1
志二十四
Treatise 24
2
時憲五
Shixian Calendar 5
3
康熙甲子元法下
Kangxi jiazi calendrical system — lower section
4
月食用數
Constants for lunar eclipse calculation
5
朔策二十九日五三０五九三。
New-moon interval constant: 29.530593 days.
6
望策十四日七六五二九六五。
Full-moon interval constant: 14.7652965 days.
7
太陽平行，朔策一十萬四千七百八十四秒，小餘三０四三二四。
Sun's mean motion per synodic month: 104,784 seconds, fractional remainder 0.304324.
8
太陽引數，朔策一十萬四千七百七十九秒，小餘三五八八六五。
Sun's equation per synodic month: 104,779 seconds, fractional remainder 0.358865.
9
太陰引數，朔策九萬二千九百四十秒，小餘二四八五九。
Moon's anomaly per synodic month: 92,940 seconds, fractional remainder 0.24859.
10
太陰交周，朔策十一萬０四百十四秒，小餘０一六五七四。
Moon's nodal motion per synodic month: 110,414 seconds, fractional remainder 0.016574.
11
太陽平行，望策十四度三十三分十二秒０九微。
Sun's mean motion per half-month: 14°33′12.09″.
12
太陽引數，望策十四度三十三分０九秒四十一微。
Sun's equation per half-month: 14°33′09.41″.
13
太陰引數，望策六宮十二度五十四分三十秒０七微。
Moon's anomaly per half-month: 6 mansions 12°54′30.07″.
14
太陰交周，望策六宮十五度二十分０七秒。
Moon's nodal motion per half-month: 6 mansions 15°20′07″.
15
太陽一小時平行一百四十七秒，小餘八四七一０四九。
Sun's hourly mean motion: 147 seconds, fractional remainder 0.8471049.
16
太陽一小時引數一百四十七秒，小餘八四０一二七。
Sun's hourly equation: 147 seconds, fractional remainder 0.840127.
17
太陰一小時引數一千九百五十九秒，小餘七四七六五四二。
Moon's hourly anomaly: 1,959 seconds, fractional remainder 0.7476542.
18
太陰一小時交周一千九百八十四秒，小餘四０二五四九。
Moon's hourly nodal motion: 1,984 seconds, fractional remainder 0.402549.
19
月距日一小時平行一千八百二十八秒，小餘六一二一一０八。
Moon's hourly mean elongation from the sun: 1,828 seconds, fractional remainder 0.6121108.
20
太陽光分半徑六百三十七。
Solar semidiameter (light-seconds): 637.
21
太陰實半徑二十七。
Lunar true semidiameter: 27.
22
地半徑一百。
Terrestrial semidiameter: 100.
23
太陽最高距地一千０十七萬九千二百０八，與地半徑之比例，為十一萬六千二百。
Solar apogee distance: 10,179,208 units; ratio to Earth's semidiameter: 116,200.
24
太陰最高距地一千０十七萬二千五百，與地半徑之比例，為五千八百一十六。
Lunar apogee distance: 10,172,500 units; ratio to Earth's semidiameter: 5,816.
25
朔應二十六日三八五二六六六。
Epoch new-moon offset: 26.3852666 days.
26
首朔太陽平行應初宮二十六度二十分四十二秒五十七微。
At the epoch new moon, the sun's mean longitude should be 26°20′42.57″ in the first mansion.
27
首朔太陽引數應初宮十九度一十分二十七秒二十一微。
At the epoch new moon, the sun's equation should be 19°10′27.21″ in the first mansion.
28
首朔太陰引數應九宮十八度三十四分二十六秒十六微。
At the epoch new moon, the moon's anomaly should be 18°34′26.16″ in the ninth mansion.
29
首朔太陰交周應六宮初度三十分五十五秒十四微，餘見日躔、月離。
At the epoch new moon, the moon's nodal longitude should be 0°30′55.14″ in the sixth mansion; other constants appear in the sections on solar motion and lunar distance.
30
推月食法
Procedure for computing lunar eclipses
31
求天正冬至，同日躔。
To find the winter solstice of the civil year, use the same method as in the section on solar motion.
32
求紀日，以天正冬至日數加一日，得紀日。
To find the cycle-day count, add one day to the winter-solstice day-number of the civil year.
33
求首朔，先求得積日同月離。 置積日減朔應，得通朔。 上考則加。 以朔策除之，得數加一為積朔。 餘數轉減朔策為首朔。 上考則除得之數即積朔，不用加一。 餘數即首朔，不用轉減。
To find the epoch new moon, first compute the accumulated days by the same method as in the lunar-distance section. Subtract the new-moon epoch offset from the accumulated days to obtain the general new-moon interval. When projecting backward into antiquity, add instead of subtract. Divide by the new-moon interval constant; add one to the quotient to obtain the count of accumulated new moons. Subtract the remainder from the new-moon interval constant to obtain the epoch new-moon fraction. For backward projection, the quotient itself is the accumulated new-moon count; do not add one. The remainder itself is the epoch new-moon fraction; no reverse subtraction is needed.
34
求太陰入食限，置積朔，以太陰交周朔策乘之，滿周天秒數去之，餘為積朔太陰交周。 加首朔太陰交周應，得首朔太陰交周。 上考則置首朔交周應減積朔交周。 又加太陰交周望策，再以交周朔策遞加十三次，得逐月望太陰平交周。 視某月交周入可食之限，即為有食之月。 交周自五宮十五度０六分至六宮十四度五十四分，自十一宮十五度０六分至初宮十四度五十四分，皆可食之限。 再於實交周詳之。
To determine whether the moon enters the eclipse zone: multiply the accumulated new-moon count by the nodal motion per synodic month, cast out full circuits of the celestial seconds, and the remainder is the nodal longitude at accumulated new moon. Add the epoch nodal longitude at new moon to obtain the nodal longitude at the first new moon. For backward projection, subtract the accumulated nodal longitude from the epoch nodal offset instead. Add the nodal motion per half-month, then add the nodal motion per synodic month thirteen times in succession to obtain the mean nodal longitude at each month's full moon. If a given month's nodal longitude falls within the possible-eclipse zone, that month may witness a lunar eclipse. The possible-eclipse zone runs from 15°06′ in the fifth mansion to 14°54′ in the sixth, and from 15°06′ in the eleventh mansion to 14°54′ in the first. Refine the result using the true nodal longitude.
35
求平望，以太陰入食限月數與朔策相乘，加望策，再加首朔日分及紀日，滿紀法去之，餘為平望日分。 自初日起甲子，得平望干支，以刻下分通其小餘，如法收之。 初時起子正，得時刻分秒。
To find the mean full moon: multiply the eclipse-month index by the new-moon interval, add the half-month interval, then add the epoch new-moon day-fraction and cycle-day count; cast out full cycle periods, and the remainder is the mean full-moon day-fraction. Count the sexagenary day from the epoch jiazi day to obtain the mean full-moon day stem-branch; convert the fractional remainder through sub-quarter parts and reduce by the standard rule. Starting from exact zi as the initial hour, obtain the hour, quarter, minute, and second.
36
求太陽平行，置積朔，加太陰入食限之月數為通月，以太陽平行朔策乘之。 滿周天秒數去之，加首朔太陽平行應，上考則減。 又加太陽平行望策，即得。
To find the sun's mean longitude: take the accumulated new-moon count, add the eclipse-month index to obtain the general month-count, and multiply by the sun's mean motion per synodic month. Cast out full circuits of celestial seconds, add the epoch sun mean longitude at new moon; for backward projection, subtract instead. Add the sun's mean motion per half-month to obtain the result.
37
求太陽平引，置通月，以太陽引數朔策乘之，去周天秒數，加首朔太陽引數應，上考則減。 又加太陽引數望策，即得。
To find the sun's mean equation: multiply the general month-count by the sun's equation per synodic month, cast out full circuits, and add the epoch sun equation; for backward projection, subtract. Add the sun's equation per half-month to obtain the result.
38
求太陰平引，置通月，以太陰引數朔策乘之，去周天秒數，加首朔太陰引數應，上考則減。 又加太陰引數望策，即得。
To find the moon's mean anomaly: multiply the general month-count by the moon's anomaly per synodic month, cast out full circuits, and add the epoch moon anomaly; for backward projection, subtract. Add the moon's anomaly per half-month to obtain the result.
39
求太陰實引，以一小時化秒為一率，太陰一小時引數為二率，距時秒為三率，求得四率為秒。 以度分收之，為太陰引弧。 依距時加減號。 以加減太陰平引，得實引。
To find the moon's true anomaly: use one hour converted to seconds as the first ratio, the moon's hourly anomaly as the second, and the interval time in seconds as the third; the fourth ratio gives the correction in seconds. Convert to degrees and minutes to obtain the moon's anomaly arc. Apply addition or subtraction according to the sign of the interval time. Add or subtract this from the moon's mean anomaly to obtain the true anomaly.
40
求實望，以太陽實引復求均數為日實均，並求得太陽距地心線。 即實均第二平三角形對正角之邊。 以太陰實引復求均數為月實均，并求得太陰距地心線。 法同太陽。 兩均相加減為實距弧。 加減與距弧同。 依前求距時法，求得時分為實距時，以加減平望，加減與距時同。 得實望。 加滿二十四時，則實望進一日，不足減者，借一日作二十四時減之，則實望退一日。
To find the true full moon: from the sun's true anomaly compute the equation of center as the sun's true equation, and also obtain the sun's geocentric distance. This is the side opposite the right angle in the second plane triangle of the true equation. From the moon's true anomaly compute the equation of center as the moon's true equation, and also obtain the moon's geocentric distance. The method is the same as for the sun. Add or subtract the two equations to obtain the true elongation arc. The sign of addition or subtraction follows that of the elongation arc. By the previous method for interval time, convert to hours and minutes as the true interval time, and add or subtract from the mean full moon with the same sign. This yields the true full moon. If the addition reaches twenty-four hours, the true full moon advances one day; if subtraction is insufficient, borrow one day as twenty-four hours, and the true full moon retreats one day.
41
求實交周，以一小時化秒為一率，太陰一小時交周為二率，實距時化秒為三率，求得四率為秒，以度分收之，為交周距弧。 以加減太陰交周，依實距時加減號。 又以月實均加減之，為實交周。 若實交周入必食之限，為有食。 自五宮十七度四十三分０五秒至六宮十二度十六分五十五秒，自十一宮十七度四十三分０五秒至初宮十二度十六分五十五秒，為必食之限。 不入此限者，不必布算。
To find the true nodal longitude: use one hour in seconds, the moon's hourly nodal motion, and the true interval time in seconds; convert the result to degrees and minutes as the nodal interval arc. Add or subtract from the moon's nodal longitude according to the sign of the true interval time. Further add or subtract the moon's true equation to obtain the true nodal longitude. If the true nodal longitude falls within the certain-eclipse zone, an eclipse will occur. The certain-eclipse zone runs from 17°43′05″ in the fifth mansion to 12°16′55″ in the sixth, and from 17°43′05″ in the eleventh mansion to 12°16′55″ in the first. If the longitude does not fall within this zone, no further calculation is required.
42
求太陽黃赤道實經度，以一小時化秒為一率，太陽一小時平行為二率，實距時化秒為三率，求得四率為秒，以度分收之，為太陽距弧。 依時距時加減號。 以加減太陽平行，又以日實均加減之，即黃道經度。 又用弧三角形求得赤道經度。 詳月離求太陰出入時刻條。
To find the sun's true ecliptic and equatorial longitude: use one hour in seconds, the sun's hourly mean motion, and the true interval time; convert to degrees and minutes as the solar interval arc. Apply addition or subtraction according to the sign of the interval time. Add or subtract from the sun's mean longitude and further apply the sun's true equation to obtain the ecliptic longitude. Use a spherical triangle to obtain the equatorial longitude as well. See the section on lunar distance for the method of finding the moon's rising and setting times.
43
求實望用時，以日實均變時為均數時差，以升度差黃赤道經度之較。 變時為升度時差，兩時差相加減為時差總，加減之法，詳月離求用時平行條。 以加減實望，為實望用時。 距日出後日入前九刻以內者，可以見食。 九刻以外者全在晝，不必算。
To find the apparent time of true full moon: convert the sun's true equation to time as the equation time correction; the rising-degree correction is the difference between ecliptic and equatorial longitudes. Convert the rising-degree difference to time; combine the two time corrections; for the rules of addition and subtraction, see the section on lunar distance concerning apparent time and mean motion. Apply these corrections to the true full moon to obtain the apparent time of true full moon. An eclipse can be observed if it falls within nine quarters of an hour after sunrise or before sunset. If it lies beyond nine quarters, the event falls entirely in daylight and need not be calculated.
44
求食甚時刻，以本天半徑為一率，黃白大距之餘弦為二率，實交周之正切為三率，求得四率為正切，檢表得食甚交周。 與實交周相減，為交周升度差。 又以太陰一小時引數與太陰實引相加，依月離求初均法算之，為後均。 以後均與月實均相加減，兩均同號相減，異號相加。 得數又與一小時月距日平行相加減，兩均同加，後均大則加，小則減。 兩均同減，後均大則減，小則加。 兩均一加一減，其加減從後均。 為月距日實行。 乃以月距日實行化秒為一率，一小時化秒為二率，交周升度差化秒為三率，求得四率為秒。 以時分收之，得食甚距時。 以加減實望用時，實交周初宮六宮為減，五宮十一宮為加。 為食甚時刻。
To find the time of greatest eclipse: use the celestial radius as the first ratio, the cosine of the great ecliptic-lunar distance as the second, and the tangent of the true nodal longitude as the third; look up the resulting tangent in the tables to obtain the nodal longitude at greatest eclipse. Subtract this from the true nodal longitude to obtain the nodal rising-degree difference. Add the moon's hourly anomaly to its true anomaly and compute by the first-equation method in the lunar-distance section to obtain the second equation. Combine the second equation with the moon's true equation: subtract if they share the same sign, add if opposite. Combine this result with the hourly mean moon-sun elongation: if both equations call for addition, add the larger second equation and subtract the smaller; if both call for subtraction, subtract the larger and add the smaller. If both equations call for subtraction, subtract the larger second equation and add the smaller. If one equation calls for addition and the other for subtraction, follow the sign of the second equation. This yields the moon's true elongation from the sun. Use the true moon-sun elongation in seconds as the first ratio, one hour in seconds as the second, and the nodal rising-degree difference in seconds as the third; the fourth ratio gives the correction in seconds. Convert to hours and minutes to obtain the interval time to greatest eclipse. Apply this to the apparent time of true full moon: subtract if the true nodal longitude is in the first or sixth mansion, add if in the fifth or eleventh. This yields the time of greatest eclipse.
45
求食甚距緯，以本天半徑為一率，黃白大距之正弦為二率，實交周之正弦為三率，求得四率為正弦，檢表得食甚距緯。 實交周初宮五宮為北，六宮十一宮為南。
To find the latitude distance at greatest eclipse: use the celestial radius, the sine of the great ecliptic-lunar distance, and the sine of the true nodal longitude; look up the resulting sine in the tables. If the true nodal longitude is in the first or fifth mansion, the latitude is north; if in the sixth or eleventh, south.
46
求太陰半徑，以太陰最高距地為一率，地半徑比例數為二率，太陰距地心線內減去次均輪半徑為三率，求得四率為太陰距地。 又以太陰距地為一率，太陰實半徑為二率，本天半徑為三率，求得四率為正弦。 檢表得太陰半徑。
To find the lunar semidiameter: use the lunar apogee distance as the first ratio, the Earth-semidiameter ratio as the second, and the moon's geocentric distance minus the deferent epicycle radius as the third; the fourth ratio gives the moon's distance from Earth. Further use the moon's distance from Earth as the first ratio, its true semidiameter as the second, and the celestial radius as the third; the fourth ratio is a sine. Look up the result in the tables to obtain the lunar semidiameter.
47
求地影半徑，以太陽最高距地為一率，地半徑比例數為二率，太陽距地心線為三率，求得四率為太陽距地。 又以太陽光分半徑內減地半徑為一率，太陽距地為二率，地半徑為三率，求得四率為地影之長。 又以地影長為一率，地半徑為二率，本天半徑為三率，求得四率為正弦，檢表得地影角。 又以本天半徑為一率，地影角之正切為二率，地影長內減太陰距地為三率，求得四率為太陰所入地影之闊。 乃以太陰距地為一率，地影之闊為二率，本天半徑為三率，求得四率為正切，檢表得地影半徑。
To find the Earth's shadow semidiameter: use the solar apogee distance, the Earth-semidiameter ratio, and the sun's geocentric distance; the fourth ratio gives the sun's distance from Earth. Use the solar light-semidiameter minus the Earth's semidiameter as the first ratio, the sun's distance as the second, and the Earth's semidiameter as the third; the fourth ratio gives the length of the Earth's shadow. Use the shadow length, Earth's semidiameter, and celestial radius; look up the resulting sine in the tables to obtain the Earth-shadow angle. Use the celestial radius, the tangent of the shadow angle, and the shadow length minus the moon's distance; the fourth ratio gives the width of shadow entered by the moon. Use the moon's distance, the shadow width, and the celestial radius; look up the resulting tangent to obtain the Earth's shadow semidiameter.
48
求食分，以太陰全徑為一率，十分為二率，並徑太陰地影兩半徑相併。 內減食甚距緯之較並徑不及減距緯即不食。 為三率，求得四率即食分。
To find the eclipse magnitude: use the moon's full diameter as the first ratio and ten as the second; the third ratio is the sum of the lunar and Earth-shadow semidiameters. Subtract the latitude distance at greatest eclipse; if the combined diameter cannot cover the latitude separation, no eclipse occurs. This serves as the third ratio; the fourth ratio is the eclipse magnitude.
49
求初虧、復圓時刻，以食甚距緯之餘弦為一率，並徑之餘弦為二率，半徑千萬為三率，求得四率為餘弦，檢表得初虧、復圓距弧。 又以月距日實行化秒為一率，一小時化秒為二率，初虧、復圓距弧化秒為三率，求得四率為秒。 以時分收之，為初虧、復圓距時。 以加減食甚時刻，得初虧、復圓時刻。 減得初虧，加得復圓。
To find the times of first and last contact: use the cosine of the latitude at greatest eclipse, the cosine of the combined diameter, and the radius of ten million; look up the arc intervals in the tables. Use the true moon-sun elongation in seconds, one hour in seconds, and the contact interval arcs in seconds; the fourth ratio gives the correction in seconds. Convert to hours and minutes to obtain the interval times to first and last contact. Apply these to the time of greatest eclipse to obtain the times of first and last contact. Subtract for first contact; add for last contact.
50
求食既、生光時刻，以食甚距緯之餘弦為一率，兩半徑較之餘弦為二率，半徑千萬為三率，求得四率為餘弦，檢表得食既、生光距弧。 又以月距日實行化秒為一率，一小時化秒為二率，食既、生光距弧化秒為三率，求得四率為秒。 以時分收之，為食既、生光距時。 以加減食甚時刻，得食既、生光時刻。 減得食既，加得生光。
To find the times when totality begins and light returns: use the cosine of the latitude at greatest eclipse, the cosine of the difference of the two semidiameters, and the radius of ten million; look up the arc intervals in the tables. Use the true moon-sun elongation in seconds, one hour in seconds, and the totality interval arcs in seconds; the fourth ratio gives the correction in seconds. Convert to hours and minutes to obtain the interval times to totality and light return. Apply these to the time of greatest eclipse to obtain the times of totality and light return. Subtract for the beginning of totality; add for the return of light.
51
求食限總時，以初虧、復圓距時倍之，即得。
To find the total duration of the eclipse, double the interval time from greatest eclipse to first or last contact.
52
求太陰黃道經緯度，置太陽黃道經度，加減六宮，過六宮則減去六宮，不及六宮，則加六宮。 再加減食甚距弧，又加減黃白升度差，求升度差法，詳月離求黃道實行條。 得太陰黃道經度。 求緯度，詳月離。
To find the moon's ecliptic longitude and latitude: take the sun's ecliptic longitude and add or subtract six mansions; if the result exceeds six mansions, subtract six; if insufficient, add six. Further add or subtract the interval arc at greatest eclipse and the ecliptic-lunar rising-degree difference; for the method of finding the rising-degree difference, see the lunar-distance section on true ecliptic motion. This yields the moon's ecliptic longitude. To find the latitude, see the section on lunar distance.
53
求太陰赤道經緯度，詳月離求太陰出入時刻條。
To find the moon's equatorial longitude and latitude, see the section on lunar distance concerning rising and setting times.
54
求宿度，同日躔。
To find the lodge position, use the same method as in the section on solar motion.
55
求黃道地平交角，以食甚時刻變赤道度，每時之四分變一度。 又於太陽赤道經度內減三宮，不及減者，加十二宮減之。 餘為太陽距春分赤道度。 兩數相加，滿全周去之。 為春分距子正赤道度。 與半周相減，得春分距午正東西赤道度。 過半周者，減去半周，為午正西。 不及半周者，去減半周，為午正東。 春分距午正東西度過象限者，與半周相減，餘為秋分距午正東西赤道度。 秋分距午東西，與春分相反。 以春秋分距午正東西度與九十度相減，餘為春秋分距地平赤道度。 乃用為弧三角形之一邊，以黃赤大距及赤道地平交角即赤道地平上高度，春分午西、秋分午東者用此。 若春分午東、秋分午西者，則以此度與半周相減用其餘。 為邊傍之兩角，求得對邊之角，為黃道地平交角。 春分午東、秋分午西者，得數即為黃道地平交角。 春分午西、秋分午東者，則以得數與半周相減，餘為黃道地平交角。
To find the ecliptic-horizon angle: convert the time of greatest eclipse to equatorial degrees, with each quarter-hour equal to one degree. Subtract three mansions from the sun's equatorial longitude; if insufficient, add twelve mansions before subtracting. The remainder is the sun's equatorial longitude measured from the spring equinox. Add the two values and cast out full circuits. This gives the spring equinox's equatorial longitude from exact zi. Subtract from a semicircle to obtain the spring equinox's east-west equatorial distance from noon. If it exceeds a semicircle, subtract the semicircle to place it west of noon. If it falls short of a semicircle, subtract it from the semicircle to place it east of noon. If the spring equinox's east-west distance from noon exceeds a quadrant, subtract from a semicircle; the remainder is the autumn equinox's east-west equatorial distance from noon. The autumn equinox's east-west relation to noon is the reverse of the spring equinox. Subtract the equinox's east-west distance from noon from ninety degrees; the remainder is the equinox's equatorial distance from the horizon. Use this as one side of a spherical triangle, with the obliquity of the ecliptic and the equator-horizon angle (the equatorial altitude); apply when the spring equinox lies west of noon or the autumn equinox east of noon. If the spring equinox lies east of noon or the autumn equinox west of noon, subtract this value from a semicircle and use the remainder. With these as the two adjacent angles, solve for the opposite angle to obtain the ecliptic-horizon angle. When the spring equinox is east of noon or the autumn equinox west of noon, the result is the ecliptic-horizon angle directly. When the spring equinox is west of noon or the autumn equinox east of noon, subtract the result from a semicircle; the remainder is the ecliptic-horizon angle.
56
求黃道高弧交角，以黃道地平交角之正弦為一率，赤道地平交角之正弦為二率，春秋分距地平赤道度之正弦為三率，求得四率為正弦，檢表得春秋分距地平黃道度。 又視春秋分在地平上者，以太陰黃道經度與三宮、九宮相減，春分與三宮相減，秋分與九宮相減。 餘為太陰距春秋分黃道度。 春秋分宮度大於太陰宮度，為距春秋分前； 反此則在後。 又以春秋分距地平黃道度與太陰距春秋分黃道度相加減，為太陰距地平黃道度，春秋分在午正西者，太陰在分後則加，在分前則減； 春秋分在午正東者反是。 隨視其距限之東西。 春秋分在午正西者，太陰距地平黃道度不及九十度為限西，過九十度為限東； 春秋分在午正東者反是。 乃以太陰距地平黃道度之餘弦為一率，本天半徑為二率，黃道地平交角之餘切為三率，求得四率為正切，檢表得黃道高弧交角。
To find the ecliptic altitude arc angle: use the sines of the ecliptic-horizon angle, the equator-horizon angle, and the equinox's equatorial distance from the horizon; look up the result to obtain the equinox's ecliptic distance from the horizon. When the equinox is on the horizon, subtract the moon's ecliptic longitude from the third or ninth mansion: subtract the third mansion at spring equinox and the ninth at autumn equinox. The remainder is the moon's ecliptic longitude measured from the equinox. If the equinox mansion longitude exceeds the moon's mansion longitude, the moon lies before the equinox; otherwise it lies after. Combine the equinox's ecliptic distance from the horizon with the moon's distance from the equinox to obtain the moon's ecliptic distance from the horizon; when the equinox is west of noon, add if the moon is after the equinox and subtract if before; when the equinox is east of noon, reverse the rule. Adjust according to whether the eclipse limit lies to the east or west. When the equinox is west of noon, if the moon's ecliptic distance from the horizon is less than ninety degrees the limit is west; if greater, east; when the equinox is east of noon, reverse the rule. Use the cosine of the moon's ecliptic distance from the horizon, the celestial radius, and the cotangent of the ecliptic-horizon angle; look up the resulting tangent to obtain the ecliptic altitude arc angle.
57
求初虧、復圓定交角，置食甚交周，以初虧、復圓距弧加減之，得初虧、復圓交周。 減得初虧，加得復圓。 乃以本天半徑為一率，黃白大距之正弦為二率，初虧交周之正弦為三率，求得四率為正弦，檢表得初虧距緯。 又以復圓交周之正弦為三率，一率二率同前。 求得四率為正弦，檢表得復圓距緯。 交周初宮、五宮為緯北，六宮、十一宮為緯南。 又以並徑之正弦為一率，初虧、復圓距緯之正弦各為二率，半徑千萬為三率，各求得四率為正弦，檢表得初虧、復圓兩緯差角。 以兩緯差角各與黃道高弧交角相加減，得初虧、復圓定交角。 初虧限東，緯南則加，緯北則減； 限西，緯南則減，緯北則加。 復圓反是。 若初虧、復圓無緯差角，即以黃道高弧交角為定交角。
To find the fixed contact angles: take the nodal longitude at greatest eclipse and add or subtract the contact interval arcs to obtain the nodal longitudes at first and last contact. Subtract for first contact; add for last contact. Use the celestial radius, the sine of the great ecliptic-lunar distance, and the sine of the nodal longitude at first contact; look up the result to obtain the latitude distance at first contact. Use the sine of the nodal longitude at last contact as the third ratio, with the first and second unchanged. Look up the resulting sine to obtain the latitude distance at last contact. If the nodal longitude is in the first or fifth mansion, latitude is north; if in the sixth or eleventh, south. Use the sine of the combined diameter and the sines of the latitude distances at first and last contact, with the radius of ten million; look up the two latitude-difference angles. Combine each latitude-difference angle with the ecliptic altitude arc angle to obtain the fixed contact angles. At first contact in the eastern limit: add for south latitude, subtract for north; in the western limit: subtract for south latitude, add for north. For last contact, reverse the rule. If there is no latitude-difference angle at contact, use the ecliptic altitude arc angle directly as the fixed contact angle.
58
求初虧、復圓方位，食在限東者，定交角在四十五度以內，初虧下偏左，復圓上偏右。 四十五度以外，初虧左偏下，復圓右偏上。 適足九十度，初虧正左，復圓正右。 過九十度，初虧左偏上，復圓右偏下。 食在限西者，定交角四十五度以內，初虧上偏左，復圓下偏右。 四十五度以外，初虧左偏上，復圓右偏下。 適足九十度，初虧正左，復圓正右。 過九十度，初虧左偏下，復圓右偏上。 京師黃平象限恆在天頂南，定方位如此。 在天頂北反是。
To find the orientation at first and last contact: when the eclipse is in the eastern limit and the fixed angle is within forty-five degrees, first contact lies lower-left and last contact upper-right. Beyond forty-five degrees, first contact is to the lower left and last contact to the upper right. At exactly ninety degrees, first contact is directly to the left and last contact directly to the right. Past ninety degrees, first contact is to the upper left and last contact to the lower right. When the eclipse is in the western limit and the fixed angle is within forty-five degrees, first contact lies upper-left and last contact lower-right. Beyond forty-five degrees, first contact is to the upper left and last contact to the lower right. At exactly ninety degrees, first contact is directly to the left and last contact directly to the right. Past ninety degrees, first contact is to the lower left and last contact to the upper right. At the capital, the ecliptic-horizon quadrant always lies south of the zenith; determine orientation accordingly. If the quadrant lies north of the zenith, reverse the orientation.
59
求帶食分秒，以本日日出或日入時分初虧或食甚在日入前者，為帶食出地，用日入分。 食甚或復圓在日出後者，為帶食入地，用日出分。 與食甚時分相減，餘為帶食距時。 以一小時化秒為一率，一小時月距日實行化秒為二率，帶食距時化秒為三率，求得四率為秒。 以度分收之，為帶食距弧。 又以半徑千萬為一率，帶食距弧之餘切為二率，食甚距緯之餘弦為三率，求得四率為餘切，檢表得帶食兩心相距之弧。 乃以太陰全徑為一率，十分為二率，並徑內減帶食兩心相距之餘為三率，求得四率，即帶食分秒。
To find the magnitude of a horizon eclipse: if first contact or greatest eclipse occurs before sunset, it is an eclipse emerging from the horizon—use the sunset time. If greatest eclipse or last contact occurs after sunrise, it is an eclipse setting below the horizon—use the sunrise time. Subtract from the time of greatest eclipse; the remainder is the interval time to the horizon. Use one hour in seconds, the hourly true moon-sun elongation in seconds, and the horizon interval time in seconds; the fourth ratio gives the correction in seconds. Convert to degrees and minutes to obtain the horizon interval arc. Use the radius of ten million, the cotangent of the horizon interval arc, and the cosine of the latitude at greatest eclipse; look up the arc between the two centers at the horizon. Use the moon's full diameter and ten as ratios; subtract the center separation from the combined diameter as the third ratio; the fourth ratio is the horizon eclipse magnitude.
60
求各省月食時刻，以各省距京師東西偏度變時，每偏一度，變時之四分。 加減京師月食時刻，即得。 東加，西減。
To find lunar eclipse times for each province: convert by the province's longitudinal offset from the capital, with each degree equal to one quarter-hour. Add or subtract this from the capital's eclipse time to obtain the provincial time. Add for provinces to the east; subtract for those to the west.
61
求各省月食方位，以各省赤道高度及月食時刻，依京師推方位法求之，即得。
To find the eclipse orientation for each province: use the provincial equatorial altitude and eclipse time, applying the same orientation method as for the capital.
62
繪月食圖，先作橫豎二線，直角相交，橫線當黃道，豎線當黃道經圈，用地影半徑度於中心作圈以象闇虛。 次以並徑為度作外虛圈，為初虧、復圓之限。 又以兩徑較為度作內虛圈，為食既、生光之限。 復於外虛圈上周豎線或左或右，取五度為識，視實交周初宮、十一宮作識於右，五宮、六宮作識於左。 乃自所識作線過圈心至外虛圈下周，即為白道經圈。 於此線上自圈心取食甚距緯作識，即食甚月心所在。 從此作十字橫線，即為白道。 割內外虛圈之點，為食甚前後四限月心所在。 末以月半徑為度，於五限月心各作小圈，五限之象具備。
To draw the lunar eclipse diagram: first draw horizontal and vertical lines at right angles—the horizontal for the ecliptic, the vertical for the ecliptic meridian—and at the center draw a circle with the Earth's shadow semidiameter to represent the umbra. Next draw an outer circle with the combined diameter as the limit for first and last contact. Draw an inner circle with the difference of the two semidiameters as the limit for totality and the return of light. On the outer circle along the vertical line, mark five degrees to the left or right: if the true nodal longitude is in the first or eleventh mansion, mark on the right; if in the fifth or sixth, on the left. Draw a line from the mark through the center to the lower edge of the outer circle; this is the lunar-path meridian. Along this line from the center, mark the latitude distance at greatest eclipse to locate the moon's center at greatest eclipse. From this point draw a horizontal cross-line representing the lunar path. The points where the lunar path cuts the inner and outer circles mark the moon's center at the four contact phases before and after greatest eclipse. Finally draw small circles at each of the five moon-center positions using the lunar semidiameter; the five phases of the eclipse are now complete.
63
日食用數
Constants for solar eclipse calculation
64
太陽實半徑五百零七，餘見月食推日食法。
Solar true semidiameter: 507; for the remaining constants and methods, see the lunar-eclipse section on computing solar eclipses.
65
求天正冬至，同日躔。
To find the winter solstice of the civil year, use the same method as in the section on solar motion.
66
求紀日，同月食。
To find the cycle-day count, use the same method as for lunar eclipses.
67
求首朔，同月食。
To find the epoch new moon, use the same method as for lunar eclipses.
68
求太陰入食限，與月食求逐月望平交周之法同，惟不用望策，即為逐月朔平交周。 視某月交周入可食之限，即為有食之月。 交周自五宮九度零八分至六宮八度五十一分，又自十一宮二十一度零九分至初宮二十度五十二分，皆為可食之限。
To determine whether the moon enters the eclipse zone, use the same method as for finding each month's mean nodal longitude at full moon in the lunar-eclipse section, but without the half-month interval—this gives the mean nodal longitude at each new moon. If a given month's nodal longitude falls within the possible-eclipse zone, that month may witness a solar eclipse. The possible-eclipse zone runs from 9°08′ in the fifth mansion to 8°51′ in the sixth, and from 21°09′ in the eleventh mansion to 20°52′ in the first.
69
求平朔，
To find the mean new moon,
70
求太陽平行，
To find the sun's mean longitude,
71
求太陽平引，
To find the sun's mean equation,
72
求太陰平引，以上四條，皆與月食求平望之法同，惟不加望策。
To find the moon's mean anomaly—the above four steps all follow the lunar-eclipse method for mean full moon, but without adding the half-month interval.
73
求太陽實引，同月食。
To find the sun's true anomaly, use the same method as for lunar eclipses.
74
求太陰實引，同月食。
To find the moon's true anomaly, use the same method as for lunar eclipses.
75
求實朔，與月食求實望之法同。
To find the true new moon, use the same method as for finding the true full moon in the lunar-eclipse section.
76
求實交周，與月食同。 視實交周入食限為有食。 自五宮十一度四十五分至六宮六度十四分，又自十一宮二十三度四十六分至初宮十八度十五分，為實朔可食限。
To find the true nodal longitude, use the same method as for lunar eclipses. If the true nodal longitude falls within the eclipse zone, an eclipse will occur. The possible-eclipse zone at true new moon runs from 11°45′ in the fifth mansion to 6°14′ in the sixth, and from 23°46′ in the eleventh mansion to 18°15′ in the first.
77
求太陽黃赤道實經度，同月食。
To find the sun's true ecliptic and equatorial longitude, use the same method as for lunar eclipses.
78
求實朔用時，同月食求實望用時。 實朔用時，在日出前或日入後。 五刻以外，則在夜，不必算。
To find the apparent time of true new moon, use the same method as for the apparent time of true full moon in the lunar-eclipse section. The apparent time of true new moon must fall before sunrise or after sunset. If it lies more than five quarters of an hour from sunrise or sunset, the event falls at night and need not be calculated.
79
求食甚用時，與月食求食甚時刻法同。
To find the apparent time of greatest eclipse, use the same method as in the lunar-eclipse section.
80
求用時春秋分距午赤道度，以太陽赤道經度減三宮，不足減者，加十二宮減之。 為太陽距春分後赤道度。 又以食甚用時變為赤道度，加減半周，過半周者減去半周，不及半周者加半周。 為太陽距午正赤道度。 兩數相加，滿全周去之。 其數不過象限者，為春分距午西赤道度。 過一象限者，與半周相減，餘為秋分距午東赤道度。 過二象限者，則減去二象限，餘為秋分距午西赤道度。 過三象限者，與全周相減，餘為春分距午東赤道度。
To find the equinox's east-west equatorial distance from noon at apparent time: subtract three mansions from the sun's equatorial longitude; if insufficient, add twelve mansions before subtracting. This gives the sun's equatorial longitude measured from the spring equinox. Convert the apparent time of greatest eclipse to equatorial degrees and add or subtract a semicircle; if it exceeds a semicircle, subtract one; if insufficient, add one. This gives the sun's equatorial longitude measured from noon. Add the two values and cast out full circuits. If the result does not exceed a quadrant, it is the spring equinox's equatorial distance west of noon. If past one quadrant, subtract from a semicircle; the remainder is the autumn equinox's equatorial distance east of noon. If past two quadrants, subtract two quadrants; the remainder is the autumn equinox's equatorial distance west of noon. If past three quadrants, subtract from the full circuit; the remainder is the spring equinox's equatorial distance east of noon.
81
求用時春秋分距午黃道度，以黃赤大距之餘弦為一率，本天半徑為二率，春秋分距午赤道度之正切為三率，求得四率為正切，檢表得用時春秋分距午黃道度。
To find the equinox's ecliptic distance from noon at apparent time: use the cosine of the obliquity, the celestial radius, and the tangent of the equatorial distance; look up the result in the tables.
82
求用時正午黃赤距緯，以本天半徑為一率，黃赤大距之正弦為二率，距午黃道度之正弦為三率，求得四率為正弦，檢表得用時正午黃赤距緯。
To find the ecliptic-equator latitude separation at noon for apparent time: use the celestial radius, the sine of the obliquity, and the sine of the ecliptic distance from noon; look up the result in the tables.
83
求用時黃道與子午圈交角，以距午黃道度之正弦為一率，距午赤道度之正弦為二率，本天半徑為三率，求得四率為正弦，檢表得用時黃道與子午圈交角。
To find the angle between the ecliptic and the meridian at apparent time: use the sines of the ecliptic and equatorial distances from noon and the celestial radius; look up the result in the tables.
84
求用時正午黃道宮度，置用時春秋分距午黃道度，春分加減三宮。 午西加三宮，午東與三宮相減。 秋分加減九宮，午西加九宮，午東與九宮相減。 得用時正午黃道宮度。
To find the ecliptic mansion at noon for apparent time: take the equinox's ecliptic distance from noon and, at spring equinox, add or subtract three mansions. If west of noon, add three mansions; if east of noon, subtract from three mansions. At autumn equinox, add or subtract nine mansions; if west of noon, add nine; if east of noon, subtract from nine. This yields the ecliptic mansion at noon for apparent time.
85
求用時正午黃道高，置赤道高度，北極高度減象限之餘。 以正午黃赤距緯加減之，黃道三宮至八宮加，九宮至二宮減。 即得。
To find the ecliptic altitude at noon for apparent time: set the equatorial altitude to the north polar altitude minus a quadrant. Add or subtract the noon ecliptic-equator latitude separation: add for mansions three through eight, subtract for nine through two. This yields the result.
86
求用時黃平象限距午，以黃道子午圈交角之餘弦為一率，本天半徑為二率，正午黃道高之正切為三率，求得四率為正切，檢表得度分。 與九十度相減，餘為黃平象限距午之度分。
To find the ecliptic-horizon quadrant's distance from noon at apparent time: use the cosine of the ecliptic-meridian angle, the celestial radius, and the tangent of the noon ecliptic altitude; look up the degrees and minutes in the tables. Subtract from ninety degrees; the remainder is the quadrant's distance from noon.
87
求用時黃平象限宮度，以黃平象限距午度分與正午黃道宮度相加減，正午黃道宮度初宮至五宮為加，六宮至十一宮為減，若正午黃道高過九十度，則反其加減。 即得。
To find the quadrant mansion at apparent time: combine the quadrant distance from noon with the noon ecliptic mansion—add for mansions one through five, subtract for six through eleven; if noon ecliptic altitude exceeds ninety degrees, reverse the signs. This yields the result.
88
求用時月距限，以太陽黃道經度與用時黃平象限宮度相減，餘為月距限度，隨視其距限之東西。 太陽黃道經度大於黃平象限宮度者為限東，小者為限西。
To find the moon's distance from the eclipse limit at apparent time: subtract the quadrant mansion from the sun's ecliptic longitude; adjust according to whether the limit lies east or west. If the sun's ecliptic longitude exceeds the quadrant mansion, the limit is east; if less, west.
89
求用時限距地高，以本天半徑為一率，黃道子午圈交角之正弦為二率，正午黃道高之餘弦為三率，求得四率為餘弦，檢表得限距地高。
To find the limit's altitude above the horizon at apparent time: use the celestial radius, the sine of the ecliptic-meridian angle, and the cosine of the noon ecliptic altitude; look up the result in the tables.
90
求用時太陰高弧，以本天半徑為一率，限距地高之正弦為二率，月距限之餘弦為三率，求得四率為正弦，檢表得太陰高弧。
To find the moon's altitude arc at apparent time: use the celestial radius, the sine of the limit altitude, and the cosine of the moon's distance from the limit; look up the result in the tables.
91
求用時黃道高弧交角，以月距限之正弦為一率，限距地高之餘切為二率，本天半徑為三率，求得四率為正切，檢表得黃道高弧交角。
To find the ecliptic altitude arc angle at apparent time: use the sine of the moon's distance from the limit, the cotangent of the limit altitude, and the celestial radius; look up the result in the tables.
92
求用時白道高弧交角，置黃道高弧交角，以黃白大距加減之，食甚交周初宮、十一宮，月距限東則加，限西則減。 五宮、六宮反是。 即得。 如過九十度，限東變為限西，限西變為限東，不足減者反減之。 則黃平象限在天頂南者，白平象限在天頂北； 黃平象限在天頂北者，白平象限在天頂南。
To find the lunar-path altitude arc angle at apparent time: take the ecliptic altitude arc angle and add or subtract the great ecliptic-lunar distance; if the nodal longitude at greatest eclipse is in the first or eleventh mansion, add when the limit is east and subtract when west. For the fifth and sixth mansions, reverse the rule. This yields the result. If the angle exceeds ninety degrees, east and west limits interchange; if subtraction is insufficient, reverse it. If the ecliptic-horizon quadrant lies south of the zenith, the lunar-path horizon quadrant lies north; if the ecliptic-horizon quadrant lies north of the zenith, the lunar-path horizon quadrant lies south.
93
求太陽距地，詳月食求地影半徑條。
To find the sun's distance from Earth, see the lunar-eclipse section on the Earth's shadow semidiameter.
94
求太陰距地，詳月食求太陰半徑條。
To find the moon's distance from Earth, see the lunar-eclipse section on the lunar semidiameter.
95
求用時高下差，用平三角形，以地半徑為一邊，太陽距地為一邊，用時太陰高弧與象限相減，餘為所夾之角，求得對太陽距地邊之角。 減去一象限，為太陽視高。 與太陰高弧相減，餘為太陽地半徑差。 又用平三角形，以地半徑為一邊，太陰距地為一邊，用時太陰高弧與象限相減，餘為所夾之角，求得對太陰距地邊之角。 減去一象限，為太陰視高。 與高弧相減，餘為太陰地半徑差。 兩地半徑差相減，得高下差。
To find the parallax in altitude at apparent time: in a plane triangle with the Earth's semidiameter and the sun's distance as sides, use the moon's altitude arc minus a quadrant as the included angle and solve for the angle opposite the sun's distance. Subtract a quadrant to obtain the sun's apparent altitude. Subtract from the moon's altitude arc; the remainder is the sun's parallax due to the Earth's semidiameter. Again in a plane triangle with the Earth's semidiameter and the moon's distance, use the moon's altitude arc minus a quadrant and solve for the angle opposite the moon's distance. Subtract a quadrant to obtain the moon's apparent altitude. Subtract from the altitude arc; the remainder is the moon's parallax due to the Earth's semidiameter. Subtract the two parallaxes to obtain the difference in altitude parallax.
96
求用時東西差，以半徑千萬為一率，白道高弧交角之餘弦為二率，高下差之正切為三率，求得四率為正切，檢表得用時東西差。
To find the east-west correction at apparent time: use the radius of ten million, the cosine of the lunar-path altitude arc angle, and the tangent of the altitude parallax; look up the result in the tables.
97
求食甚近時，以月距日實行化秒為一率，一小時化秒為二率，東西差化秒為三率，求得四率為秒。 以時分收之，為近時距分。 以加減食甚用時，月距限西則加，限東則減，仍視白道高弧交角變限不變限為定。 得食甚近時。
To find the near time of greatest eclipse: use the true moon-sun elongation in seconds, one hour in seconds, and the east-west correction in seconds; the fourth ratio gives the correction in seconds. Convert to hours and minutes as the near-time interval. Apply to the apparent time of greatest eclipse: add if the limit is west, subtract if east—still governed by whether the lunar-path altitude arc angle reverses the limit. This yields the near time of greatest eclipse.
98
求近時春秋分距午赤道度，以食甚近時變赤道度求之，餘與前用時之法同。 後諸條仿此，但皆用近時度分立算。
To find the equinox's equatorial distance from noon at near time: convert using the near time of greatest eclipse; the remaining steps follow the apparent-time method. The following steps follow the same pattern, but all calculations use the near-time values.
99
求近時春秋分距午黃道度。
Find the equinox's ecliptic distance from noon at near time.
100
求近時正午黃赤距緯。
Find the noon ecliptic-equator latitude separation at near time.
101
求近時黃道與子午圈交角。
Find the ecliptic-meridian angle at near time.
102
求近時正午黃道宮度。
Find the noon ecliptic mansion at near time.
103
求近時正午黃道高。
Find the noon ecliptic altitude at near time.
104
求近時黃平象限距午。
Find the ecliptic-horizon quadrant's distance from noon at near time.
105
求近時黃平象限宮度。
Find the ecliptic-horizon quadrant mansion at near time.
106
求近時月距限，置太陽黃道經度，加減用時東西差，依近時距分加減號。 為近時太陰黃道經度。 與近時黃平象限宮度相減，為近時月距限。 餘同用時。
To find the moon's distance from the limit at near time: take the sun's ecliptic longitude and add or subtract the apparent-time east-west correction according to the sign of the near-time interval. This gives the moon's ecliptic longitude at near time. Subtract the near-time quadrant mansion to obtain the moon's distance from the limit. The remaining steps follow the apparent-time method.
107
求近時限距地高。
Find the limit's altitude above the horizon at near time.
108
求近時太陰高弧。
Find the moon's altitude arc at near time.
109
求近時黃道高弧交角。
Find the ecliptic altitude arc angle at near time.
110
求近時白道高弧交角。
Find the lunar-path altitude arc angle at near time.
111
求近時高下差。
Find the parallax in altitude at near time.
112
求近時東西差。
Find the east-west correction at near time.
113
求食甚視行，倍用時東西差減近時東西差，即得。
To find the apparent motion at greatest eclipse: double the apparent-time east-west correction and subtract the near-time correction.
114
求食甚真時，以視行化秒為一率，近時距分化秒為二率，用時東西差化秒為三率，求得四率為秒。 以時分收之，為真時距分，以加減食甚用時，得食甚真時。 加減與近時距分同。
To find the true time of greatest eclipse: use the apparent motion in seconds, the near-time interval in seconds, and the apparent-time east-west correction in seconds; the fourth ratio gives the correction in seconds. Convert to hours and minutes as the true-time interval, apply to the apparent time of greatest eclipse, and obtain the true time of greatest eclipse. The sign of addition or subtraction follows that of the near-time interval.
115
求真時春秋分距午赤道度，以食甚真時變赤道度求之，餘與用時之法同。 後諸條仿此，但皆用真時度分立算。
To find the equinox's equatorial distance from noon at true time: convert using the true time of greatest eclipse; the remaining steps follow the apparent-time method. The following steps follow the same pattern, but all calculations use the true-time values.
116
求真時春秋分距午黃道度。
Find the equinox's ecliptic distance from noon at true time.
117
求真時正午黃赤距緯。
Find the noon ecliptic-equator latitude separation at true time.
118
求真時黃道與子午圈交角。
Find the ecliptic-meridian angle at true time.
119
求真時正午黃道宮度。
Find the noon ecliptic mansion at true time.
120
求真時正午黃道高。
Find the noon ecliptic altitude at true time.
121
求真時黃平象限距午。
Find the ecliptic-horizon quadrant's distance from noon at true time.
122
求真時黃平象限宮度。
Find the ecliptic-horizon quadrant mansion at true time.
123
求真時月距限，置太陽黃道經度，加減近時東西差，依真時距分加減號。 為真時太陰黃道經度。 餘同用時。
To find the moon's distance from the limit at true time: take the sun's ecliptic longitude and add or subtract the near-time east-west correction according to the sign of the true-time interval. This gives the moon's ecliptic longitude at true time. The remaining steps follow the apparent-time method.
124
求真時限距地高。
Find the limit's altitude above the horizon at true time.
125
求真時太陰高弧。
Find the moon's altitude arc at true time.
126
求真時黃道高弧交角。
Find the ecliptic altitude arc angle at true time.
127
求真時白道高弧交角。
Find the lunar-path altitude arc angle at true time.
128
求真時高下差。
Find the parallax in altitude at true time.
129
求真時東西差。
Find the east-west correction at true time.
130
求真時南北差，以半徑千萬為一率，真時白道高弧交角之正弦為二率，真時高下差之正弦為三率，求得四率為正弦，檢表得真時南北差。
To find the north-south correction at true time: use the radius of ten million, the sine of the lunar-path altitude arc angle, and the sine of the altitude parallax; look up the result in the tables.
131
求食甚視緯，依月食求食甚距緯法推之，得實緯。 以真時南北差加減之，為食甚視緯。 白平象限在天頂南者，緯南則加，而視緯仍為南； 緯北則減，而視緯仍為北。 若緯北而南北差大於實緯，則反減而視緯變為南。 限在天頂北者反是。
To find the apparent latitude at greatest eclipse: apply the lunar-eclipse method for latitude at greatest eclipse to obtain the true latitude. Add or subtract the true-time north-south correction to obtain the apparent latitude at greatest eclipse. If the lunar-path horizon quadrant lies south of the zenith, add for south latitude and the apparent latitude remains south; subtract for north latitude and the apparent latitude remains north. If the true latitude is north but the north-south correction exceeds it, reverse the subtraction and the apparent latitude becomes south. If the quadrant lies north of the zenith, reverse the rule.
132
求太陽半徑，以太陽距地為一率，太陽實半徑為二率，本天半徑為三率，求得四率為正弦，檢表得太陽半徑。
To find the solar semidiameter: use the sun's distance from Earth, its true semidiameter, and the celestial radius; look up the resulting sine in the tables.
133
求太陰半徑，詳月食。
To find the lunar semidiameter, see the lunar-eclipse section.
134
求食分，以太陽全徑為一率，十分為二率，並徑太陽太陰兩半徑並。 減去視緯為三率，求得四率即食分。
To find the eclipse magnitude: use the sun's full diameter as the first ratio and ten as the second; the third ratio is the sum of the solar and lunar semidiameters. Subtract the apparent latitude as the third ratio; the fourth ratio is the eclipse magnitude.
135
求初虧、復圓用時，以食甚視緯之餘弦為一率，並徑之餘弦為二率，半徑千萬為三率，求得四率為餘弦，檢表得初虧、復圓距弧。 又以月距日實行化秒為一率，一小時化秒為二率，初虧、復圓距弧化秒為三率，求得四率為秒。 以時分收之，為初虧、復圓距時。 以加減食甚真時，得初虧、復圓用時。 減得初虧，加得復圓。
To find the apparent times of first and last contact: use the cosine of the apparent latitude at greatest eclipse, the cosine of the combined diameter, and the radius of ten million; look up the arc intervals in the tables. Use the true moon-sun elongation in seconds, one hour in seconds, and the contact interval arcs in seconds; the fourth ratio gives the correction in seconds. Convert to hours and minutes to obtain the interval times to first and last contact. Apply these to the true time of greatest eclipse to obtain the apparent times of first and last contact. Subtract for first contact; add for last contact.
136
求初虧春秋分距午赤道度，以初虧用時變赤道度求之，餘與用時同。 後諸條仿此，但皆用初虧度分立算。
To find the equinox's equatorial distance from noon at first contact: convert using the apparent time of first contact; the remaining steps follow the apparent-time method. The following steps follow the same pattern, but all calculations use the first-contact values.
137
求初虧春秋分距午黃道度。
Find the equinox's ecliptic distance from noon at first contact.
138
求初虧正午黃赤距緯。
Find the noon ecliptic-equator latitude separation at first contact.
139
求初虧黃道與子午圈交角。
Find the ecliptic-meridian angle at first contact.
140
求初虧正午黃道宮度。
Find the noon ecliptic mansion at first contact.
141
求初虧正午黃道高。
Find the noon ecliptic altitude at first contact.
142
求初虧黃平象限距午。
Find the ecliptic-horizon quadrant's distance from noon at first contact.
143
求初虧黃平象限宮度。
Find the ecliptic-horizon quadrant mansion at first contact.
144
求初虧月距限，置太陽黃道經度，減初虧、復圓距弧，又加減真時東西差，依真時距分加減號。 得初虧太陰黃道經度。 餘同用時。
To find the moon's distance from the limit at first contact: take the sun's ecliptic longitude, subtract the contact interval arc, and add or subtract the true-time east-west correction according to the sign of the true-time interval. This gives the moon's ecliptic longitude at first contact. The remaining steps follow the apparent-time method.
145
求初虧限距地高。
Find the limit's altitude above the horizon at first contact.
146
求初虧太陰高弧。
Find the moon's altitude arc at first contact.
147
求初虧黃道高弧交角。
Find the ecliptic altitude arc angle at first contact.
148
求初虧白道高弧交角。
Find the lunar-path altitude arc angle at first contact.
149
求初虧高下差。
Find the parallax in altitude at first contact.
150
求初虧東西差。
Find the east-west correction at first contact.
151
求初虧南北差。
Find the north-south correction at first contact.
152
求初虧視行，以初虧、東西差與真時東西差相減並初虧食甚同限則減，初虧限東食甚限西則並。 為差分，以加減初虧、復圓距弧為視行。 相減為差分者，食在限東，初虧東西差大則減，小則加。 食在限西反是。 相併為差分者恆減。
To find the apparent motion at first contact: combine the first-contact and true-time east-west corrections—subtract if first contact and greatest eclipse share the same limit, add if the first-contact limit is east and the greatest-eclipse limit is west. This difference is applied to the contact interval arc to obtain the apparent motion. When the difference is obtained by subtraction: if the eclipse is in the eastern limit, subtract if the first-contact east-west correction is larger, add if smaller. If the eclipse is in the western limit, reverse the rule. When the difference is obtained by addition, always subtract.
153
求初虧真時，以初虧、視行化秒為一率，初虧、復圓距時化秒為二率，初虧、復圓距弧化秒為三率，求得四率為秒。 以時分收之，為初虧距分。 以減食甚真時，得初虧真時。
To find the true time of first contact: use the apparent motion in seconds, the contact interval time in seconds, and the contact interval arc in seconds; the fourth ratio gives the correction in seconds. Convert to hours and minutes as the first-contact interval. Subtract from the true time of greatest eclipse to obtain the true time of first contact.
154
求復圓春秋分距午赤道度，以復圓用時變赤道度求之。 餘同用時。 後諸條仿此，但皆用復圓度分立算。
To find the equinox's equatorial distance from noon at last contact: convert using the apparent time of last contact. The remaining steps follow the apparent-time method. The following steps follow the same pattern, but all calculations use the last-contact values.
155
求復圓春秋分距午黃道度。
Find the equinox's ecliptic distance from noon at last contact.
156
求復圓正午黃赤距緯。
Find the noon ecliptic-equator latitude separation at last contact.
157
求復圓黃道與子午圈交角。
Find the ecliptic-meridian angle at last contact.
158
求復圓正午黃道宮度。
Find the noon ecliptic mansion at last contact.
159
求復圓正午黃道高。
Find the noon ecliptic altitude at last contact.
160
求復圓黃平象限距午。
Find the ecliptic-horizon quadrant's distance from noon at last contact.
161
求復圓黃平象限宮度。
Find the ecliptic-horizon quadrant mansion at last contact.
162
求復圓月距限，置太陽黃道經度，加初虧、復圓距弧，又加減真時東西差，依真時距分加減號。 得復圓太陰黃道經度。 餘同用時。
To find the moon's distance from the limit at last contact: take the sun's ecliptic longitude, add the contact interval arc, and add or subtract the true-time east-west correction according to the sign of the true-time interval. This gives the moon's ecliptic longitude at last contact. The remaining steps follow the apparent-time method.
163
求復圓限距地高。
Find the limit's altitude above the horizon at last contact.
164
求復圓太陰高弧。
Find the moon's altitude arc at last contact.
165
求復圓黃道高弧交角。
Find the ecliptic altitude arc angle at last contact.
166
求復圓白道高弧交角。
Find the lunar-path altitude arc angle at last contact.
167
求復圓高下差。
Find the parallax in altitude at last contact.
168
求復圓東西差。
Find the east-west correction at last contact.
169
求復圓南北差。
Find the north-south correction at last contact.
170
求復圓視行，以復圓東西差與真時東西差相減並為差分，復圓食甚同限，則減； 食甚限東，復圓限西，則並。 以加減初虧、復圓距弧為視行。 相減為差分者，食在限東，復圓東西差大則加，小則減。 食在限西反是，相併為差分者恆減。
To find the apparent motion at last contact: combine the last-contact and true-time east-west corrections—subtract if last contact and greatest eclipse share the same limit; add if the greatest-eclipse limit is east and the last-contact limit is west. Apply this to the contact interval arc to obtain the apparent motion. When the difference is obtained by subtraction: if the eclipse is in the eastern limit, add if the last-contact east-west correction is larger, subtract if smaller. If in the western limit, reverse the rule; when obtained by addition, always subtract.
171
求復圓真時，以復圓視行化秒為一率，初虧、復圓距時化秒為二率，初虧、復圓距弧化秒為三率，求得四率為秒。 以時分收之，為復圓距分。 以加食甚真時，得復圓真時。
To find the true time of last contact: use the apparent motion in seconds, the contact interval time in seconds, and the contact interval arc in seconds; the fourth ratio gives the correction in seconds. Convert to hours and minutes as the last-contact interval. Add to the true time of greatest eclipse to obtain the true time of last contact.
172
求食限總時，以初虧距分與復圓距分相併，即得。
To find the total duration of the eclipse, add the first-contact and last-contact intervals.
173
求太陽黃道宿度，同日躔。
To find the sun's ecliptic lodge position, use the same method as in the section on solar motion.
174
求太陽赤道宿度，依恆星求赤道經緯法求得本年赤道宿鈐，餘同日躔求黃道法。
To find the sun's equatorial lodge position: use the fixed-star method to obtain the year's equatorial lodge register; the remaining steps follow the ecliptic method in the solar-motion section.
175
求初虧、復圓定交角，求得初虧、復圓各視緯，與食甚法同。 以求各緯差角。 各與黃道高弧交角相加減，為初虧及復圓之定交角。 法與月食同。
To find the fixed contact angles: compute the apparent latitude at first and last contact by the same method as at greatest eclipse. Then obtain each latitude-difference angle. Combine each with the ecliptic altitude arc angle to obtain the fixed contact angles at first and last contact. The method is the same as for lunar eclipses.
176
求初虧、復圓方位，食在限東者，定交角在四十五度以內，初虧上偏右，復圓下偏左。 四十五度以外，初虧右偏上，復圓左偏下。 適足九十度，初虧正右，復圓正左。 過九十度，初虧右偏下，復圓左偏上。 食在限西者，定交角在四十五度以內，初虧下偏右，復圓上偏左。 四十五度以外，初虧右偏下，復圓左偏上。 適足九十度，初虧正右，復圓正左。 過九十度，初虧右偏上，復圓左偏下。 京師黃平象限恆在天頂南，定方位如此，在天頂北反是。
To find the orientation at first and last contact: when the eclipse is in the eastern limit and the fixed angle is within forty-five degrees, first contact lies upper-right and last contact lower-left. Beyond forty-five degrees, first contact is to the upper right and last contact to the lower left. At exactly ninety degrees, first contact is directly to the right and last contact directly to the left. Past ninety degrees, first contact is to the lower right and last contact to the upper left. When the eclipse is in the western limit and the fixed angle is within forty-five degrees, first contact lies lower-right and last contact upper-left. Beyond forty-five degrees, first contact is to the lower right and last contact to the upper left. At exactly ninety degrees, first contact is directly to the right and last contact directly to the left. Past ninety degrees, first contact is to the upper right and last contact to the lower left. At the capital, the ecliptic-horizon quadrant always lies south of the zenith; determine orientation accordingly; if north of the zenith, reverse the rule.
177
求帶食分秒，以本日日出或日入時分初虧或食甚在日出前者，為帶食出地，用日出分； 食甚或復圓在日入後者，為帶時入地，用日入分。 與食甚真時相減，餘為帶食距時。 乃以初虧、復圓距時化秒為一率，初虧、復圓視行化秒為二率，帶食在食甚前，用初虧視行； 帶食在食甚後，用復圓視行。 帶食距時化秒為三率，求得四率為秒。 以度分收之，為帶食距弧。 又以半徑千萬為一率，帶食距弧之餘切為二率，食甚距緯之餘弦為三率，求得四率為餘切，檢表得帶食兩心相距。 乃以太陽全徑為一率，十分為二率，並徑內減帶食兩心相距為三率，求得四率，為帶食分秒。
To find the magnitude of a horizon eclipse: if first contact or greatest eclipse occurs before sunrise, it is an eclipse emerging at sunrise—use the sunrise time; if greatest eclipse or last contact occurs after sunset, it is an eclipse setting at sunset—use the sunset time. Subtract from the true time of greatest eclipse; the remainder is the interval time to the horizon. Use the contact interval time in seconds as the first ratio and the contact apparent motion in seconds as the second; if the horizon event precedes greatest eclipse, use the apparent motion at first contact; if it follows greatest eclipse, use the apparent motion at last contact. Use the horizon interval time in seconds as the third ratio; the fourth ratio gives the correction in seconds. Convert to degrees and minutes to obtain the horizon interval arc. Use the radius of ten million, the cotangent of the horizon interval arc, and the cosine of the latitude at greatest eclipse; look up the separation between the two centers at the horizon. Use the sun's full diameter and ten as ratios; subtract the center separation from the combined diameter as the third ratio; the fourth ratio is the horizon eclipse magnitude.
178
求各省日食時刻及食分，以京師食甚用時，按各省東西偏度加減之，得各省食甚用時。 乃按各省北極高度，如京師法求之，即得。
To find the time and magnitude of the solar eclipse for each province: start from the capital's apparent time of greatest eclipse and adjust by each province's longitudinal offset. Then apply each province's polar altitude using the same method as for the capital.
179
求各省日食方位，以各省黃道高弧交角及初虧、復圓視緯，求其定交角，即得。
To find the eclipse orientation for each province: use the provincial ecliptic altitude arc angle and the apparent latitudes at first and last contact to obtain the fixed contact angle.
180
繪日食圖法同月食，但只用日月兩半徑為度，作一大虛圈，為初虧、復圓月心所到。 不用內虛圈，無食既、生光二限。
The method for drawing the solar eclipse diagram is the same as for lunar eclipses, but use only the solar and lunar semidiameters to draw one large circle marking where the moon's center reaches at first and last contact. No inner circle is used, as there are no totality or light-return phases.
181
凌犯用數，具七政恆星行及交食。
Constants for occultation and close approach, covering the motions of the seven planets, fixed stars, and eclipses.
182
推凌犯法，求凌犯入限，太陰凌犯恆星，以太陰本日次日經度，查本年忄互星經緯度表，某星緯度不過十度，經度在此限內，為凌犯入限。 復查太陰在入限各星之上下，如星月兩緯同在黃道北者，緯多為在上，緯少為在下。 同在黃道南者反是。 一南一北者，北為在上，南為在下。 太陰在上者，兩緯相距二度以內取用； 太陰在下者，一度以內取用。 相距十七分以內為凌，十八分以外為犯，緯同為掩。 太陰凌犯五星，以本日太陰經度在星前、次日在星後為入限，餘與凌犯恆星同。 五星凌犯恆星，以兩緯相距一度內取用。 相距三分以內為凌，四分以外為犯，餘與太陰同。 五星自相凌犯，以行速者為凌犯之星，行遲者為受凌犯之星。 如遲速相同而有順逆，則為順行之星凌犯逆行之星，皆以此星經度本日在彼星前、次日在彼星後為入限。 餘同凌犯恆星。
Procedure for occultation and close approach: to determine entry into the zone, when the moon occults a fixed star, compare the moon's longitude today and tomorrow against the year's fixed-star table; if a star's latitude does not exceed ten degrees and its longitude falls within the limit, the event qualifies. Further determine whether the moon passes above or below each qualifying star: if both latitudes are north of the ecliptic, the higher latitude is above and the lower below. If both are south of the ecliptic, reverse the rule. If one is north and one south, the northern body is above and the southern below. When the moon passes above, apply if the latitudes are within two degrees; when it passes below, within one degree. A separation of seventeen arcminutes or less is an occultation; eighteen or more is a close approach; identical latitude is a covering. When the moon occults a planet: if today's lunar longitude lies before the planet and tomorrow's after, it enters the zone; the remaining rules match those for fixed stars. When a planet occults a fixed star, apply if the latitudes are within one degree. Three arcminutes or less is an occultation; four or more is a close approach; the remaining rules match those for the moon. When planets occult one another, the faster planet is the occulting body and the slower the body occulted. If speeds are equal but motions differ in direction, the direct-moving planet occults the retrograde one; entry requires today's longitude before the other body and tomorrow's after. The remaining rules match those for fixed stars.
183
求日行度，太陰凌犯恆星，即以太陰一日實行度為日行度。 凌犯五星，以太陰一日實行度與本星一日實行度相加減，星順行則減，逆行則加。 為日行度。 五星凌犯恆星，以本星一日實行度為日行度。 五星自相凌犯，以兩星一日實行度相加減，順逆同行則減，異行則加。 為日行度。
To find the daily motion: for the moon occulting a fixed star, use the moon's true motion over one day. For occulting a planet: combine the moon's and planet's true daily motions—subtract if the planet is direct, add if retrograde. This yields the daily motion. When a planet occults a fixed star, use that planet's true motion over one day. When planets occult each other: combine their true daily motions—subtract if both move in the same direction, add if opposite. This yields the daily motion.
184
求凌犯時刻，以日行度化秒為一率，刻下分為二率，本日子正相距度化秒為三率，求得四率為分。 以時刻收之，初時起子正，即得。
To find the time of occultation: use the daily motion in seconds, sub-quarter parts, and the separation from exact zi at today's epoch in seconds; the fourth ratio gives minutes. Convert to clock time starting from exact zi to obtain the result.
185
求太陰凌犯視差，五星視差甚微，可以不計。 以刻下分為一率，太陽一日實行度化秒為二率，凌犯時刻化分為三率，求得四率為秒。 以度分收之，與本日子正太陽實行相加，為本時太陽黃道度。 依日食法求東西差及南北差。
To find parallax for lunar occultation: planetary parallax is negligible and may be ignored. Use sub-quarter parts, the sun's true daily motion in seconds, and the occultation time in minutes; the fourth ratio gives seconds. Convert to degrees and minutes and add to the sun's true motion from exact zi to obtain the solar ecliptic longitude at that moment. Apply the solar-eclipse method to find the east-west and north-south corrections.
186
求太陰視緯，置太陰實緯，以南北差加減之，加減之法，與日食同。 即得。 求太陰距星，以太陰視緯與星緯相加減，南北相同則減，一南一北則加。 得太陰距星。 取相距一度以內者用。
To find the moon's apparent latitude: take its true latitude and apply the north-south correction by the same rules as for solar eclipses. This yields the result. To find the moon's latitude separation from the star: combine the moon's apparent latitude with the star's latitude—subtract if both lie on the same side of the ecliptic, add if opposite. This gives the moon's latitude separation from the star. Apply if the separation is within one degree.
187
求凌犯視時，以太陰一小時實行化秒為一率，一小時化秒為二率，東西差化秒為三率，求得四率為秒。 收為分，以加減凌犯時刻，太陰距限西則加，東則減。 得凌犯視時。
To find the apparent time of occultation: use the moon's hourly true motion in seconds, one hour in seconds, and the east-west correction in seconds; the fourth ratio gives seconds. Convert to minutes and apply to the occultation time: add if the limit is west, subtract if east. This yields the apparent time of occultation.
188
求太陽實引，以太陽平引，依日躔法求得太陽均數，以太陰平引，依月離法求得太陰初均數，兩均數相加減為距弧。 兩均同號相減，異號相加。 以月距日一小時平行為一率，一小時化秒為二率，距弧化秒為三率，求得四率為距時秒，隨定其加減號。 兩均同號，日大仍之，日小反之； 兩均一加一減，其加減從日。 又以一小時化秒為一率，太陽一小時引數為二率，距時秒為三率，求得四率為秒。 以度分收之，為太陽引弧。 依距時加減號。 以加減太陽平引，得實引。
To find the sun's true anomaly: from the sun's mean anomaly compute its equation by the solar-motion method; from the moon's mean anomaly compute its first equation by the lunar-distance method; combine the two equations to obtain the elongation arc. Subtract if they share the same sign, add if opposite. Use the moon's hourly mean elongation from the sun, one hour in seconds, and the elongation arc in seconds; the fourth ratio gives the interval time in seconds, with sign determined accordingly. If both equations share the same sign: if the sun's is larger, keep the sign; if smaller, reverse it; if one calls for addition and one for subtraction, follow the sign of the sun's equation. Further use one hour in seconds, the sun's hourly equation, and the interval time in seconds; the fourth ratio gives the correction in seconds. Convert to degrees and minutes to obtain the sun's anomaly arc. Apply addition or subtraction according to the sign of the interval time. Add or subtract this from the sun's mean anomaly to obtain the true anomaly.