, use the lunar-motion method.
114
求三星晨夕伏見定限度，視本星黃道實行與太陽實行同宮同度為合伏。 合伏後距太陽漸遠，為晨見東方順行。 順行漸遲，遲極而退為留退。 初退行距太陽半周為退沖，退沖之次日為夕見。 退行漸遲，遲極而順為留順。 初順行漸疾復近太陽，以至合伏，為夕不見。 其伏見限度，土星十一度，木星十度，火星十一度半。 合伏前後某日，太陽實行與本星實行相距近此限度，即以本星本日黃道實行，用弧三角形，以赤道地平交角為所知一角，夕，春分後用內角，秋分後用外角； 晨反是。 實行距春秋分度為對邊，黃赤大距為所知又一角，求得不知之對邊。 乃用所知兩邊對所知兩角，求得不知之又一角，夕，秋分後用內角，春分後用外角； 晨反是。 為限距地高。 乃用弧三角形，有正角，有黃道地平交角，即限距地高。 有本星伏見限度，為對交角之弧，求得對正角之弧，為距日黃道度。 若星當黃道無距緯，即為定限度。 又用弧三角形，有正角，有黃道地平交角，以本星距緯為對交角之弧，求得兩角間之弧，為加減差。 以加減距日黃道度，緯南加，緯北減。 得伏見定限度。 視本星距太陽度與定限度相近，如在合伏前某日，即為某日夕不見； 在合伏後某日，即為某日晨見。
For the fixed morning/evening visibility limits of the three planets: conjunction and invisibility occur when the planet's and sun's ecliptic true longitudes share the same mansion and degree. After conjunction/invisibility, increasing solar separation brings morning visibility in the east under direct motion. Direct motion slows until it stations and turns retrograde. Early retrograde at semicircular solar separation is opposition in retrograde; evening visibility begins the next day. Retrograde motion slows until it stations and turns direct again. Direct motion accelerates as the planet again approaches the sun until conjunction/invisibility — evening invisibility. Visibility limits: Saturn 11°, Jupiter 10°, Mars 11°30′. When solar and planetary true longitudes on some day near conjunction/invisibility approach this limit, take the planet's ecliptic true longitude that day and solve a spherical triangle with the equator-horizon angle known — for evening visibility after the spring equinox use the interior angle, after the autumn equinox the exterior angle; For morning visibility, reverse the rule. Take the true longitude's distance from the equinoxes as one opposite side and the great yellow-red distance as a known angle; solve for the other opposite side. With two known sides opposite two known angles, solve for the remaining angle — for evening, use the interior angle after the autumn equinox and the exterior angle after the spring equinox; For morning visibility, reverse the rule. This gives the limit height above the horizon. Next form a spherical right triangle with the ecliptic-horizon angle equal to the limit height above the horizon. With the planet's visibility limit as the arc opposite the crossing angle, the arc opposite the right angle is the ecliptic elongation from the sun. If the star lies on the ecliptic with no latitude offset, that value is the fixed limit. Again use a spherical right triangle with the ecliptic-horizon angle; take the planet's ecliptic latitude as the arc opposite the crossing angle; the arc between the angles is the correction. Apply this correction to the ecliptic elongation from the sun — add for south latitude, subtract for north. This yields the fixed visibility limit. When the planet's solar separation nears the fixed limit: if this occurs before conjunction/invisibility on a given day, that day is evening invisibility; if after conjunction/invisibility on a given day, that day is morning visibility.
115
求三星合伏時刻，視太陽實行將及本星實行，為合伏本日； 已過本星實行，為合伏次日。 求時刻，於太陽一日之實行即本日次日兩實行之較。 內減本星一日之實行為一率，餘同月離求朔、望。
For the moment of conjunction/invisibility: if the sun's true longitude is about to reach the planet's, the event falls on that day; if it has already passed the planet's true longitude, the next day. For the time fraction, use the sun's daily true motion — the difference between this day's and the next day's true longitudes. Use the planet's daily true motion as the divisor for the first ratio; proceed as in the lunar-motion section for new and full moon.
116
求三星退沖時刻，視本星黃道實行與太陽實行相距將半周，為退沖本日； 已過半周，為退沖次日。 求時刻之法，以太陽一日之實行與本星一日之實行相加為一率，餘同前。
For opposition in retrograde: when ecliptic separation of planet and sun approaches a semicircle, the event falls on that day; if a semicircle has already been exceeded, the next day. For the time fraction, add the sun's and planet's daily true motions as the first ratio; proceed as before for the remainder.
117
求同度時刻，以兩星一日之實行相加減兩星同行則減。 一順一逆則加。 為一率，刻下分為二率，兩星相距為三率，求得四率為距子正之分數，以時刻收之即得。 五星並同。
For the moment of equal longitude: combine the two bodies' daily true motions — subtract when both move in the same direction. Add when one is direct and the other retrograde. That sum is the first ratio, fractional parts under a quarter the second, the bodies' separation the third; the fourth ratio is the fraction from midnight — convert to hours and quarters. The same method applies to all five planets.
118
金星用數
Constants for Venus
119
每日平行三千五百四十八秒，小餘三三０五一六九。
Mean motion 3,548 seconds per day, fractional remainder 0.3305169.
120
最高日行十分秒之二又二七一０九五。
Apogee daily rate: two parts in ten thousand of a second plus 0.271095.
121
伏見每日平行二千二百十九秒，小餘四三一一八八六。
Visibility mean motion 2,219 seconds per day, fractional remainder 0.4311886.
122
本輪半徑二十三萬一千九百六十二。
Primary epicycle radius: 231,962.
123
均輪半徑八萬八千八百五十二。
Equating epicycle radius: 88,852.
124
次輪半徑七百二十二萬四千八百五十。
Secondary epicycle radius: 7,224,850.
125
次輪面與黃道交角三度二十九分。
Secondary epicycle plane/ecliptic angle: 3°29′.
126
金星平行應初宮初度二十分十九秒十八微。
Venus mean-motion epoch: 0 mansions 0°20′19.18″.
127
最高應六宮一度三十三分三十一秒四微。
Apogee epoch: 6 mansions 1°33′31.04″.
128
伏見應初宮十八度三十八分十三秒六微。
Visibility epoch: 0 mansions 18°38′13.06″.
129
水星用數
Constants for Mercury
130
每日平行與金星同。
Daily mean motion matches Venus.
131
最高日行十分秒之二又八八一一九三。
Apogee daily rate: two parts in ten thousand of a second plus 0.881193.
132
伏見每日平行一萬一千一百八十四秒，小餘一一六五二四八。
Visibility mean motion 11,184 seconds per day, fractional remainder 0.1165248.
133
本輪半徑五十六萬七千五百二十三。
Primary epicycle radius: 567,523.
134
均輪半徑一十一萬四千六百三十二。
Equating epicycle radius: 114,632.
135
次輪半徑三百八十五萬。
Secondary epicycle radius: 3,850,000.
136
次輪心在大距，與黃道交角五度四十分。
At greatest elongation the secondary epicycle center/ecliptic angle is 5°40′.
137
次輪心在正交，與黃道交角北五度五分十秒，其交角較三十四分五十秒。 與大距交角相較，後仿此。 南六度三十一分二秒，其交角較五十一分二秒。
At the node the ecliptic angle is 5°05′10″ north; the crossing-angle differential is 34′50″. Compare with the greatest-elongation angle; follow this pattern below. South 6°31′02″; crossing-angle differential 51′02″.
138
次輪心在中交，與黃道交角北六度十六分五十秒，其交角較三十六分五十秒。 南四度五十五分三十二秒，其交角較四十四分二十八秒。
At the middle node the ecliptic angle is 6°16′50″ north; crossing-angle differential 36′50″. South 4°55′32″; crossing-angle differential 44′28″.
139
水星平行應與金星同。
Mercury mean-motion epoch matches Venus.
140
最高應十一宮三度三分五十四秒五十四微。
Apogee epoch: 11 mansions 3°03′54.54″.
141
伏見應十宮一度十三分十一秒十七微，餘見日躔。
Visibility epoch: 10 mansions 1°13′11.17″; see solar motion for the remainder.
142
推金、水星法
Methods for Venus and Mercury
143
求天正冬至，同日躔。
For the civil-year winter solstice, use the same method as for solar motion.
144
求金、水本星平行，同土、木、火星。
For Venus and Mercury mean longitudes, use the Saturn, Jupiter, and Mars method.
145
求金、水最高行，同土、木、火星。
For Venus and Mercury apogee longitudes, use the Saturn, Jupiter, and Mars method.
146
求金、水伏見平行，同本星平行。
For Venus and Mercury visibility mean motion, use each planet's mean-longitude method.
147
求金、水正交行，置本星最高平行，金星減十六度，水星加減六宮，即得。
For node longitude: from the planet's apogee mean longitude, subtract 16° for Venus or add/subtract six mansions for Mercury.
148
求金星初實行，用本星引數求初均數，以加減本星平行，為本星初實行。 及求次輪心距地心線，並同土、木、火星。
For Venus's first true longitude: derive the first equation from the equation argument and apply it to mean longitude. Finding the secondary epicycle center's geocentric distance follows the Saturn, Jupiter, and Mars method.
149
求水星初實行，用平三角形，以本輪半徑為一邊，均輪半徑為一邊，以引數三倍之為所夾之外角，過半周者與全周相減，用其餘。 求其對角之邊，並對均輪半徑之角。 又用平三角形，以本天半徑為大邊，以對角之邊為小邊，以對均輪半徑之角與均輪心距最卑度相加減，引數不及半周者，與半周相減； 過半周者，減去半周，即均輪心距最卑度。 加減之法，視三倍引數不過半周則加，過半周則減。 為所夾之角，求得對小邊之角為初均數，並求得對角之邊為次輪心距地心線。 以初均數加減水星平行，引數初宮至五宮為減，六宮至十一宮為加。 得水星初實行。
For Mercury's first true longitude: form a plane triangle with primary and equating epicycle radii as sides and triple the equation argument as the exterior angle; if it exceeds a semicircle, subtract from a full circuit and use the remainder. Solve for the side opposite the angle and the angle opposite the equating epicycle radius. Next form a plane triangle with primary heaven radius as the large side and the opposite side as the small side; combine the angle opposite the equating epicycle radius with the equating epicycle center's perigee distance — if the argument is less than a semicircle, subtract from a semicircle; if it exceeds a semicircle, subtract a semicircle to obtain the perigee distance in degrees. Add when triple the equation argument is less than a semicircle; subtract when it exceeds one. With that as the contained angle, the angle opposite the small side is the first equation and the side opposite the angle is the secondary epicycle center's geocentric distance. Apply the first equation to Mercury's mean longitude — subtract for argument mansions 0–5, add for mansions 6–11. This yields Mercury's first true longitude.
150
求金、水伏見實行，置本星伏見平行，加減本星初均數，引數初宮至五宮為加，六宮至十一宮為減。 即得。
For Venus and Mercury visibility true longitude: take visibility mean motion and apply the first equation, adding for mansions 1–5 and subtracting for mansions 6–11. This gives the result.
151
求金、水黃道實行，用平三角形，以本星次輪心距地心線為一邊，本星次輪半徑為一邊，本星伏見實行為所夾之外角，過半周者與全周相減，用其餘。 求得對次輪半徑之角為次均數，並求得對角之邊為本星距地心線。 以次均數加減初實行，伏見實行初宮至五宮為加，六宮至十一宮為減。 得本星黃道實行。
For ecliptic true longitude of Venus and Mercury: form a plane triangle with the line to the secondary epicycle center and its radius as sides and visibility true longitude as the exterior angle, reducing by 360° if over 180°. The angle opposite the secondary epicycle radius is the second equation, and the opposite side is the planet's geocentric distance. Apply the second equation to the first true longitude, adding for visibility mansions 1–5 and subtracting for 6–11. This yields the planet's ecliptic true longitude.
152
求金、水距次交實行，置本星初實行，減本星正交行，為距交實行。 與本星伏見實行相加，得本星距次交實行。
For distance from the node: subtract the node's longitude from the first true longitude. Add visibility true longitude to obtain distance from the secondary node.
153
求金、水視緯，以本天半徑為一率，本星次輪與黃道交角之正弦為二率，金星交角惟一，水星交角則時時不同，須求實交角用之，法詳後。 本星距次交實行之正弦為三率，求得四率為正弦，檢表得本星次緯。 又以本天半徑為一率，本星次緯之正弦為二率，本星次輪半徑為三率，求得四率為本星距黃道線。 乃以本星距地心線為一率，本星距黃道線為二率，本天半徑為三率，求得四率為正弦，檢表得本星視緯，隨定其南北。 初宮至五宮為黃道北，六宮至十一宮為黃道南。
For apparent latitude: use deferent radius as the first ratio and the sine of the secondary epicycle–ecliptic crossing angle as the second; Venus uses a fixed angle, but Mercury needs the true crossing angle computed below. Use the sine of distance from the secondary node as the third ratio, look up the fourth ratio in the table for secondary latitude. Next use deferent radius, sine of secondary latitude, and secondary epicycle radius to obtain the planet's distance from the ecliptic. Then use geocentric distance, ecliptic distance, and deferent radius to look up apparent latitude and assign north or south. Mansions 1–5 are north of the ecliptic, mansions 6–11 are south.
154
求水星實交角，以半徑一千萬為一率，交角較化秒為二率，距交實行九宮至二宮用正交交角較，三宮至八宮用中交交角較，仍視其南北用之。 距交實行之正弦為三率，求得四率為交角差。 置交角，用交角之法與用交角較同。 以交角差加減之，距交實行九宮至二宮，星在黃道北則加，南則減； 三宮至八宮反是。 得實交角。
For Mercury's true orbital inclination: use radius 10,000,000 and the crossing-angle difference in seconds; apply the ascending-node difference for mansions 9–2 and the mid-node difference for 3–8, according to hemisphere. The sine of distance from the node gives the crossing-angle correction. Take the nominal crossing angle; apply the correction by the same procedure as the crossing-angle difference. Apply the correction: for mansions 9–2, add if north of the ecliptic and subtract if south; reverse for mansions 3–8. This yields the true crossing angle.
155
求黃道宿度及紀日，同日躔。
For ecliptic lodge longitude and sexagenary day, use the solar-motion method.
156
求交宮時刻，同月離。
For node-crossing time, use the lunar-motion method.
157
求金、水晨夕伏見定限度，本星實行與太陽實行同宮同度為合伏，合伏後距太陽漸遠； 夕見西方順行，順行漸遲，遲極而退為留退。 初退行漸近太陽，則夕不見，復與太陽同度為合退伏。 自是又漸遠太陽，晨見東方。 仍退行漸遲，遲極而順為留順。 初順行漸疾，復近太陽，以至合伏，為晨不見。 其伏見限度，金星為五度，水星為十度。 其求定限度之法，與土、木、火星同，視本星距太陽度與定限相近。 如在合伏前某日，即為某日晨不見； 合伏後某日，即為某日夕見； 合退伏前某日，即為某日夕不見； 合退伏後某日，即為某日晨見。
For Venus and Mercury visibility limits: conjunction-invisibility occurs when planet and sun share the same longitude; afterward the planet moves away from the sun; it appears in the west at dusk, moving direct but ever slower until it stations and turns retrograde. Retrograde motion brings it back toward the sun until evening visibility ends at conjunction-retrograde-invisibility. It then separates from the sun again and appears in the east at dawn. Still retrograde, it slows until it stations and resumes direct motion. Direct motion accelerates as it nears the sun again to conjunction-invisibility, ending morning visibility. Visibility limits are 5° for Venus and 10° for Mercury. Find the fixed limit as for the outer planets by comparing solar elongation to the limit. The day before conjunction-invisibility when the limit is reached is morning invisibility; the day after conjunction-invisibility when the limit is reached is evening visibility; before conjunction-retrograde-invisibility is evening invisibility; after conjunction-retrograde-invisibility is morning visibility.
158
求金、水合伏時刻，視本星實行將及太陽實行為合伏本日，已過太陽實行為合伏次日。 求時刻之法，與月離求朔、望時刻之法同。
For conjunction-invisibility time: if the planet is about to catch the sun, the event is today; if it has passed, tomorrow. Compute the moment by the same method used for new and full moons.
159
求金、水合退伏時刻，視太陽實行將及本星實行為合退伏本日，已過本星實行為合退伏次日。 求時刻之法，與土、木、火星求退沖時刻之法同。
For conjunction-retrograde-invisibility: if the sun is about to catch the planet, the event is today; if it has passed, tomorrow. Compute the moment by the outer-planet retrograde-opposition method.
160
恆星用數
Constants for fixed stars
161
見日躔。
See under solar motion.
162
推恆星法求黃道經度，以距康熙壬子年數減一，得積年歲差，乘之。 收為度分，與康熙壬子年恆星表經度相加，得各恆星本年經度。 求赤道經緯度，用弧三角形，以星距黃極為一邊，黃赤大距為一邊，本年星距夏至前後為所夾之角，求得對星距黃極邊之角。 夏至前用本度，夏至後與周天相減用其餘度。 自星紀宮初度起算，為各恆星赤道經度。 又求得對原角之邊，與象限相減，餘為赤道緯度。 減象限為北，減去象限為南。
For fixed-star ecliptic longitude: multiply years since Kangxi renzi (minus one) by annual precession. Reduce to degrees and minutes and add to the Kangxi renzi star-table longitudes for each star's current ecliptic longitude. For equatorial coordinates: use a spherical triangle with polar distance, obliquity, and the star's distance from the summer solstice as the included angle. Before the summer solstice use the value directly; after, subtract from 360°. Count equatorial longitude from the start of the Xingji mansion. The side opposite the original angle minus 90° gives equatorial latitude. If the star is inside the quadrant the latitude is north; if outside, south.
163
求中星，以刻下分為一率，本日太陽實行與次日太陽實行相減餘為二率，以所設時刻化分為三率，求得四率，與本日太陽實行相加，得本時太陽黃道經度。 用弧三角形，推得太陽赤道經度，以所設時刻變赤道度一時變為十五度，一分變為十五分，一秒變為十五秒。 加減半周，不及半周則加半周，過半周則減半周。 得本時太陽距午後度。 與太陽赤道經度相加，得本時正午赤道經度。 視本年恆星赤道經度同者，即為中星。
For the culminating star: proportion from sub-quarter parts, the sun's daily motion, and the given time to obtain the sun's ecliptic longitude at that moment. Derive solar equatorial longitude by spherical triangle, converting time to arc at 15° per hour, 15′ per minute, and 15″ per second. Adjust by 180°: add if under 180°, subtract if over. This gives the sun's hour-angle from noon. Add to solar equatorial longitude to obtain the meridian's equatorial longitude at that moment. The fixed star whose equatorial longitude matches is the star on the meridian.