1
志二十六
Treatise 26
2
時憲七
Shixian Calendar 7
3
雍正癸卯元法下
Yongzheng guimao calendrical system — lower section
4
月食用數
Constants for lunar eclipse calculation
5
朔策二十九日五三０五九０五三。
New-moon interval constant: 29.53059053 days.
6
望策一十四日七六五二九五二六五。
Full-moon interval constant: 14.765295265 days.
7
太陰交周朔策一十一萬零四百一十三秒，小餘九二四四一三三四。
Moon's nodal motion per synodic month: 110,413 seconds, fractional remainder 0.92441334.
8
太陰交周望策六宮一十五度二十分零六秒五十八微。
Moon's nodal motion per half-month: 6 mansions 15°20′06.58″.
9
中距太陰地半徑差五十七分三十秒。
Moon's diurnal parallax at mean distance: 57′30″.
10
太陽最大地半徑差一十秒。
Sun's maximum diurnal parallax: 10″.
11
中距太陽距地心一千萬。
Sun's geocentric distance at mean distance: 10,000,000.
12
中距太陰距地心一千萬。
Moon's geocentric distance at mean distance: 10,000,000.
13
中距太陽視半徑一十六分六秒。
Sun's apparent semidiameter at mean distance: 16′06″.
14
中距太陰視半徑一十五分四十秒三十微。
Moon's apparent semidiameter at mean distance: 15′40.30″.
15
朔應一十五日一二六三三。
Epoch new-moon offset: 15.12633 days.
16
首朔太陰交周應六宮二十三度三十六分五十二秒四十九微。 餘見日躔、月離。
At the epoch new moon, the moon's nodal longitude should be 23°36′52.49″ in the sixth mansion. For the remaining constants, see the sections on solar motion and lunar distance.
17
推月食法
Procedure for computing lunar eclipses
18
求天正冬至，
To find the winter solstice of the civil year (as in the jiazi epoch method),
19
求紀日，
To find the cycle-day count (as in the jiazi epoch method),
20
求首朔，
To find the epoch new moon (as in the jiazi epoch method),
21
求太陰入食限，並同甲子元法。 視某月太陰平交周入可食之限，即為有食之月。 交周自五宮十四度五十一分至六宮十五度九分，自十一宮十四度五十一分至初宮十五度九分，皆可食之限。 再於實時距正交詳之。
To determine whether the moon enters the eclipse zone, use the same method as in the jiazi epoch system. If in a given month the moon's mean nodal longitude falls within the eclipse zone, that month will have an eclipse. The eclipse zone extends from 14°51′ in the fifth mansion to 15°09′ in the sixth, and from 14°51′ in the eleventh mansion to 15°09′ in the first. Refine this at the true time by the distance from the true node.
22
求平望，同甲子元法。
To find the mean full moon, use the same method as in the jiazi epoch system.
23
求實望實時，先求泛時，用兩日實行較，同甲子元求朔望法。 次設前、後兩時，各求日、月黃道實行。 復用兩時實行較，得實望實時。 又以實時各求日、月黃道實行，視本時月距正交入限為有食。 自五宮十七度四十三分至六宮十二度十七分，自十一宮十七度四十三分至初宮十二度十七分，皆有食之限。
To find the true time of full moon, first find the approximate time by comparing true motions over two days—by the same method as in the jiazi epoch system for new and full moon. Next set two times before and after, and compute the true ecliptic longitudes of the sun and moon at each. By comparing true motions at the two times, obtain the true time of full moon. At the true time, compute the true ecliptic longitudes of the sun and moon; if the moon's distance from the true node at that instant falls within the limit, an eclipse occurs. At true time the eclipse limits are from 17°43′ in the fifth mansion to 12°17′ in the sixth, and from 17°43′ in the eleventh mansion to 12°17′ in the first.
24
求實望用時，用實時太陽均數及升度求法，同甲子元法。 比視日出入亦同。
To find the apparent time of true full moon, compute the sun's equation of center at true time and the rising-degree correction by the same method as in the jiazi epoch system. The comparison with sunrise and sunset is the same as well.
25
求食甚時刻，用平三角形，以一小時太陰白道實行化秒為一邊，本時次時二實行較。 一小時太陽黃道實行化秒為一邊，實望黃白大距為所夾之角，求得對小邊之角為斜距交角差。 以加實時黃白大距，為斜距黃道交角。 又以斜距交角差之正弦為一率，一小時太陽實行為二率，實望黃白大距之正弦為三率，求得四率，為一小時兩經斜距。 又以半徑千萬為一率，斜距黃道交角之餘弦、正弦各為二率，實望月離黃道實緯為三率，各求得四率，為食甚實緯南北與實望黃道實緯同。 及距弧。 又以一小時兩經斜距為一率，一小時化秒為二率，食甚距弧為三率，求得四率為食甚距時。 以加減實望用時，月距正交初宮、六宮為減，五宮、十一宮為加。 得食甚時刻。
To find the time of greatest eclipse, by plane triangle: convert the moon's hourly true motion along the white path to seconds as one side, and compare true motions at the current and following hours. Use the sun's hourly true ecliptic motion in seconds as the other side and the great ecliptic-lunar distance at true full moon as the included angle; the angle opposite the shorter side is the oblique-separation nodal-angle correction. Add this to the true-time great ecliptic-lunar distance to obtain the oblique-separation ecliptic nodal angle. Use the sine of the oblique-separation nodal-angle correction, the sun's hourly true motion, and the sine of the great ecliptic-lunar distance at true full moon; the fourth ratio is the hourly oblique separation of the two longitudes. Use the radius of ten million, the cosine and sine of the oblique-separation ecliptic nodal angle, and the moon's true ecliptic latitude at true full moon; the results give the north-south true latitude at greatest eclipse, equal to the true ecliptic latitude at true full moon. And the interval arc. Use the hourly oblique separation of the two longitudes, one hour in seconds, and the interval arc at greatest eclipse; the fourth ratio is the interval time to greatest eclipse. Add or subtract this from the apparent time of true full moon: subtract when the moon is at the first or sixth mansion from the true node, add at the fifth or eleventh. This yields the time of greatest eclipse.
26
求太陽太陰實引，置實望太陽引數，加減本時太陽均數，得太陽實引。 又置實望太陰引數，加減本時太陰初均數，得太陰實引。
To find the sun's and moon's true anomalies: take the sun's anomaly at true full moon and apply the equation of center at the current time to obtain the sun's true anomaly. Take the moon's anomaly at true full moon and apply the moon's first equation at the current time to obtain the moon's true anomaly.
27
求實影半徑，以太陰距地為一率，中距太陰距地為二率，中距太陰最大地半徑差為三率，求得四率為本時太陰最大地半徑差。 又以六十九除之，為影差。 又以太陽距地為一率，中距太陽距地為二率，中距太陽視半徑為三率，求得四率為太陽視半徑，與本時太陰最大地半徑差相減。 又加太陽最大地半徑差，為影半徑，又加影差，為實影半徑。
To find the true umbral radius: use the moon's geocentric distance, the mean-distance lunar distance, and the moon's maximum diurnal parallax at mean distance; the fourth ratio is the moon's maximum parallax at the current time. Divide by sixty-nine to obtain the shadow correction. Use the sun's geocentric distance, the mean-distance solar distance, and the sun's apparent semidiameter at mean distance; subtract the result from the moon's maximum parallax at the current time. Add the sun's maximum parallax to obtain the shadow semidiameter; add the shadow correction to obtain the true umbral semidiameter.
28
求太陰視半徑，以太陰距地為一率，中距太陰距地為二率，中距太陰視半徑為三率，求得四率，為太陰視半徑。 知
To find the moon's apparent semidiameter: use the moon's geocentric distance, the mean-distance lunar distance, and the moon's apparent semidiameter at mean distance. With these known:
29
求食分，以太陰全徑為一率，十分化作六百秒為二率，並徑實影視太陰兩半徑並。 內減食甚實緯，餘化秒為三率，求得四率為秒，以分收之，即食分。
To find the eclipse magnitude: use the moon's full diameter as the first ratio and ten tenths (600 seconds) as the second; the third ratio is the sum of the true shadow and lunar semidiameters. Subtract the true latitude at greatest eclipse; convert the remainder to seconds as the third ratio; reduce the fourth ratio to minutes to obtain the eclipse magnitude.
30
求初虧、復圓時刻，以並徑與食甚實緯相加化秒為首率，相減化秒為末率，求得中率為秒，以分收之，為初虧、復圓距弧。 又以一小時兩經斜距為一率，一小時化秒為二率，初虧、復圓距弧為三率，求得四率為初虧、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: add and subtract the combined diameter and the true latitude at greatest eclipse, convert to seconds as first and last ratios; convert the middle ratio to minutes as the contact interval arcs. Use the hourly oblique separation, one hour in seconds, and the contact interval arcs; apply the resulting interval times to the time of greatest eclipse to obtain the times of first and last contact. Subtract for first contact; add for last contact.
31
求食既、生光時刻，以兩徑較實影視太陰兩半徑相減之餘。 與食甚實緯相加化秒為首率，相減化秒為末率，求得中率為秒，以分收之，為食既、生光距弧。 求距時時刻，與初虧、復圓法同。 食在十分以內，則無此二限。
To find the times when totality begins and light returns: take the difference of the two semidiameters—the true shadow and the lunar semidiameter. Add and subtract this from the true latitude at greatest eclipse and convert to seconds as first and last ratios; convert the middle ratio to minutes as the totality interval arcs. To find the interval times, use the same method as for first and last contact. If the eclipse magnitude is less than ten tenths, these two limits do not apply.
32
求食限總時，同甲子元法。
To find the total duration of the eclipse, use the same method as in the jiazi epoch system.
33
求食甚太陰黃道經緯宿度，以一小時化秒為一率，一小時太陰白道實行為二率，食甚距時化秒為三率，求得四率，為距時月實行。 以加減實望太陰白道實行，加減與食甚距時同。 得食甚太陰白道經度。 又置實望月距正交，加減距時月實行，得食甚月距正交。 再求黃道經緯宿度，同月離。
To find the moon's ecliptic longitude, latitude, and mansion at greatest eclipse: use one hour in seconds, the moon's hourly true motion along the white path, and the interval time to greatest eclipse in seconds; the fourth ratio is the moon's true motion over that interval. Add or subtract this from the moon's true white-path longitude at true full moon, with the same sign as the interval time to greatest eclipse. This yields the moon's white-path longitude at greatest eclipse. Take the moon's distance from the true node at true full moon and apply the interval motion to obtain the distance from the true node at greatest eclipse. Then compute the ecliptic longitude, latitude, and mansion by the same method as in the lunar-distance section.
34
求食甚太陰赤道經緯宿度，以半徑千萬為一率，食甚太陰距春、秋分黃道經度正弦為二率，食甚太陰黃道經度不及三宮者，與三宮相減； 過三宮者，減三宮； 過六宮者，與九宮相減； 過九宮者，減九宮。 食甚太陰黃道緯度餘切為三率，求得四率為餘切，檢表得太陰距二分弧與黃道交角，以加減黃赤大距，食甚太陰黃道經度九宮至三宮，緯南加，緯北減，皆在赤道南，反減則在北。 三宮至九宮加減反是。 為太陰距二分弧與赤道交角。 又以太陰距二分弧與黃道交角之餘弦為一率，半徑千萬為二率，食甚太陰距春、秋分黃道經度之正切為三率，求得四率，為太陰距二分弧之正切。 又以半徑千萬為一率，太陰距二分弧與赤道交角之餘弦為二率，太陰距二分弧正切為三率，求得四率為正切，檢表為距春、秋分赤道經度。 加減三宮九宮，食甚太陰黃道經度不及三宮，與三宮相減，過三宮者加三宮。 過六宮者，與九宮相減，過九宮者加九宮。 得食甚太陰赤道經度。 求緯度宿度，同甲子元法。
To find the moon's equatorial longitude, latitude, and mansion at greatest eclipse: use the radius of ten million and the sine of the moon's ecliptic longitude measured from the equinox; if the moon's ecliptic longitude is less than three mansions, subtract from three mansions; if greater than three mansions, subtract three mansions; if greater than six mansions, subtract from nine mansions; if greater than nine mansions, subtract nine mansions. Use the cotangent of the moon's ecliptic latitude at greatest eclipse as the third ratio; look up the moon's angle with the solstitial arc in the tables; add or subtract the great ecliptic-equator distance—for ecliptic longitudes from the ninth to the third mansion, add for south latitude and subtract for north (all on the south side of the equator; reverse for the north). From the third to the ninth mansion, reverse the addition and subtraction. This is the moon's angle with the equator measured from the solstitial arc. Use the cosine of the moon's angle with the solstitial arc on the ecliptic, the radius of ten million, and the tangent of the moon's ecliptic longitude from the equinox; the fourth ratio is the tangent of the moon's distance from the solstitial arc. Use the radius of ten million, the cosine of the moon's angle with the solstitial arc on the equator, and the tangent of the distance from the solstitial arc; look up the result to obtain the equatorial longitude from the equinox. Apply the three- and nine-mansion corrections: if the ecliptic longitude is less than three mansions, subtract from three mansions; if greater, add three mansions. If greater than six mansions, subtract from nine; if greater than nine mansions, add nine. This yields the moon's equatorial longitude at greatest eclipse. To find the latitude and mansion, use the same method as in the jiazi epoch system.
35
求初虧、復圓黃道高弧交角，以半徑千萬為一率，黃赤大距正弦為二率，影距春、秋分黃道經度正弦為三率，求得四率為正弦，檢表得影距赤道度。 影距春、秋分度數與太陽同，太陽在赤道北，影在南，太陽在赤道南，影在北。 又以影距春、秋分黃道經度餘弦為一率，黃赤大距餘切為二率，半徑千萬為三率，求得四率為正切，檢表為黃道赤經交角。 乃用弧三角形，以北極距天頂為一邊，影距赤道與九十度相加減為一邊，北則減，南則加。 初虧、復圓各子正時刻過十二時者，與二十四時相減。 變赤道度，各為所夾之角，求得對北極距天頂之角。 各為赤經高弧交角，以加減黃道赤經交角，太陰在夏至前六宮，食在子正後則減，為限西。 食在子正前則加，加過九十度，與半周相減，為限東。 不及九十度，則不與半周相減，變為限西。 在夏至後六宮反是。 各得黃道高弧交角。 若食在子正，影在正午，無赤經高弧交角，則黃道赤經交角即黃道高弧交角。 太陰在夏至前為限西，後為限東。
To find the ecliptic altitude arc angles at first and last contact: use the radius of ten million, the sine of the great ecliptic-equator distance, and the sine of the shadow's ecliptic longitude from the equinox; look up the result to obtain the shadow's equatorial distance. The shadow's distance from the equinox matches the sun's: when the sun is north of the equator the shadow is south, and when the sun is south the shadow is north. Use the cosine of the shadow's ecliptic longitude from the equinox, the cotangent of the great ecliptic-equator distance, and the radius of ten million; look up the resulting tangent to obtain the ecliptic-equator right-ascension angle. Then by spherical triangle: use the distance from the north pole to the zenith as one side and the shadow's equatorial distance plus or minus ninety degrees as the other—subtract for north, add for south. For first and last contact, if the midnight time exceeds twelve hours, subtract from twenty-four hours. Convert to equatorial degrees as the included angle and obtain the angle opposite the north-pole-to-zenith side. Each gives an equatorial altitude arc angle; add or subtract the ecliptic-equator right-ascension angle—for the moon before summer solstice in the first six mansions, subtract if the eclipse is after midnight (western limit). Add if the eclipse is before midnight; if the sum exceeds ninety degrees, subtract from a semicircle for the eastern limit. If less than ninety degrees, do not subtract from a semicircle; this becomes the western limit. After summer solstice in the last six mansions, reverse the rule. This yields the ecliptic altitude arc angle for each contact. If the eclipse is at midnight and the shadow at noon, there is no equatorial altitude arc angle; the ecliptic-equator right-ascension angle is then the ecliptic altitude arc angle. Before summer solstice the limit is west; after, east.
36
求初虧、復圓並徑高弧交角，以並徑為一率，食甚實緯為二率，半徑千萬為三率，求得四率為餘弦，檢表為並徑交實緯角。 如無食甚實緯，即無此角，亦無並徑黃道交角。 又置九十度，加減斜距黃道交角，得初虧、復圓黃道交實緯角。 食甚月距正交初宮、六宮，初虧減，復圓加。 五宮、十一宮，初虧加，復圓減。 各與並徑交實緯角相減，為初虧、復圓並徑黃道交角。 並徑初交實緯角小，距緯南北與食甚同。 大則反是。 以加減黃道高弧交角，虧限東，復圓限西，緯南加，緯北減。 初虧限西，復圓限東，加減反是。 各得並徑高弧交角。 如無並徑黃道交角，則黃道高弧交角即並徑高弧交角。
To find the combined-diameter altitude arc angles at first and last contact: use the combined diameter, the true latitude at greatest eclipse, and the radius of ten million; look up the cosine to obtain the angle between the combined diameter and the true latitude. If there is no true latitude at greatest eclipse, this angle and the combined-diameter ecliptic angle do not exist. Set ninety degrees and add or subtract the oblique-separation ecliptic nodal angle to obtain the ecliptic angles with true latitude at first and last contact. At greatest eclipse, if the moon is at the first or sixth mansion from the true node: subtract for first contact, add for last contact. At the fifth or eleventh mansion: add for first contact, subtract for last contact. Subtract each from the combined-diameter with true-latitude angle to obtain the combined-diameter ecliptic angles at first and last contact. If the initial combined-diameter with true-latitude angle is small, the north-south latitude distance matches that at greatest eclipse. If large, the reverse holds. Add or subtract the ecliptic altitude arc angle: for the eastern eclipse limit and western last-contact limit, add for south latitude and subtract for north. For the western limit at first contact and the eastern limit at last contact, reverse the addition and subtraction. Each yields the combined-diameter altitude arc angle. If there is no combined-diameter ecliptic angle, the ecliptic altitude arc angle equals the combined-diameter altitude arc angle.
37
求初虧、復圓方位，即以並徑高弧交角為定交角，求法同甲子元。 但以並徑高弧交角初度初虧在限東為正下，限西為正上； 復圓在限東為正上，限西為正下。 據京師北極高度定，與甲子元法同。
To find the orientation at first and last contact, take the combined-diameter altitude arc angle as the fixed contact angle; the method is the same as in the jiazi epoch system. But at the initial degree of the combined-diameter altitude arc angle: first contact at the eastern limit is directly below, at the western limit directly above; At last contact, the eastern limit is directly above and the western limit directly below. Determine the orientation from the capital's polar altitude, as in the jiazi epoch method.
38
求帶食分秒，用兩經斜距，不用月距日實行，餘與甲子元法同。
To find the magnitude of a horizon eclipse, use the two-nodal oblique separation rather than the moon-sun true motion; the remaining steps follow the jiazi epoch method.
39
求帶食方位，用帶食兩心相距，不用並徑求諸交角，如初虧、復圓定方位。 食甚前與初虧同，食甚後與復圓同。
To find the horizon-eclipse orientation, use the center separation at the horizon rather than deriving the various nodal angles from the combined diameter, as for fixing the orientation at first and last contact. Before greatest eclipse, use the same rules as for first contact; after greatest eclipse, the same as for last contact.
40
求各省月食時刻方位，理同甲子元法。
To find the time and orientation of the lunar eclipse for each province, the principle is the same as in the jiazi epoch method.
41
繪月食圖，同甲子元法。
The method for drawing the lunar eclipse diagram is the same as in the jiazi epoch system.
42
日食用數
Constants for solar eclipse calculation
43
太陽光分一十五秒，餘見日躔、月離、月食。
Solar limb (light-semidiameter): 15″; for the remaining constants, see the sections on solar motion, lunar distance, and lunar eclipse.
44
推日食法
Procedure for computing solar eclipses
45
求天正冬至，
To find the winter solstice of the civil year,
46
求紀日，
To find the cycle-day count,
47
求首朔，
To find the epoch new moon,
48
求太陰入食限，並同月食，惟不用望策，即為逐月朔太陰交周。 視某月入可食之限，即為有食之月。 交周自五宮八度四十二分至六宮九度一十四分，又自十一宮二十度四十六分至初宮二十一度一十八分，皆可食之限。
To determine whether the moon enters the eclipse zone, use the same method as for lunar eclipses, but without the half-month interval—this gives the moon's nodal longitude at each new moon. If a given month falls within the possible-eclipse zone, that month will have an eclipse. The possible-eclipse zone runs from 8°42′ in the fifth mansion to 9°14′ in the sixth, and from 20°46′ in the eleventh mansion to 21°18′ in the first.
49
求平朔，
To find the mean new moon,
50
求實朔實時，並同月食求望法，惟不加望策。 視本時月距正交入食限為有食。 自五宮十一度三十四分至六宮六度二十二分，又自十一宮二十三度三十八分至初宮十八度二十六分，為有食之限。
To find the true time of new moon, use the same method as for finding true full moon in the lunar-eclipse section, but without adding the half-month interval. If at that instant the moon's distance from the true node falls within the eclipse limit, an eclipse occurs. At true new moon the eclipse limits run from 11°34′ in the fifth mansion to 6°22′ in the sixth, and from 23°38′ in the eleventh mansion to 18°26′ in the first.
51
求實朔用時，與月食求實望用時同。 比視日出入，同甲子元法。
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 comparison with sunrise and sunset follows the jiazi epoch method.
52
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 for the time of greatest eclipse in the lunar-eclipse section.
53
求太陽太陰實引，
To find the true anomalies of the sun and moon,
54
求太陽太陰距地，並同月食。
To find the geocentric distances of the sun and moon, use the same method as for lunar eclipses.
55
求地平高下差，先求本日太陰最大地半徑差，法同月食。 乃減太陽最大地半徑差，得地平高下差。
To find the horizon altitude-parallax difference, first compute the moon's maximum diurnal parallax for that day by the lunar-eclipse method. Subtract the sun's maximum diurnal parallax to obtain the horizon altitude-parallax difference.
56
求太陽實半徑，先求太陽視半徑，法同月食。 內減太陽光分，得太陽實半徑。
To find the sun's true semidiameter, first compute its apparent semidiameter by the lunar-eclipse method. Subtract the solar limb from within to obtain the sun's true semidiameter.
57
求太陰視半徑，法同月食。
To find the moon's apparent semidiameter, use the same method as for lunar eclipses.
58
求食甚太陽黃道經度宿度，求經度與月食求太陰白道法同； 求宿度同日躔。
To find the sun's ecliptic longitude and lodge at greatest eclipse: the longitude is found by the same method as the moon's white-path longitude in the lunar-eclipse section; the lodge position by the same method as in the solar-motion section.
59
求食甚太陰赤道經緯宿度，用黃赤大距，法同月食求太陰黃道。
To find the moon's equatorial longitude, latitude, and lodge at greatest eclipse, use the great ecliptic-equatorial distance and the same method as for the moon's ecliptic in the lunar-eclipse section.
60
求黃赤及黃白、赤白二經交角，以食甚太陽距春、秋分黃道經度餘弦為一率，黃赤大距餘切為二率，半徑千萬為三率，求得四率為餘切，檢表得黃赤二經交角。 冬至後黃經在赤經西，夏至後在赤經東，如太陽在二至，則無此角。 又以前所得斜距黃道交角，即為黃白二經交角。 實朔月距正交初宮、十一宮，白經在黃經西； 五宮、六宮，在黃經東。 二交角相加減，為赤白二經交角。 二交角同為東同為西者相加，白經在赤經之東西仍之。 一為東一為西者相減。 東西從大角。 如減盡，則無此角。 如無黃赤二經交角，則黃白即赤白，東西並同。
To find the ecliptic-equatorial, ecliptic-lunar, and equatorial-lunar nodal angles: use the cosine of the sun's ecliptic longitude from the equinox at greatest eclipse, the cotangent of the great ecliptic-equatorial distance, and the radius of ten million; look up the resulting cotangent in the tables to obtain the ecliptic-equatorial nodal angle. After the winter solstice the ecliptic longitude lies west of the equatorial longitude; after the summer solstice, east. If the sun is at either solstice, this angle does not exist. The oblique-separation ecliptic nodal angle obtained above is the ecliptic-lunar nodal angle. At true new moon, if the moon is at the first or eleventh mansion from the true node, the white-path longitude lies west of the ecliptic longitude; At the fifth or sixth mansion, it lies east of the ecliptic longitude. Combine the two nodal angles by addition or subtraction to obtain the equatorial-lunar nodal angle. If both angles lie on the same side (both east or both west), add them; the white-path longitude remains on the same side of the equatorial longitude. If one is east and one west, subtract. The east-west designation follows the larger angle. If subtraction reduces the angle to zero, this angle does not exist. If there is no ecliptic-equatorial nodal angle, the ecliptic-lunar angle is the equatorial-lunar angle, with the same east-west designation.
61
求用時太陽距午赤道度，以食甚用時與十二時相減，餘數變赤道度，得用時太陽距午赤道度。
To find the sun's equatorial longitude from the meridian at apparent time: subtract the greatest-eclipse apparent time from twelve hours, convert the remainder to equatorial degrees, and obtain the sun's equatorial longitude from the meridian.
62
求用時赤經高弧交角，用弧三角形，以北極距天頂為一邊，太陽距北極為一邊，赤緯在南，加九十度； 在北，與九十度相減。 用時太陽距午赤道度為所夾之角，求得對北極距天頂之角，為用時赤經高弧交角。 午前赤經在高弧東，午後赤經在高弧西。 若太陽在正午，則無此角。
To find the right-ascension altitude arc angle at apparent time, by spherical triangle: use the polar distance to the zenith as one side and the sun's distance from the north pole as the other; if the declination is south, add ninety degrees; If north, subtract from ninety degrees. Use the sun's equatorial longitude from the meridian at apparent time as the included angle; the angle opposite the polar distance to the zenith is the right-ascension altitude arc angle at apparent time. Before noon the right ascension lies east of the altitude arc; after noon, west. If the sun is at noon, this angle does not exist.
63
求用時太陽距天頂，以用時赤經高弧交角正弦為一率，北極距天頂之正弦為二率，用時太陽距午赤道度之正弦為三率，求得四率為正弦，檢表得太陽距天頂。
To find the sun's zenith distance at apparent time: use the sine of the right-ascension altitude arc angle, the sine of the polar distance to the zenith, and the sine of the sun's equatorial longitude from the meridian; look up the resulting sine in the tables to obtain the sun's zenith distance.
64
求用時高下差，以半徑千萬為一率，地平高下差化秒為二率，用時太陽距天頂之正弦為三率，求得四率為秒，以分收之，為用時高下差。
To find the altitude-parallax difference at apparent time: use the radius of ten million, the horizon altitude-parallax difference in seconds, and the sine of the sun's zenith distance; convert the result to minutes to obtain the altitude-parallax difference at apparent time.
65
求用時白經高弧交角，以用時赤經高弧交角與赤白二經交角相加減，得用時白經高弧交角。 東西同者相加，白經在高弧之東西仍之。 一東一西者相減，東西從大角。 如無赤白二經交角，或無赤經高弧交角，則即以所有一角命之，東西並同。 如二角俱無，或同度減盡，則無此角。 食甚用時即真時。 用時高下差與食甚實緯，南加北減，即食甚兩心視相距。
To find the white-path altitude arc angle at apparent time, add or subtract the equatorial-lunar nodal angle from the right-ascension altitude arc angle. If both lie on the same side, add; the white-path longitude remains on the same side of the altitude arc. If one is east and one west, subtract; the east-west designation follows the larger angle. If either the equatorial-lunar nodal angle or the right-ascension altitude arc angle is absent, use whichever angle exists, with the same east-west designation. If both angles are absent, or subtraction reduces them to zero, this angle does not exist. The apparent time of greatest eclipse is the true time. Add the altitude-parallax difference at apparent time to the true latitude at greatest eclipse for south latitude and subtract for north; the result is the apparent center separation at greatest eclipse.
66
求用時對兩心視相距角，月在黃道北，取用時白經高弧交角； 月在黃道南，取用時白經高弧交角之外角，實距在高弧之東西，月在北則與白經同，在南則相反。 皆為用時對兩心視相距角。 若自經高弧交角過九十度，緯南如緯北，緯北如緯南。
To find the angle opposite the apparent center separation at apparent time: if the moon is north of the ecliptic, use the white-path altitude arc angle at apparent time; if south of the ecliptic, use the exterior angle of the white-path altitude arc angle. For the east-west position of the true separation relative to the altitude arc: if the moon is north, it matches the white path; if south, it is reversed. In each case this is the angle opposite the apparent center separation at apparent time. If the nodal altitude arc angle exceeds ninety degrees, treat south latitude as north and north as south.
67
求用時對兩心實相距角，用平三角形，以食甚用時兩心實相距為一邊，即食甚實緯。 用時高下差為一邊，用時對兩心視相距角為所夾之角，即求得用時對兩心實相距角。
To find the angle opposite the true center separation at apparent time, by plane triangle: use the true center separation at greatest-eclipse apparent time as one side—that is, the true latitude at greatest eclipse. Use the altitude-parallax difference at apparent time as the other side and the angle opposite the apparent center separation as the included angle; this yields the angle opposite the true center separation at apparent time.
68
求用時兩心視相距，以用時對兩心實相距角之正弦為一率，用時兩心實相距為二率，用時對兩心視相距角之正弦為三率，求得四率，即用時兩心視相距。 白經在高弧西，兩心視相距大於並徑者，或無食或未及等者，用時即初虧真時，在高弧東為已過及復圓真時。 若小於並徑，高弧西為初虧食甚之間，東為復圓食甚之間。
To find the apparent center separation at apparent time: use the sine of the angle opposite the true center separation, the true center separation, and the sine of the angle opposite the apparent center separation; the fourth ratio is the apparent center separation at apparent time. If the white path is west of the altitude arc and the apparent center separation exceeds the combined diameter—indicating no eclipse or not yet contact—the apparent time is the true time of first contact; if east of the altitude arc, it is past greatest eclipse or the true time of last contact. If less than the combined diameter, west of the altitude arc falls between first contact and greatest eclipse; east, between last contact and greatest eclipse.
69
求食甚設時，用時白經高弧交角東向前取，西向後取，角大遠取，角小近取，遠不過九刻，近或數分。 量距用時前後若干分，為食甚設時。
To find the provisional time of greatest eclipse: if the white-path altitude arc angle at apparent time is east, take a time forward; if west, backward. A large angle calls for a distant time, a small angle a near one—distant not beyond nine quarters of an hour, near perhaps only a few minutes. Measure an interval of several minutes before or after the apparent time to obtain the provisional time of greatest eclipse.
70
求設時距分，以食甚設時與食甚用時相減，得設時距分。
To find the interval in minutes from provisional to apparent time: subtract the greatest-eclipse apparent time from the provisional time.
71
求設時距弧，以一小時化秒為一率，一小時兩經斜距為二率，設時距分化秒為三率，求得四率，為設時距弧。
To find the interval arc at provisional time: use one hour in seconds, the hourly two-nodal oblique separation, and the provisional interval in seconds; the fourth ratio is the interval arc.
72
求設時對距弧角，以食甚實緯為一率，設時距弧為二率，半徑千萬為三率，求得四率為正切，檢表得設時對距弧角。
To find the angle opposite the interval arc at provisional time: use the true latitude at greatest eclipse, the interval arc, and the radius of ten million; look up the resulting tangent in the tables.
73
求設時兩心實相距，以設時對距弧角之正弦為一率，設時距弧為二率，半徑千萬為三率，求得四率，即設時兩心實相距。
To find the true center separation at provisional time: use the sine of the angle opposite the interval arc, the interval arc, and the radius of ten million; the fourth ratio is the true center separation.
74
求設時太陽距午赤道度，
To find the sun's equatorial longitude from the meridian at provisional time,
75
求設時赤經高弧交角，
To find the right-ascension altitude arc angle at provisional time,
76
求設時太陽距天頂，
To find the sun's zenith distance at provisional time,
77
求設時高下差，
To find the altitude-parallax difference at provisional time,
78
求設時白經高弧交角，以上五條，皆與用時同，但皆用設時度分立算。
To find the white-path altitude arc angle at provisional time—the above five steps all follow the apparent-time method, but each is computed separately using provisional-time values.
79
求設時對兩心視相距角，月在黃道北，以設時白經高弧交角與設時對距弧角相減，月在黃道南則相加，又與半周相減，餘為設時對兩心視相距角。 相減者，對距弧角小，實距在高弧之東西與白經同； 對距弧角大則相反。 相加又減半周者，實距在高弧之東西，恆與白經反。 如兩角相等而減盡無餘，或相加適足一百八十度，則無交角，亦無對設時兩心實相距角，即以設時高下差與設時兩心實相距相減，餘為設時兩心視相距。 若白經高弧交角過九十度，緯南如緯北，緯北如緯南。
To find the angle opposite the apparent center separation at provisional time: if the moon is north of the ecliptic, subtract the angle opposite the interval arc from the white-path altitude arc angle; if south, add them, then subtract from a semicircle; the remainder is the angle opposite the apparent center separation. When subtracting: if the angle opposite the interval arc is small, the east-west position of the true separation relative to the altitude arc matches the white path; If large, it is reversed. When adding and then subtracting a semicircle, the east-west position of the true separation is always opposite to the white path. If the two angles are equal and subtraction leaves no remainder, or their sum exactly equals 180°, there is no nodal angle and no angle opposite the true center separation at provisional time; subtract the altitude-parallax difference from the true center separation to obtain the apparent center separation. If the white-path altitude arc angle exceeds ninety degrees, treat south latitude as north and north as south.
80
求設時對兩心實相距角，
To find the angle opposite the true center separation at provisional time,
81
求設時兩心視相距，皆與用時同。
To find the apparent center separation at provisional time, use the same method as at apparent time.
82
求設時白經高弧交角較，以設時白經高弧交角與用時白經高弧交角相減，即得。
To find the difference in white-path altitude arc angles: subtract the apparent-time angle from the provisional-time angle.
83
求設時高弧交用時視距角，以設時白經高弧交角較與用時對兩心實相距角相加減，即得。 緯北為減，緯南為加。 若白經高弧交角過九十度，反是。
To find the angle between the altitude arc at provisional time and the apparent separation at apparent time: add or subtract the white-path altitude arc angle difference from the angle opposite the true center separation at apparent time. Subtract for north latitude, add for south. If the white-path altitude arc angle exceeds ninety degrees, reverse the addition and subtraction.
84
求對設時視行角，以設時高弧交用時視距角與設時對兩心實相距角相加減，即得。 兩實距同在高弧東，或同在西，則減； 一東一西者，則加； 加過半周者，與全周相減，用其餘。 如無設時對兩心實相距角，設時高下差大於設時兩心實相距，則設時高弧交用時視距角即對設時視行角； 設時高下差小於設時兩心實相距，則以設時高弧交用時視距角與半周相減，餘為對設時視行角。
To find the angle opposite the apparent motion at provisional time: add or subtract the provisional-time angle opposite the true center separation from the angle between the altitude arc and the apparent separation. If both true separations lie on the same side of the altitude arc (both east or both west), subtract; If one is east and one west, add; If the sum exceeds a semicircle, subtract from the full circle and use the remainder. If there is no angle opposite the true center separation at provisional time: when the altitude-parallax difference at provisional time exceeds the true center separation at provisional time, the angle between the altitude arc at provisional time and the apparent separation at apparent time is the angle opposite the apparent motion at provisional time; When the altitude-parallax difference is less than the true center separation, subtract the angle between the altitude arc and the apparent separation from a semicircle; the remainder is the angle opposite the apparent motion at provisional time.
85
求對設時視距角，用平三角形，以用時兩心視相距為一邊，設時兩心視相距為一邊，對設時視行角為所夾之角，即求得對設時視距角。
To find the angle opposite the apparent separation at provisional time, by plane triangle: use the apparent center separation at apparent time as one side, the apparent center separation at provisional time as the other, and the angle opposite the apparent motion at provisional time as the included angle; this yields the angle opposite the apparent separation at provisional time.
86
求設時視行，以對設時視距角之正弦為一率，設時兩心視相距為二率，對設時視行角正弦為三率，求得四率，為設時視行。
To find the apparent motion at provisional time: use the sine of the angle opposite the apparent separation, the apparent center separation at provisional time, and the sine of the angle opposite the apparent motion; the fourth ratio is the apparent motion at provisional time.
87
求真時視行，以半徑千萬為一率，對設時視距角餘弦為二率，用時兩心視相距為三率，求得四率，為真時視行。
To find the apparent motion at true time: use the radius of ten million, the cosine of the angle opposite the apparent separation at provisional time, and the apparent center separation at apparent time; the fourth ratio is the apparent motion at true time.
88
求真時兩心視相距。 以半徑千萬為一率，對設時視距角正弦為二率，用時兩心視相距為三率，求得四率，為真時兩心視相距。
To find the apparent center separation at true time. Use the radius of ten million, the sine of the angle opposite the apparent separation at provisional time, and the apparent center separation at apparent time; the fourth ratio is the apparent center separation at true time.
89
求食甚真時，以設時視行為一率，設時距分為二率，真時視行為三率，求得四率，為真時距分，以加減食甚用時，白經在高弧西則加，在高弧東則減。 得食甚真時。
To find the true time of greatest eclipse: use the apparent motion at provisional time, the provisional-time interval in minutes, and the apparent motion at true time; the fourth ratio gives the true-time interval in minutes. Add or subtract this from the apparent time of greatest eclipse—add if the white path is west of the altitude arc, subtract if east. This yields the true time of greatest eclipse.
90
求真時距弧，
To find the interval arc at true time,
91
求真時對距弧角，
To find the angle opposite the interval arc at true time,
92
求真時兩心實相距，以上三條，法與設時同，但皆用真時度分立算。
To find the true center separation at true time—the above three steps follow the provisional-time method, but all computations use true-time degrees separately.
93
求真時太陽距午赤道度，
To find the sun's equatorial longitude from the meridian at true time,
94
求真時赤經高弧交角，
To find the right-ascension altitude arc angle at true time,
95
求真時太陽距天頂，
To find the sun's zenith distance at true time,
96
求真時高下差，
To find the altitude-parallax difference at true time,
97
求真時白經高弧交角，
To find the white-path altitude arc angle at true time,
98
求真時對兩心視相距角，
To find the angle opposite the apparent center separation at true time,
99
求真時對兩心實相距角，
To find the angle opposite the true center separation at true time,
100
求考真時兩心視相距，以上八條，法與用時同，但皆用真時度分立算。
To find the apparent center separation at verified true time—the above eight steps follow the apparent-time method, but all computations use true-time degrees separately.
101
求真時白經高弧交角較，法同設時，但用真時度分立算。
To find the difference in white-path altitude arc angles at true time, use the same method as at provisional time, but compute with true-time degrees separately.
102
求真時高弧交設時視距角，法同設時，加減有異。 月在黃道北，設時真時兩實距在高弧東西同，惟白經異。 設時白經高弧交角小則加，大則減。 若白經亦同，反是。 若兩實距一東一西，則皆相減。 月在黃道南，設時交角小則加，大則減。 如無設時對兩心實相距角，設時高下差大於設時兩心實相距，則真時白經高弧交角較，即真時高弧交設時視距角； 設時高下差小於設時兩心實相距，則以真時白經高弧交角較與半周相減，餘為真時高弧交設時視距角。 若白經高弧交角過九十度，緯南如緯北，緯北如緯南。
To find the angle between the altitude arc at true time and the apparent separation at provisional time: the method matches provisional time, but the addition and subtraction differ. If the moon is north of the ecliptic and the two true separations at provisional and true time lie on the same side of the altitude arc, only the white-path positions differ. If the white-path altitude arc angle at provisional time is small, add; if large, subtract. If the white-path positions also match, reverse the addition and subtraction. If the two true separations lie on opposite sides (one east, one west), subtract in both cases. If the moon is south of the ecliptic, add when the nodal angle at provisional time is small and subtract when it is large. If there is no angle opposite the true center separation at provisional time: when the altitude-parallax difference at provisional time exceeds the true center separation at provisional time, the white-path altitude arc angle difference at true time equals the angle between the altitude arc at true time and the apparent separation at provisional time; when the altitude-parallax difference is less than the true center separation, subtract the white-path altitude arc angle difference at true time from a semicircle; the remainder is the angle between the altitude arc at true time and the apparent separation at provisional time. If the white-path altitude arc angle exceeds ninety degrees, treat south latitude as north and north as south.
103
求對考真時視行角，法同設時。 如設時實距與高弧合，無東西者，設時高下差大於設時兩心實相距，則相減，小則加。 如真時白經高弧交角較與設時對兩心實相距角相等，而減盡無餘，則真時對兩心實相距角，即對考真時視行角。 或相加適足半周，則真時對兩心實相距角與半周相減，即對考真時視行角。
To find the angle opposite the apparent motion at verified true time, use the same method as at provisional time. If at provisional time the true separation coincides with the altitude arc with no east-west distinction: subtract when the altitude-parallax difference exceeds the true center separation, add when it is smaller. If the white-path altitude arc angle difference at true time equals the angle opposite the true center separation at provisional time and subtraction leaves no remainder, the angle opposite the true center separation at true time is the angle opposite the apparent motion at verified true time. Or if the sum exactly equals a semicircle, subtract the angle opposite the true center separation at true time from a semicircle; the result is the angle opposite the apparent motion at verified true time.
104
求對考真時視距角，
To find the angle opposite the apparent separation at verified true time,
105
求考真時視行，以上二條，法同設時，但用考真時度分立算。
To find the apparent motion at verified true time—the above two steps follow the provisional-time method, but compute with verified-true-time degrees separately.
106
求定真時視行，如定真時視行與考真時視行等，則食甚真時即為定真時。 如或大或小，再用下法求之。
To find the apparent motion at fixed true time: if the fixed-true-time apparent motion equals the verified-true-time apparent motion, the true time of greatest eclipse is the fixed true time. If it is greater or smaller, apply the method below again.
107
求定真時兩心視相距，以上二條，法同真時，用考真時度分立算。
To find the apparent center separation at fixed true time—the above two steps follow the true-time method, using verified-true-time degrees separately.
108
求食甚定真時，以考真時視行為一率，設時距分與真時距分相減餘為二率，定真時視行為三率，求得四率，為定真時距分。 以加減食甚設時，白經在高弧東，設時距分小測減，大則加。 白經在高弧西，反是。 得食甚定真時。
To find the fixed true time of greatest eclipse: use the apparent motion at verified true time as the first ratio, the difference between the provisional-time and true-time interval minutes as the second ratio, and the apparent motion at fixed true time as the third ratio; the fourth ratio gives the fixed-true-time interval in minutes. Add or subtract this from the provisional time of greatest eclipse: if the white path is east of the altitude arc, subtract when the provisional-time interval is small and add when it is large. If the white path is west of the altitude arc, reverse the addition and subtraction. This yields the fixed true time of greatest eclipse.
109
求食分，以太陽實半徑倍之為一率，十分為二率，並徑內減定真時兩心視相距餘為三率，求得四率，即食分。
To find the eclipse magnitude: use twice the solar true semidiameter as the first ratio and ten as the second; subtract the apparent center separation at fixed true time from the combined diameter as the third ratio; the fourth ratio is the eclipse magnitude.
110
求初虧、復圓前設時，白經在高弧西，食甚用時兩心視相距與並徑相去不遠，即以食甚用時為初虧前設時，小則向前取，大則向後取，量距食甚用時前後若干分，為初虧前設時。 與食甚定真時相減，餘數與食甚定真時相加，為復圓前設時，白經在高弧東，先取復圓，後得初虧，理並同。
To find the provisional time before first or last contact: if the white path is west of the altitude arc and the apparent center separation at greatest-eclipse apparent time is close to the combined diameter, use the apparent time of greatest eclipse as the provisional time before first contact; if smaller, take a time before; if larger, take a time after—measure several minutes before or after the apparent time of greatest eclipse to obtain the provisional time before first contact. Subtract this from the fixed true time of greatest eclipse and add the remainder to the fixed true time of greatest eclipse to obtain the provisional time before last contact. If the white path is east of the altitude arc, take last contact first and then first contact—the reasoning is the same.
111
求初虧前設時距分，
To find the interval minutes at the provisional time before first contact,
112
求初虧前設時距弧，
To find the interval arc at the provisional time before first contact,
113
求初虧前設時對距弧角，初虧前設時在食甚用時前為西，在食甚用時後為東。
To find the angle opposite the interval arc at the provisional time before first contact: if that provisional time precedes the apparent time of greatest eclipse, it is west; if it follows, it is east.
114
求初虧前設時兩心實相距，以上四條，法同食甚設時，但用初虧前設時度分立算。
To find the true center separation at the provisional time before first contact—the above four steps follow the greatest-eclipse provisional-time method, but compute with degrees at the provisional time before first contact.
115
求初虧前設時太陽距午赤道度，
To find the sun's equatorial longitude from the meridian at the provisional time before first contact,
116
求初虧前設時赤經高弧交角，
To find the right-ascension altitude arc angle at the provisional time before first contact,
117
求初虧前設時太陽距天頂，
To find the sun's zenith distance at the provisional time before first contact,
118
求初虧前設時高下差，
To find the altitude-parallax difference at the provisional time before first contact,
119
求初虧前設時白經高弧交角，以上五條，法同食甚用時。
To find the white-path altitude arc angle at the provisional time before first contact—the above five steps follow the greatest-eclipse apparent-time method.
120
求初虧前設時對兩心視相距角，法同食甚用時，加減有異，月在黃道北，二角東西同，則相加； 一東一西，相減。 月在黃道南，反是。 又與半周相減。 若白經高弧交角過九十度，則緯南、緯北互異。 餘同食甚設時。
To find the angle opposite the apparent center separation at the provisional time before first contact: the method matches greatest-eclipse apparent time, but addition and subtraction differ. If the moon is north of the ecliptic and both angles lie on the same side, add; if one is east and one west, subtract. If the moon is south of the ecliptic, reverse the rule. Then subtract from a semicircle. If the white-path altitude arc angle exceeds ninety degrees, interchange south and north latitude. The remaining steps follow the greatest-eclipse provisional-time method.
121
求初虧前設時對兩心實相距角，
To find the angle opposite the true center separation at the provisional time before first contact,
122
求初虧前設時兩心視相距，以上二條，法同食甚用時，但用初虧前設時度分立算。
To find the apparent center separation at the provisional time before first contact—the above two steps follow the greatest-eclipse apparent-time method, but compute with degrees at the provisional time before first contact.
123
求初虧後設時，視初虧前設時兩心視相距小於並徑，則向前取，大則向後取，察其較之多寡，量取前後若干分，為初虧後設時。 以下逐條推算，皆與前設時同，但用後設時度分立算。
To find the provisional time after first contact: if the apparent center separation at the provisional time before first contact is less than the combined diameter, take a time before; if greater, take a time after. Observe the magnitude of the difference and measure several minutes before or after to obtain the provisional time after first contact. For each step below, follow the before-provisional-time method, but compute with degrees at the after-provisional time.
124
求初虧視距較，以前後設時兩心視相距相減，即得。
To find the first-contact apparent-separation difference: subtract the apparent center separations at the before- and after-provisional times.
125
求初虧設時較，以前後設時距分相減，即得。
To find the first-contact provisional-time difference: subtract the interval minutes at the before- and after-provisional times.
126
求初虧視距並徑較，以初虧後設時兩心視相距與並徑相減，即得。
To find the first-contact apparent-separation combined-diameter difference: subtract the combined diameter from the apparent center separation at the provisional time after first contact.
127
求初虧定真時，以初虧視距較為一率，初虧設時較為二率，初虧視距並徑較為三率，求得四率，為初虧真時距分。 以加減初虧後設時，後設時兩心視相距大於並徑為加，小為減。 得初虧真時。 乃以初虧真時依前法求其兩心視相距，果與並徑等，則初虧真時即初虧定真時。 初虧真時對兩心實相距角即初虧方位角。 如或大或小，則以初虧前後設時兩心視相距與並徑尤近者，與考真時兩心視相距相較，依法比例，得初虧定真時。
To find the fixed true time of first contact: use the apparent-separation difference, the provisional-time difference, and the apparent-separation combined-diameter difference as ratios; the fourth ratio gives the true-time interval minutes for first contact. Add or subtract this from the provisional time after first contact: add if the apparent center separation at the after-provisional time exceeds the combined diameter, subtract if it is smaller. This yields the true time of first contact. Then, at the true time of first contact, compute the apparent center separation by the previous method; if it equals the combined diameter, the true time of first contact is the fixed true time of first contact. The angle opposite the true center separation at the true time of first contact is the orientation angle at first contact. If it is too large or too small, compare whichever of the before- and after-provisional-time apparent center separations is nearest the combined diameter with the apparent center separation at verified true time, and by proportion obtain the fixed true time of first contact.
128
求復圓前設時諸條，法同初虧，但用復圓前設時度分立算。
To find all quantities at the provisional time before last contact, use the first-contact method, but compute with degrees at the provisional time before last contact.
129
求復圓後設時，視復圓前設時兩心視相距小於並徑，則向後取，大於並徑，則向前取，察其較之多寡，量取前後若干分，為復圓後設時。 逐條推算，皆與前設時同，但用後設時度分立算。
To find the provisional time after last contact: if the apparent center separation at the provisional time before last contact is less than the combined diameter, take a time after; if greater than the combined diameter, take a time before. Observe the magnitude of the difference and measure several minutes before or after to obtain the provisional time after last contact. Compute each step in sequence by the before-provisional-time method, but use degrees at the after-provisional time.
130
求復圓視距較，
To find the last-contact apparent-separation difference,
131
求復圓設時較，
To find the last-contact provisional-time difference,
132
求復圓視距並徑較，
To find the last-contact apparent-separation combined-diameter difference,
133
求復圓定真時，以上四條，皆與初虧法同，但用復圓度分立算。
To find the fixed true time of last contact—the above four steps all follow the first-contact method, but compute with last-contact degrees separately.
134
求食限總時，置初虧定真時，減復圓定真時，即得。
To find the total duration of the eclipse: subtract the fixed true time of last contact from the fixed true time of first contact.
135
求初虧、復圓定交角，初虧白經在高弧之東，以初虧方位角與半周相減，在高弧之西，即用初虧方位角； 復圓反是：皆為定交角。
To find the fixed contact angles at first and last contact: if the white path at first contact is east of the altitude arc, subtract the orientation angle at first contact from a semicircle; if west, use the orientation angle at first contact directly; for last contact, reverse the rule. Each result is a fixed contact angle.
136
求初虧、復圓方位，法與甲子元同，但以定交角初度初虧白經在高弧東為正上，在西為正下； 復圓在東為正下，在西為正上。
To find the orientation at first and last contact, use the jiazi epoch method, but at the initial degree of the fixed contact angle: if the white path at first contact is east of the altitude arc, the orientation is directly above; if west, directly below; at last contact, east is directly below and west is directly above.
137
求帶食用日出入分，同甲子元法。
To find the sunrise and sunset times for a horizon eclipse, use the same method as in the jiazi epoch system.
138
求帶食距時，以日出入分與食甚用時相減，即得。
To find the interval time to the horizon: subtract the sunrise or sunset time from the apparent time of greatest eclipse.
139
求帶食距弧，法同食甚設時，但用帶食距時立算。
To find the horizon interval arc, use the greatest-eclipse provisional-time method, but compute from the horizon interval time.
140
求帶食赤經高弧交角，以黃赤距緯之餘弦為一率，北極高度之正弦為二率，半徑千萬為三率，求得四率為餘弦，檢表得帶食赤經高弧交角。
To find the right-ascension altitude arc angle at the horizon: use the cosine of the ecliptic-equatorial latitude distance, the sine of the polar altitude, and the radius of ten million; look up the resulting cosine in the tables to obtain the right-ascension altitude arc angle at the horizon.
141
求帶食白經高弧交角，法與食甚用時同，但用帶食度分立算。
To find the white-path altitude arc angle at the horizon, use the greatest-eclipse apparent-time method, but compute with horizon-eclipse degrees separately.
142
求帶食對距弧角，
To find the angle opposite the interval arc at the horizon,
143
求帶食兩心實相距，
To find the true center separation at the horizon,
144
求帶食對兩心視相距角，以上三條，法與食甚設時同，但用帶食度分立算。
To find the angle opposite the apparent center separation at the horizon—the above three steps follow the greatest-eclipse provisional-time method, but compute with horizon-eclipse degrees separately.
145
求帶食對兩心實相距角，用地平高下差，餘法同食甚用時。
To find the angle opposite the true center separation at the horizon, use the horizon altitude-parallax difference; the remaining steps follow the greatest-eclipse apparent-time method.
146
求帶食兩心視相距，法同食甚用時，但用帶食度分立算。
To find the apparent center separation at the horizon, use the greatest-eclipse apparent-time method, but compute with horizon-eclipse degrees separately.
147
求帶食分秒，與求食分同，用帶食相距立算。
To find the magnitude of the horizon eclipse, use the same method as for eclipse magnitude, computing from the separation at the horizon.
148
求帶食方位，在食甚前者，用初虧法； 在食甚後者，用復圓法。
To find the orientation of a horizon eclipse: if the event occurs before greatest eclipse, use the first-contact method; if after greatest eclipse, use the last-contact method.
149
求各省日食時刻方位，理同甲子元法。
To find the time and orientation of the solar eclipse for each province, the principle is the same as in the jiazi epoch method.
150
繪日食圖，同甲子元法。
To draw the solar eclipse diagram, use the jiazi epoch method.
151
繪日食坤輿圖，取見食極多之分，每分為一限。 止於二十一限。 又取見食時刻早晚，每刻為一限。 止於九十六限。 交錯相求，反推得見食各地北極高下度、東西偏度。 乃按度聯為一圖。 又按坤輿全圖所當高度偏度各地名，遂一填註。
To draw the terrestrial map of a solar eclipse, take the magnitude of greatest visibility and make each unit of magnitude one tier. This extends to twenty-one tiers. Also take the time of visibility, early or late, with each quarter-hour as one tier. This extends to ninety-six tiers. Cross the two scales and work backward to obtain the polar altitude and east-west deviation for each place of visibility. Then join the results by degree into a single map. Then, using the full terrestrial map, enter each place-name at its corresponding altitude and deviation.
152
相距用數，見月離及五星、恆星行。
Constants for close approach: see the sections on lunar distance and the motions of the five planets and fixed stars.
153
推相距法，同甲子元推凌犯法。
The procedure for close approach is the same as the jiazi epoch method for occultation and conjunction.
154
推步用表
Tables for computational astronomy
155
甲子元及癸卯元二法，除本法外，皆有用表推算之法，約其大旨著於篇。
Besides the present method, both the jiazi and guimao epoch systems include table-based procedures; their main features are summarized here.
156
甲子元法：
Jiazi epoch method:
157
一曰年根表，以紀年、紀日、值宿為綱，由法元之年順推三百年，各得其年天正冬至次日子正太陽及最卑平行，列為太陽年根表； 太陰及最高、正交平行，列為太陰年根表； 五星及最高、正交、伏見諸平行，為各星年根表。
First, epoch root tables: keyed by year count, day count, and lodge assignment, extend forward three hundred years from the epoch origin; for each year obtain the sun's mean motion and perigee mean motion at midnight after the civil winter solstice, and tabulate these as the solar epoch root table; the moon's mean motion together with apogee and true-node mean motions, tabulated as the lunar epoch root table; and the five planets' mean motions together with apogee, true node, and visibility values, each as a planetary epoch root table.
158
一曰周歲平行表，以日數為綱，由一日至三百六十六日，積累日、月、五星及最卑、最高、正交、伏見諸平行，各列為周歲平行表。
Second, tropical-year mean-motion tables: keyed by day count from 1 to 366, accumulate the mean motions of the sun, moon, five planets, and their perigee, apogee, true node, and visibility values, each tabulated separately.
159
一曰周日平行表，以時分秒為綱，與度分秒對列三層，自一至六十，積累日、月、五星及最高、正交、伏見、月距日、太陰引數、交周諸平行，各列為周日平行表。
Third, diurnal mean-motion tables: keyed by hours, minutes, and seconds and paired in three tiers with degrees, minutes, and seconds from 1 to 60, accumulate the mean motions of the sun, moon, five planets, apogee, true node, visibility, moon-sun distance, lunar anomaly, and nodal cycle, each tabulated separately.
160
一曰均數表，以引數為綱，豫推得逐度逐分盈縮遲疾，備列於表。 太陰別有二三均數表，以引數及月距日為綱，縱橫對列，推得二三均數，備列於表。 土、木、金、水四星，則以初均及中分、次均及較分，同列為一表。 火星則以初均及次輪心距地數、次輪半徑本數、太陽高卑差數，同列為一表。 皆為均數表。
Fourth, equation tables: keyed by anomaly, pre-compute the fast-slow variation for each degree and minute and tabulate the results. The moon has separate second- and third-equation tables keyed by anomaly and moon-sun distance, arranged in crosswise pairs. For Saturn, Jupiter, Venus, and Mercury, the first equation and middle term together with the second equation and correction term are combined in one table. For Mars, the first equation together with the epicycle-center geocentric distance, epicycle radius, and solar distance variation are combined in one table. These are all equation tables.
161
一曰距度表，以黃道宮度為綱，列所對赤道南北距緯，為黃赤距度表。 以月距正交為綱，分黃白大距為六限，列所對黃道南北距緯，為黃白距度表。
Fifth, latitude-distance tables: keyed by ecliptic mansion degree, list the corresponding north-south equatorial latitude distance as the ecliptic-equator latitude-distance table. Keyed by the moon's distance from the true node and divided into six tiers of great ecliptic-lunar distance, list the corresponding north-south ecliptic latitude distance as the ecliptic-lunar latitude-distance table.
162
一曰升度表，以黃道宮度為綱，列所對赤道度，為黃赤升度表。
Sixth, right-ascension tables: keyed by ecliptic mansion degree, list the corresponding equatorial degree as the ecliptic-equator right-ascension table.
163
一曰黃道赤經交角表，以黃道宮度為綱，取所對黃道赤經交角列於表。
Seventh, ecliptic right-ascension angle table: keyed by ecliptic mansion degree, list the corresponding ecliptic right-ascension angle.
164
一曰升度差表，以月、五星距交宮度為綱，各列所當黃道度之較，各為升度差表。
Eighth, right-ascension difference tables: keyed by the moon's and five planets' distance from the node in mansion degrees, list the corresponding ecliptic degree correction for each body.
165
一曰時差表，以黃道為綱，取所當赤道度之較變時，列為升度時差表。 又以引數為綱，取所當均數變時，列為均數時差表。
Ninth, time-equation tables: keyed by ecliptic degree, convert the corresponding equatorial degree difference to time and tabulate it as the right-ascension time-equation table. Also keyed by anomaly, convert the corresponding equation to time and tabulate it as the equation time-equation table.
166
一曰地半徑差表，以實高度為綱，取所當太陽、太陰及火、金、水三星諸地半徑差，各列為表。
Tenth, diurnal-parallax tables: keyed by true altitude, list the corresponding diurnal parallax for the sun, moon, Mars, Venus, and Mercury.
167
一曰清蒙氣差表，以實高度為綱，取所當清蒙氣差，列為表。
Eleventh, refraction tables: keyed by true altitude, list the corresponding atmospheric refraction.
168
一曰實行表，以引數為綱，取所當太陽、太陰及月距日實行，各列為表。
Twelfth, true-motion tables: keyed by anomaly, list the corresponding true motions of the sun, moon, and moon-sun distance.
169
一曰交均距限表，以月距日為綱，取所當之交均及距限，同列為一表。
Thirteenth, nodal-equation and limit-distance table: keyed by moon-sun distance, list the corresponding nodal equation and limit distance together.
170
一曰首朔諸根表，以紀年、紀日、值宿為綱，由法元之年順推三百年，取所當之首朔日時分秒及太陽平行，太陽、太陰引數，太陰交周，五者同列為一表。
Fourteenth, first-new-moon root tables: keyed by year count, day count, and lodge assignment, extend forward three hundred years from the epoch origin; for each year list the first new moon's date and time, solar mean motion, solar and lunar anomaly, and lunar nodal cycle in one table.
171
一曰朔望策表，以月數為綱，自一至十三，取所當之朔、望策及太陽平行朔、望策，太陽、太陰引數朔、望策，太陰交周朔、望策，十事同列為一表。
Fifteenth, new- and full-moon tabulation tables: keyed by month count from 1 to 13, list the new- and full-moon tabulations together with solar mean motion, solar and lunar anomaly, and lunar nodal cycle at new and full moon—ten items in one table.
172
一曰視半徑表，以引數為綱，取所當之日半徑、月半徑、月距地影半徑、影差，五者同列為一表。
Sixteenth, apparent-semidiameter tables: keyed by anomaly, list the solar semidiameter, lunar semidiameter, lunar distance from the earth-shadow semidiameter, and shadow correction together.
173
一曰交食月行表，以食甚距緯分為綱，自初分至六十四分，與太陽、太陰、地影，凡兩半徑之和分，自二十五分至六十四分，縱橫對列，取所當之月行分秒列為表。 其太陰、地影兩半徑之較分與和分同用。
Seventeenth, eclipse lunar-motion tables: keyed by latitude distance at greatest eclipse from 1 to 64 minutes and crosswise paired with the sum of the two semidiameters of the sun, moon, and earth shadow from 25 to 64 minutes, list the corresponding lunar motion in minutes and seconds. The difference and sum of the lunar and earth-shadow semidiameters are used interchangeably.
174
一曰黃平象限表，以正午黃道宮度為綱，分北極高自十六度至四十六度為三十一限，取所當之春分距午、黃平象限、限距地高，三者同列為一表。
Eighteenth, ecliptic horizon-quadrant tables: keyed by the ecliptic mansion degree at noon and polar altitude from 16° to 46° in thirty-one tiers, list the vernal-equinox meridian distance, ecliptic horizon quadrant, and limit geocentric altitude together.
175
一曰黃道高弧交角表、以日距限為綱，自初度至九十度，分限距地高自二十度至八十九度為七十限，取所當之黃道高弧交角列為表。
Nineteenth, ecliptic altitude-arc angle table: keyed by solar limit distance from 0° to 90° and limit geocentric altitude from 20° to 89° in seventy tiers, list the corresponding ecliptic altitude-arc angle.
176
一曰太陽高弧表，列法與黃道高弧交角表同。
Twentieth, solar altitude-arc table: arranged by the same method as the ecliptic altitude-arc angle table.
177
一曰東西南北差表，以交角度為綱，自初度至九十度，與高下差一分至六十三分，縱橫對列，取所當之東西差及南北差，同列為表。
Twenty-first, east-west and north-south difference tables: keyed by nodal angle from 0° to 90° and crosswise paired with altitude-parallax difference from 1 to 63 minutes, list the corresponding east-west and north-south differences.
178
一曰緯差角表，以並徑為綱，自三十一分至六十四分，與距緯一分至六十四分，縱橫對列，取所當之緯差角列為表。
Twenty-second, latitude-difference angle tables: keyed by combined diameter from 31 to 64 minutes and crosswise paired with latitude distance from 1 to 64 minutes, list the corresponding latitude-difference angle.
179
一曰星距黃道表，以距交宮度為綱，取所當星距黃道數各列為表，水星獨分交角自四度五十五分三十二秒至六度三十一分二秒為二十限。
Twenty-third, star ecliptic-distance tables: keyed by distance from the node in mansion degrees, list each star's ecliptic distance; for Mercury alone, divide the nodal angle from 4°55′32″ to 6°31′2″ into twenty tiers.
180
一曰星距地表，以星距日宮度為綱，取所當之星距地列於表。
Twenty-fourth, star geocentric-distance tables: keyed by the star's distance from the sun in mansion degrees, list the corresponding geocentric distance.
181
一曰水星距限表，以距交宮度為綱，取所當之距限列為表。
Twenty-fifth, Mercury limit-distance table: keyed by distance from the node in mansion degrees, list the corresponding limit distance.
182
一曰五星伏見距日黃道度表，以星行黃道經表為綱，分晨夕上下列之，取各星所當距日黃道度，同列為一表。
Twenty-sixth, five-planet heliacal visibility solar ecliptic-degree tables: keyed by the planetary ecliptic-longitude tables and divided into morning and evening, upper and lower positions, list each planet's corresponding solar ecliptic distance.
183
一曰五星伏見距日加減差表，列法同黃道度表，但不分五星，別黃道南北自一度至八度。
Twenty-seventh, five-planet heliacal visibility solar correction tables: arranged like the ecliptic-degree tables, but without separating the five planets and instead dividing ecliptic latitude from 1° to 8° north and south.
184
癸卯元法所增：
Additions in the guimao epoch method:
185
一曰太陽距地心表，以太陽實引為綱，取所對之太陽距地心真數對數，並列於表。
First, solar geocentric-distance table: keyed by true solar anomaly, list the corresponding true geocentric distance and its logarithm.
186
一曰太陰一平均表，以太陽引數為綱，取所當之太陰一平均、最高平均、正交平均，並列於表。
Second, lunar first mean-motion table: keyed by solar anomaly, list the corresponding first lunar mean motion together with apogee and true-node mean motions.
187
一曰太陰二平均表，以日距月最高宮度為綱，取所當太陽在最高之二平均及高卑較秒，並列於表。
Third, lunar second mean-motion table: keyed by the sun's distance from the lunar apogee in mansion degrees, list the second mean motion when the sun is at apogee together with the distance variation in seconds.
188
一曰太陰三平均表，以月距正交宮度為綱，取所當之三平均列為表。
Fourth, lunar third mean-motion table: keyed by the moon's distance from the true node in mansion degrees, list the corresponding third mean motion.
189
一曰太陰最高均及本天心距地表，以日距月天最高宮度為綱，取所當最高均及本天心距地數，並列於表。
Fifth, lunar apogee equation and deferent-center geocentric-distance table: keyed by the sun's distance from the lunar celestial apogee in mansion degrees, list the apogee equation and deferent-center geocentric distance.
190
一曰太陰二均表，以月距日宮度為綱，取所當太陽在最高時二均及高卑較數，並列於表。
Sixth, lunar second-equation table: keyed by the moon's distance from the sun in mansion degrees, list the second equation when the sun is at apogee together with the distance variation.
191
一曰太陰三均表，以相距總數為綱，取所對之三均列於表。
Seventh, lunar third-equation table: keyed by the total conjunction number, list the corresponding third equation.
192
一曰太陰末均表，以實月距日宮度為綱，與日月最高相距，縱橫對列，取所當之末均列為表。
Eighth, lunar final-equation table: keyed by true moon-sun distance in mansion degrees and crosswise paired with sun-moon apogee separation, list the corresponding final equation.
193
一曰太陰正交實均表，以日距正交宮度為綱，取所對之正交實均列為表。
Ninth, lunar true-node true-equation table: keyed by the sun's distance from the true node in mansion degrees, list the corresponding true-node equation.
194
一曰交角加分表，以日距正交宮度為綱，取所當之距交加分加差，並列於表。
Tenth, nodal-angle addition table: keyed by the sun's distance from the true node in mansion degrees, list the corresponding nodal addition and correction.
195
一曰黃白距緯表，列法與升度差表同。
Eleventh, ecliptic-lunar latitude-distance table: arranged by the same method as the right-ascension difference table.
196
一曰太陰距地心表，以太陰實引為綱，取所當最大、最小兩心差各太陰距地心數及倍分，並列於表。 其名同而實異者，太陰初均表分大、中、小三限，黃、白升度差表列最小交角及大、小較秒，太陰地半徑差表、太陰實行表俱分大、小二限。
Twelfth, lunar geocentric-distance table: keyed by true lunar anomaly, list the geocentric distance and double minutes for the maximum and minimum two-center differences. Tables that share a name but differ in content: the lunar first-equation table is divided into large, middle, and small tiers; the ecliptic- and lunar right-ascension difference tables list the minimum nodal angle and large and small comparison seconds; the lunar diurnal-parallax and true-motion tables are each divided into large and small tiers.
197
求太陽太陰距地，用平三角形，以日躔倍兩心差為對正角之邊，以太陽實引為又一角，三宮內用本度，過三宮與六宮相減，過九宮與全周相減，用其餘。 求得對太陽實引之邊為勾。 又求得對原不知角之邊為分股，與二千萬相加減，實引三宮內九宮外加，三宮外九宮內減。 為股弦和與勾，求得股。 與分股相加減，實引三宮內九宮外減，三宮外九宮內加。 得太陽距地。 又以實望月離倍兩心差如法求之，得太陰距地。
To find the sun's and moon's geocentric distance, by plane triangle: take the solar double eccentricity as the side opposite the right angle and the true solar anomaly as the other angle; within three mansions use the degree as given, beyond three mansions subtract six mansions, beyond nine mansions subtract the full circle, and use the remainder. The side opposite the true solar anomaly is the base leg. The side opposite the originally unknown angle is the fractional leg; add or subtract it from 20,000,000—add when true anomaly is within three mansions or beyond nine mansions, subtract between three and nine mansions. With the sum of the leg and hypotenuse and the base leg, find the leg. Add or subtract the fractional leg—subtract when true anomaly is within three mansions or beyond nine mansions, add between three and nine mansions. This yields the sun's geocentric distance. Apply the same method using the moon's double eccentricity at true full moon to obtain the moon's geocentric distance.