1
志第九歷三
Treatise Nine: Calendrics, Part Three.
2
▲大統曆法一下法原
▲ Grand Concordance Calendar Method, Part One (Lower): Origins of the Method.
3
日月五星平定三差
The mean, upright, and fixed three differences for the sun, moon, and five planets.
4
太陽盈縮平立定三差之原。
Origins of the solar equation's mean, upright, and fixed three differences.
5
冬至前後盈初縮末限，八十八日九十一刻，就整。 離爲六段，每段各得一十四日八十二刻。 就整。 各段實測日躔度數，與平行相較，以爲積差。
About the winter solstice, the expansion-initial and contraction-final interval is 88 days 91 quarter-hours; round to integers. Divide it into six segments of 14 days 82 quarter-hours each. Round to integers. For each segment, compare the observed daily solar advance with the mean motion to obtain the accumulated difference.
6
第一段
Segment 1
7
一十四日八二
14 days 82 quarter-hours
8
七千零五十八分零二五
7,058.025 parts
9
第二段
Segment 2
10
二十九日六四
29 days 64 quarter-hours
11
一萬二千九百七十六三九二
12,976.392 parts
12
第三段
Segment 3
13
四十四日四六
44 days 46 quarter-hours
14
一萬七千六百九十三七四六二
17,693.7462 parts
15
第四段
Segment 4
16
五十九日二八
59 days 28 quarter-hours
17
二萬一千一百四十八七三二八
21,148.7328 parts
18
第五段
Segment 5
19
七十四日一零
74 days 10 quarter-hours
20
二萬三千二百七十九九九七
23,279.997 parts
21
第六段
Segment 6
22
八十八日九二
88 days 92 quarter-hours
23
二萬四千零二十六一八四
24,026.184 parts
24
各置其段積差，以其段積日除之，爲各段日平差。 置各段日平差，與後段日平差相減，爲一差。 置一差，與後段一差相減，爲二差。
Place each segment's accumulated difference and divide by that segment's accumulated days to obtain the segment's daily mean difference. Subtract each segment's daily mean difference from the next segment's to obtain the first difference. Subtract each segment's first difference from the next segment's to obtain the second difference.
25
日平差
Daily mean differences
26
第一段
Segment 1
27
四百七十六分二五
476′25″
28
三十八分四五
38′45″
29
一分三八
1′38″
30
第二段
Segment 2
31
四百三十七分八零
437′80″
32
三十九分八三
39′83″
33
一分三八
1′38″
34
第三段
Segment 3
35
三百九十七分九七
397′97″
36
四十一分二一
41′21″
37
一分三八
1′38″
38
第四段
Segment 4
39
三百五十六分七六
356′76″
40
四十一分五九
41′59″
41
一分三八
1′38″
42
第五段
Segment 5
43
三百一十四分一七
314′17″
44
四十三分九七
43′97″
45
第六段
Segment 6
46
二百七十零分二零
270′20″
47
置第一段日平差，四百七十六分二十五秒，爲凡平積。 以第二段二差一分三十八秒，去減第一段一差十八分四十五秒，餘三十七分零七秒，不凡平積差。 另置第一段二差一分三十八秒，折半得六十九秒，爲凡立積差。 以凡平積差三十七分零七秒，加入凡平積四百七十六分二十五秒，共得五百一十三分三十二秒，爲定差。
Take the first segment's daily mean difference of 476′25″ as the general mean accumulation. Subtract the first segment's first difference of 18′45″ from the second segment's second difference of 1′38″, leaving 37′07″ as the general mean accumulation difference. Halve the first segment's second difference of 1′38″ to obtain 69″ as the general upright accumulation difference. Add the general mean accumulation difference of 37′07″ to the general mean accumulation of 476′25″, giving 513′32″ as the fixed difference.
48
以凡立積差六十九秒，去減凡平積差三十七分零七秒，餘三十六分三十八秒爲實，以段日一十四日八十二刻爲法除之，得二分四十六秒爲平差。 置凡立積差六十九秒爲實，以段日爲法除二次，得三十一微，爲立差。
Subtract the general upright accumulation difference of 69″ from the general mean accumulation difference of 37′07″; the remainder of 36′38″ is the dividend. Dividing by the segment length of 14 days 82 quarter-hours yields 2′46″ as the mean difference. Take the general upright accumulation difference of 69″ as dividend and divide twice by the segment days to obtain 31 wei as the upright difference.
49
夏至前後縮初盈末限，九十三日七十一刻，就整。 離爲六段，每段各得一十五日六十二刻。 就整。 各段實測日躔度數，與平行相較，以爲積差。
About the summer solstice, the contraction-initial and expansion-final interval is 93 days 71 quarter-hours; round to integers. Divide it into six segments of 15 days 62 quarter-hours each. Round to integers. For each segment, compare the observed daily solar advance with the mean motion to obtain the accumulated difference.
50
第一段
Segment 1
51
一十五日六二
15 days 62 quarter-hours
52
七千零五十八分九九零四
7,058.9904 parts
53
第二段
Segment 2
54
三十一日二四
31 days 24 quarter-hours
55
一萬二千九百七十八六五八
12,978.658 parts
56
第三段
Segment 3
57
四十六日八六
46 days 86 quarter-hours
58
一萬七千六百九十六六七九
17,696.679 parts
59
第四段
Segment 4
60
六十二日四八
62 days 48 quarter-hours
61
二萬萬一千一百五十零七二九六
21,150.7296 parts
62
第五段
Segment 5
63
七十八日一零
78 days 10 quarter-hours
64
二萬三千二百七十八四八六
23,278.486 parts
65
第六段
Segment 6
66
九十三日七二
93 days 72 quarter-hours
67
二萬四千零百一十七六二四四
24,017.6244 parts
68
推日平差、一差、二差術，與盈初縮末同。
The procedure for deriving daily mean, first, and second differences is identical to that for the expansion-initial and contraction-final interval.
69
日平差
Daily mean differences
70
第一段
Segment 1
71
四百五十一分九二
451′92″
72
三十六分四七
36′47″
73
一分三三
1′33″
74
第二段
Segment 2
75
四百一十五分四五
415′45″
76
三十七分八零
37′80″
77
一分三三
1′33″
78
第三段
Segment 3
79
三百七十七分六五
377′65″
80
三十九分一二
39′12″
81
一分三三
1′33″
82
第四段
Segment 4
83
三百三十八分五二
338′52″
84
四十零分四六
40′46″
85
一分三三
1′33″
86
第五段
Segment 5
87
二百九十八分零六
298′06″
88
四十一分七九
41′79″
89
第六段
The sixth segment.
90
二百五十六分二七
256 parts 27.
91
置第一段日平差，四百五十一分九十二秒，爲凡平積。 以第一段二差一分三十三秒，去減第一段一差三十六分四十七秒，餘三十一分一十四秒，爲凡平積差。 另置第一段二差一分三十三秒折半，得六十六秒五十微，爲凡立積差。 以凡平積差三十五分一十四秒，加入凡平積四百五十一分九十二秒，共四百八十七分零六秒，爲定差。 以凡『立積差六十六秒五十微，去減凡平差三十五分一十四秒，餘三十四分四十七秒五十微爲實，以段日一十五日六二爲法除之，得二分二十一秒，爲平差。 置凡立積差六十六秒五十微爲實，以段日爲法，除二次，得二十七微，爲立差。
Take the first segment's daily mean difference of 451 parts 92 seconds as the general mean accumulation. Subtract the first segment's second difference (1 part 33 seconds) from its first difference (36 parts 47 seconds); the remainder, 31 parts 14 seconds, is the general mean accumulated difference. Halve the first segment's second difference of 1 part 33 seconds to obtain 66 seconds 50 micro-units as the general standing accumulated difference. Add the general mean accumulated difference of 35 parts 14 seconds to the general mean accumulation of 451 parts 92 seconds; the sum, 487 parts 06 seconds, is the fixed difference. Subtract the general standing accumulated difference (66 seconds 50 micro-units) from the general mean accumulated difference (35 parts 14 seconds); the remainder, 34 parts 47 seconds 50 micro-units, is the dividend. Dividing by the segment length of 15 days 62 ke yields 2 parts 21 seconds as the mean difference. Take the general standing accumulated difference of 66 seconds 50 micro-units as dividend and divide twice by the segment days; the quotient, 27 micro-units, is the standing difference.
92
凡求盈縮，以入歷初末日乘立差，得數以加平差，再以初末日乘之，得數以減定差，餘數以初末日乘之，爲盈縮積。
To find expansion and contraction: multiply the standing difference by the days from the start or end of ephemeris entry, add that to the mean difference, multiply again by those days, subtract from the fixed difference, and multiply the remainder by those days to obtain the expansion–contraction accumulation.
93
凡盈歷以八十日九零九二二五爲限，縮歷以九十三日七一二零二五爲限。 在其限已下爲初，以上轉減半歲周餘不末。 盈初是人冬至後順推，縮末是從冬至前逆溯，其距冬至同，故其盈積同。 縮初是從夏至後順推，盈末是從夏至前逆溯，其距夏至同，故其縮積同。
Expansion ephemerides use a limit of 80 days 909225; contraction ephemerides use a limit of 93 days 712025. Within the limit counts as the initial phase; beyond it, by cyclic subtraction against the half-year circuit remainder, one reaches the terminal phase. Expansion-initial is reckoned forward after the winter solstice; contraction-final is traced backward before the winter solstice. Because the distance from the winter solstice is the same, the expansion accumulation is the same. Contraction-initial is reckoned forward after the summer solstice; expansion-final is traced backward before the summer solstice. Because the distance from the summer solstice is the same, the contraction accumulation is the same.
94
表格略
[Table omitted.]
95
▲盈縮招差圖說
▲ Expansion–contraction interpolation: diagram and explanation
96
盈縮招生，本爲一象限之法。 如盈歷則以八十八日九十一刻爲象限，縮歷則以九十三日七十一刻爲象限。 今止作九限者，舉此爲例也。 其空格九行定差本數，爲實也。 其斜綿以上平差立差之數，爲法也。 斜綿以下空格之定差，乃餘實也。 假如定差爲一萬，平差爲一百，立差爲單一。 今求九限法，以九限乘定差得九萬爲實。 另置平差，以九限乘二次，得八千一百。 置立差，以九限乘三次，得七百二十九。 並兩數得八百二十九爲法。 以法減實，餘八萬一千一百七十一，爲九限積。 又法，以九限乘平差行九百，又以九限乘立差二次得八十一，並兩數得九進八十一爲法，定差一萬爲實，以法減實，餘矣千零一十九，即九限末位所書之定差也。 於是瑞以九限乘餘實，得八萬一千一百七十一，爲九限積，與前所不所得不同。 蓋前法是先乘後減，又法是先減後乘，其理一也。
Expansion–contraction interpolation derives from a method that treats a single quadrant. For expansion ephemerides the quadrant is 88 days 91 ke; for contraction ephemerides it is 93 days 71 ke. Here only nine steps are worked out, by way of illustration. The fixed-difference base numbers in the nine blank rows are the dividend. Above the oblique mesh, the mean-difference and standing-difference values form the divisor. The fixed differences in the blanks below the oblique mesh are the remainder-dividend. Suppose the fixed difference is 10,000, the mean difference 100, and the standing difference 1. For the nine-step method, multiply the fixed difference by nine to obtain 90,000 as dividend. Multiply the mean difference by nine twice to obtain 8,100. Multiply the standing difference by nine three times to obtain 729. Add the two products to obtain 8,829 as divisor. Subtract the divisor from the dividend; the remainder, 81,171, is the nine-step accumulation. By the second method: nine times the mean difference gives 900; nine squared times the standing difference gives 81; together 981 is the divisor. With fixed difference 10,000 as dividend, subtract to obtain 9,019—the fixed difference entered at the ninth step. Then multiply the remainder-dividend by nine to obtain 81,171 as the nine-step accumulation—numerically different from the first procedure's intermediate steps. The first method multiplies then subtracts; the second subtracts then multiplies—the principle is the same.
97
按《授時曆》於七政盈縮，並以垛積招差立算，其污七巧合天行，與西人用小輪推步之法，殊途同歸。 然世所傳《九章》諸書，不載其術，《歷草》載其術，而不言其故。 宣城梅文鼎爲之圖解，於平差、立差之理，垛積之法，皆有以發明其所以然。 有專書行於世，不能備錄，謹錄《招生圖說》，以明立法之大意云。
Commentary: In the Season Granting calendar, expansion and contraction for the seven regulators were all computed by pile-summation interpolation. Its fit to celestial motion parallels the Western epicycle methods—different routes, one outcome. Yet the transmitted Nine Chapters and kindred works do not record the technique; the Calendar Draft gives the procedure but not the reasoning behind it. Mei Wencheng of Xuancheng supplied illustrated explanations that set forth why mean and standing differences work and how pile summation operates. His monographs circulate widely and cannot all be quoted here; only the Diagram Explanation of Interpolation is recorded, to show the intent of the legislation.
98
盈初縮末置立差三十一微，以六因之，得一秒八十六微，爲加分立差。 置平差二分四十六秒，倍之，得四分九十二秒，加入加分立差，得四分九十二秒八十六微，爲平立合差。
Expansion-initial and contraction-final: take the standing difference of 31 micro-units, multiply by six, and obtain 1 second 86 micro-units as the incremental standing difference. Double the mean difference of 2 parts 46 seconds to get 4 parts 92 seconds; add the incremental standing difference for 4 parts 92 seconds 86 micro-units as the combined mean–standing difference.
99
置定差五百一十三分三十二秒，內減平差二分四十六秒，再減立差三十一微，餘五百一十零分八十五秒六十九微，爲加分。
From the fixed difference of 513 parts 32 seconds subtract the mean difference (2 parts 46 seconds) and the standing difference (31 micro-units); the remainder, 510 parts 85 seconds 69 micro-units, is the incremental correction.
100
縮初盈末置立差二十七微，以六因之，得一秒六十二微，爲加分立差。 置平差二分二十一秒，倍之，得四分四十二秒，加入加分立差，得四分四十三秒六十二微，爲平立合差。
Contraction-initial and expansion-final: take the standing difference of 27 micro-units, multiply by six, and obtain 1 second 62 micro-units as the incremental standing difference. Double the mean difference of 2 parts 21 seconds to get 4 parts 42 seconds; add the incremental standing difference for 4 parts 43 seconds 62 micro-units as the combined mean–standing difference.
101
置定差四百八十七分零六秒，內減平差二分二十一秒，再減立差二十七微，餘四百八十四分八十四秒七十三微，爲加分。
From the fixed difference of 487 parts 06 seconds subtract the mean difference (2 parts 21 seconds) and the standing difference (27 micro-units); the remainder, 484 parts 84 seconds 73 micro-units, is the incremental correction.
102
已上所推，皆初日之數。 其推次日，皆以加分立差，累加平立合差，爲次日平立合差。 以平立合差減其日加分，爲次日加分，盈縮並同。 其加分累積之，即盈縮積，其數並見立成。
The figures derived above are all for the first day. For each following day, add the incremental standing difference cumulatively to the combined mean–standing difference to obtain that day's combined difference. Subtract the combined mean–standing difference from the day's incremental correction to obtain the next day's incremental correction; the rule is the same for expansion and contraction. Summing the incremental corrections yields the expansion–contraction accumulation; the values are given in the ready-made tables.
103
▲太陰遲疾平立三差之原
▲ Lunar slow–fast motion: origin of the three mean, standing, and fixed differences
104
太陰轉週二十七日五十五刻四六。 測分四象，象各七段，四象二十八段，每段十二限，每象八十四限，凡三百三十六限，而四象一週。 以四象爲法，除轉週日，得每象六日八八八六五，分爲七段，每段下實測月行遲疾之數，與平行相較，以求積差。
The Moon's rotation period is 27 days 55 ke 46. Observation divides it into four quadrants of seven segments each—twenty-eight segments in all. Each segment has twelve limits, each quadrant eighty-four limits, 336 limits in total, completing one circuit in four quadrants. Dividing the rotation period by four quadrants gives 6 days 88865 per quadrant; divided into seven segments, each segment's observed slow–fast motion is compared with mean motion to obtain the accumulated difference.
105
第一段
The first segment.
106
一十二
12.
107
一度二十八分七一二
1°28′712.
108
第二段
The second segment.
109
二十四
24.
110
二度四十五分九六一六
2°45′9616.
111
第三段
The third segment.
112
三十六
36.
113
三度四十八分三七九二
3°48′3792.
114
第四段
The fourth segment.
115
四十八
48.
116
四度三十二分五九五二
4°32′5952.
117
第五段
The fifth segment.
118
四度九十五分二四
4°95′24.
119
第六段
The sixth segment.
120
七十二
72.
121
五度三十二分九四四
5°32′944.
122
第七段
The seventh segment.
123
八十四
84.
124
五度四十二分三三七六
5°42′3376.
125
各置其段積差，以其段積限爲法除之，爲各段限平差。 置各段限平差，與後段相減爲一差。 置一差，與後段一差相減爲二差。
For each segment, divide its accumulated difference by its accumulated limits to obtain the per-limit mean difference. Take each segment's per-limit mean difference and subtract the next segment's to obtain a first difference. Take a first difference and subtract the next segment's first difference to obtain a second difference.
126
限平差
Per-limit mean difference
127
第一段
The first segment.
128
一十零分七二六零
10 parts 7260.
129
四十七秒七六
47 seconds 76.
130
九秒三六
9 seconds 36.
131
第二段
The second segment.
132
一十零分二四八四
10 parts 2484.
133
五十七秒一二
57 seconds 12.
134
九秒本六
9 seconds 36.
135
第三段
The third segment.
136
九分六七七二
9 parts 6772.
137
六十六秒四八
66 seconds 48.
138
九秒三六
9 seconds 36.
139
第四段
The fourth segment.
140
九分零一二四
9 parts 0124.
141
七十五秒八四
75 seconds 84.
142
九秒三六
9 seconds 36.
143
第五段
The fifth segment.
144
八分二五四零
8 parts 2540.
145
八十五秒二零
85 seconds 20.
146
九秒三六
9 seconds 36.
147
第六段
The sixth segment.
148
七分四零二零
7 parts 4020.
149
九十四秒五六
94 seconds 56.
150
第七段
The seventh segment.
151
六分四五六四
6 parts 4564.
152
置第一段限平差一十零分七二六爲凡平積。 置第一段一差四十七秒七六，以第一段二差九秒三六減之，餘三十八秒四十微，爲凡平積差。 另置第一段二差九秒三十六微折半，得四秒六十八微，爲凡立積差。 以凡平積差三十八秒四十微，加凡平積一十零分七二六，得一十一分一十一秒，爲定差。 置凡平積差三十八秒四十微，以凡立積差四秒六十八微減之，餘三十三秒七十二微爲實，以十二限爲法除之，得二秒八十一微，爲平差。 置凡立積差四秒六十八微爲實，十二限爲法，除二次，得三微二十五纖，爲立差。
Take the first segment's per-limit mean difference of 10 parts 726 as the general mean accumulation. Take the first segment's first difference of 47 seconds 76 and subtract its second difference of 9 seconds 36; the remainder, 38 seconds 40 micro-units, is the general mean accumulated difference. Halve the first segment's second difference of 9 seconds 36 micro-units to obtain 4 seconds 68 micro-units as the general standing accumulated difference. Adding the general mean accumulated difference of 38″40 wei to the general mean accumulation of 10′726 yields 11′11″ as the fixed difference. Take the general mean accumulated difference of 38″40 wei and subtract the general upright accumulated difference of 4″68 wei; the remainder, 33″72 wei, is the dividend; dividing by 12 limits gives 2″81 wei as the mean difference. Set the general upright accumulated difference of 4″68 wei as dividend; dividing twice by 12 limits yields 3 wei 25 xian as the upright difference.
153
凡求遲疾，皆以入歷日乘十二限二十分，以在八十四限已下爲初，已上轉減一百六十八限餘爲末。 各以初末限乘立差，得數以加平差，再以初末限乘之，得數以減定差，餘以初末限乘之，爲遲疾積。 其初限是從最遲最疾處順推至後，末限是從最遲最疾處逆溯至前，其距其距最遲疾處同，故其積度同。 太陰與太陽立法同，但太陽以定氣立限，故盈縮異數。 太陰以平行立限，故遲疾同原。
To find slow-fast in every case, multiply the days entered into the ephemeris by 12 limits 20 parts; below 84 limits counts as the beginning, above that subtract from 168 limits and use the remainder as the end. For each case, multiply the beginning or end limit by the upright difference and add to the mean difference; multiply again by that limit, subtract from the fixed difference, and multiply the remainder by the limit to obtain the slow-fast accumulation. The beginning limit runs forward from the point of greatest slowness and greatest speed; the end limit runs backward from it; the distance from that point is the same in both cases, so the accumulated degrees are the same. The moon and sun follow the same legislative pattern, but the sun sets its limits by fixed qi, so the surplus-deficit numbers differ. The moon sets its limits by mean motion, so the slow-fast values match the original constants.
154
布立成法置立差三微二十五纖，以六因之，得一十九微五十纖，爲損益立差。 置平差二秒八十一微，倍之，得五秒六十二微，再加損益立差一十九微五十纖，共得五秒八十一微，爲初限平立合差。 自此以損益立差，累加之，即每限平立合差。 至八十限下，積至二十一秒四一五，爲平立合差之極。 八十一限下差一秒七八零九，八十二限下一秒七八零八，至八十三限下，平立合差，與益分中分，爲益分之終。 八十四限下差，亦與損分中分，爲損分之始。 至八十六限下差，亦二十一秒四一五，自此以損益立差累減之，即每限平立合差，至末限與初限同。 置定差一十一分一十一秒，內減平差二秒八十一微，再減立差三微二十五纖，餘一十一分零八秒一十五微七十五纖爲加分定差，即初限損益分。 置損益分，以其限平立合差益減損加之。 即爲次限損益分。 以益分積之，損分減之，便爲其下遲疾度。 以八百二十分爲一限日率，累加八百二十分爲每限日率。 以上俱詳立成。
To deploy the ready-table method: set the upright difference at 3 wei 25 xian; multiplying by 6 gives 19 wei 50 xian as the surplus-deficit upright difference. Take the mean difference of 2″81 wei and double it to 5″62 wei; adding the surplus-deficit upright difference of 19 wei 50 xian gives 5″81 wei as the beginning-limit mean-upright combined difference. From here, each limit's mean-upright combined difference is obtained by cumulatively adding the surplus-deficit upright difference. By the eightieth limit the accumulation reaches 21″415, the maximum of the mean-upright combined difference. At the eighty-first limit the difference is 1″7809; at the eighty-second, 1″7808; by the eighty-third limit the mean-upright combined difference reaches the midpoint of the surplus portion—the end of the surplus part. At the eighty-fourth limit the difference likewise reaches the midpoint of the deficit portion—the beginning of the deficit part. By the eighty-sixth limit the difference is again 21″415; from here each limit's mean-upright combined difference is obtained by cumulatively subtracting the surplus-deficit upright difference, until the end limit matches the beginning. Set the fixed difference at 11′11″; subtract the mean difference of 2″81 wei and the upright difference of 3 wei 25 xian; the remainder, 11′08″15 wei 75 xian, is the additive fixed difference—the beginning-limit surplus-deficit parts. Take the surplus-deficit part and add or subtract according to that limit's mean-upright combined difference—adding for surplus portions and subtracting for deficit portions. The result is the next limit's surplus-deficit part. Accumulate the surplus portions and subtract the deficit portions to obtain the slow-fast degree for the limit below. Take 820 parts as one limit's day-rate; cumulatively adding 820 parts yields each limit's day-rate. All of the above is set out in detail in the ready-made table.
155
五星平立定三差之原凡五星各以實測，分其行度爲八段，以求積差，略如日月法。
Origin of the five planets' mean, upright, and fixed three differences: for each planet, actual measurement divides its motion into eight segments to derive the accumulated difference, broadly parallel to the sun-and-moon procedure.
156
木星立差加，平差減。
Jupiter: the upright difference is added and the mean difference subtracted.
157
第一段
First segment
158
一十一日五十刻
11 days 50 quarters
159
一度二一五二九七一一二
1.215297112°
160
第二段
Second segment
161
二十三日
23 days
162
二度三四零五二一四
2.3405214°
163
第三段
Third segment
164
三十四日五十刻
34 days 50 quarters
165
三度三五四一三七二六五
3.354137265°
166
第四段
Fourth segment
167
四十六日
46 days
168
四度二三四六零九一二
4.23460912°
169
第五段
Fifth segment
170
五十七日五十刻
57 days 50 quarters
171
四度九六零四零一三七五
4.960401375°
172
第六段
Sixth segment
173
六十九日
69 days
174
五度五零九九七八四四
5.50997844°
175
第七段
Seventh segment
176
八十零日五十刻
80 days 50 quarters
177
五度八六一八零四七二五
5.861804725°
178
第八段
Eighth segment
179
九十二日
92 days
180
五度九九四三四四六四
5.99434464°
181
凡平差
General mean difference
182
凡平較
General mean comparison
183
凡立較
General upright comparison
184
第一段一十分五六七八零一三十九秒一六二一六秒二四二二
First segment: 10′567801; 39″1621; 6″2422
185
第二段一十分一七六一八
Second segment: 10′17618
186
四十五秒四零四三六秒二四二二
45″4043; 6″2422
187
第三段九分七二二一三七
Third segment: 9′722137
188
五十一秒六四六五六秒二四二二
51″6465; 6″2422
189
第四段九分二零五六七二
Fourth segment: 9′205672
190
五十七秒八八八七六秒二四二二
57″8887; 6″2422
191
第五段八分六二六七八五
Fifth segment: 8′626785
192
六十四秒一三零九六秒二四二二
64″1309; 6″2422
193
第六段七分九八五四七六
Sixth segment: 7′985476
194
七十零秒三七二一六秒二四二二
70″3721; 6″2422
195
第七段七分二八一七四五
Seventh segment: 7′281745
196
七十六秒六一五三
76″6153
197
第八段六分五一五五九二
Eighth segment: 6′515592
198
各置其段所測積差度爲實，以段日爲法除之，爲凡平差。 各以凡平差與次段凡平差相較，爲凡平較。 又以凡平較與次段凡平較相較，爲凡立較。 置第一段凡平較三十九秒一六二一，減其下凡立較六秒二四二二，餘三十二秒九一九九，爲初段平立較。 加初段凡平差一十分五六七八零一，共得一十零分八十九秒七十零微，爲定差。 秒置萬位。 置初段平立較差三十二秒九一九九，內減凡立較之半，三秒一二一一，餘二十九秒七九八八，以段日一十一日五十刻除之，得二秒五十九微一十二纖爲平差。 置凡立差之半，三秒一二一一，以段日爲法除二次，得二微三十六纖爲立差。
For each segment, set the measured accumulated-difference in degrees as dividend and divide by the segment days to obtain the general mean difference. For each segment, compare its general mean difference with the next segment's to obtain the general mean comparison. Compare each general mean comparison with the next segment's to obtain the general upright comparison. Take the first segment's general mean comparison of 39″1621 and subtract the general upright comparison of 6″2422; the remainder, 32″9199, is the first-segment mean-upright comparison. Adding the first segment's general mean difference of 10′567801 gives 10′89″70 wei as the fixed difference. Seconds are carried into the ten-thousands place. Set the first-segment mean-upright comparison of 32″9199 and subtract half the general upright comparison, 3″1211; the remainder, 29″7988, divided by the segment length of 11 days 50 quarters gives 2″59 wei 12 xian as the mean difference. Take half the general upright comparison, 3″1211; dividing twice by the segment days yields 2 wei 36 xian as the upright difference.
199
已上爲木星平立定三差之原。
The above constitutes the origin of Jupiter's mean, upright, and fixed three differences.
200
火星盈初縮末。 立差減，平差減。
Mars: surplus at the beginning, deficit at the end. The upright difference is subtracted and the mean difference subtracted.
201
第一段
First segment
202
七日六十二刻五十分
7 days 62 quarters 50 parts
203
第二段
Second segment
204
一十五日二十五刻
15 days 25 quarters
205
第三段
Third segment
206
二十二日八十七刻五十分
22 days 87 quarters 50 parts
207
第四段
Fourth segment
208
三十零日五十零刻
30 days 50 quarters
209
第五段
Fifth segment
210
三十八日一十二刻五十分
38 days 12 quarters 50 parts
211
第六段
Sixth segment
212
四十五日七十五刻
45 days 75 quarters
213
第七段
Seventh segment
214
五十三日三十七刻五十分
53 days 37 quarters 50 parts
215
第八段
Eighth segment
216
六十一日
61 days
217
第一段
First segment
218
六度二六八二五一二二八一八五五九三七五
6.26825122818559375°
219
第二段
Second segment
220
一十一度六零零一七五七四三五九三七五
11.60017574359375°
221
第三段
Third segment
222
一十六度零二五九六三七九二五一九五三一二五
16.02596379251953125°
223
第四段
Fourth segment
224
一十九度六六九零一三六二一二五
19.66901362125°
225
第五段
Fifth segment
226
二十二度二七九八九一四七六零七四二一八七五
22.27989147607421875°
227
第六段
Sixth segment
228
二十四度一六八二二八六零三二八一二五
24.16822860328125°
229
第七段
Seventh segment
230
二十五度三三一五五六二四九二六零一五六二五
25.33155624926015625°
231
第八段
Eighth segment
232
二十五度六一九五一五六六
25.61951566°
233
凡平差
General mean difference
234
第一段
First segment
235
八十二分零六五七三四八四三七五
82.06573484375′
236
第二段
Second segment
237
七十六分零六六七二六一六七五
76.0667261675′
238
第三段
Third segment
239
七十零分零五八八五八一零九三七五
70.058858109375′
240
第四段
Fourth segment
241
六十四分一八二九六九二五
64.18296925′
242
第五段
Fifth segment
243
五十八分四三九零五九六零九三七五
58.439059609375′
244
第六段
Sixth segment
245
五十二分八二七一二九一八七五
52.8271291875′
246
第七段
Seventh segment
247
四十七分三四七一七七九八四三七五
47.347177984375′
248
第八段
Eighth segment
249
四十一分九九九二零六
41.999206′
250
凡平較
General mean comparison
251
第一段
First segment
252
六分一三九八四七二九六八七五
6.139847296875′
253
第二段
Second segment
254
六分零零七八六八零七八一二五
6.007868078125′
255
第三段
Third segment
256
五分八七五八八八八五九三七五
5.875888859375′
257
第四段
Fourth segment
258
五分七四三九零九六四零六二五
5.743909640625′
259
第五段
Fifth segment
260
五分六一一九三零四二一八七五
5.611930421875′
261
第六段
Sixth segment
262
五分四七九九五一二零三一二五
5.479951203125′
263
第七段
Seventh segment
264
五分三四七九七一九八四三七五
5.347971984375′
265
凡立較
General upright comparison
266
第一段
First segment
267
一十三秒一九七九二一八七五
13.197921875″
268
第二段
Second segment
269
一十三秒一九七九二一八七五
13.197921875″
270
第三段
Third segment
271
一十三秒一九七九二一八七五
13.197921875″
272
第四段
Fourth segment
273
一十三秒一九七九二一八七五
13.197921875″
274
第五段
Fifth segment
275
一十三秒一九七九二一八七五
13.197921875″
276
第六段
Sixth segment
277
一十三秒一九七九二一八七五
13.197921875″
278
凡平較前多後少，應加凡立較。 置初段下凡平較六分一三九八四七二九六八七五，加凡立較一十三秒一九七九二一八七五，得六分二七一八二六五一五六二五，爲初日下平立較。 置初段凡平差八十二分二十零秒六五七三四八四三七五，加初日下平立較六分二七一八二六五一五六二五，得八十八分四十七秒八十四微，爲定差。 置初日下平立較六分二七一八二六五一五六二五，加凡立較之半，六秒五九八九六零九三七五，得分三三七八一六一二五爲實，以段日而一，得八十三秒一十一微八十九纖爲平差。 置凡立較之半，六秒五九八九六零九三七五，以段日七日六十二刻五十分爲法除二次，得一十一微三十五纖爲立差。
When the general mean comparisons are greater in front and less behind, add the general upright comparison. Set the first segment's lower general mean comparison at 6.139847296875′; add the general upright comparison of 13.197921875″ to obtain 6.271826515625′ as the first day's lower mean-upright comparison. Set the first segment's general mean difference at 82′20″6573484375; add the first day's lower mean-upright comparison of 6.271826515625′ to obtain 88′47″84 wei as the fixed difference. Take the first day's lower mean-upright comparison of 6.271826515625′, add half the general upright comparison (6.5989609375″), yielding 3.337816125′ as dividend; dividing by the segment days gives 83″11 wei 89 xian as the mean difference. Take half the general upright comparison, 6.5989609375″; dividing twice by the segment length of 7 days 62 quarters 50 parts yields 11 wei 35 xian as the upright difference.
279
火星縮初盈末平差負減，立差減。
Mars: deficit at the beginning and surplus at the end—subtract the mean difference as negative and subtract the upright difference.
280
第一段
First segment
281
一十五日二十五刻
15 days 25 quarters
282
第二段
Second segment
283
三十零日五十刻
30 days 50 quarters
284
第三段
Third segment
285
四十五日七十五刻
45 days 75 quarters
286
第四段
Fourth segment
287
六十一日
61 days
288
第五段
Fifth segment
289
七十六日二十五刻
76 days 25 quarters
290
第六段
Sixth segment
291
九十一日五十刻
91 days 50 quarters
292
第七段
Seventh segment
293
一百零六日七十五刻
106 days 75 quarters
294
第八段
Eighth segment
295
一百二十二日
122 days
296
第一段
First segment
297
四度五三一二五一八五七九六八七五
4.53125185796875°
298
第二段
Second segment
299
九度一零二九六一四五一二五
9.10296145125°
300
第三段
Third segment
301
一十三度五三一六七零九零一七七三七五
13.53167090177375°
302
第四段
Fourth segment
303
一十七度四七八九七九零四
17.47897904°
304
第五段
Fifth segment
305
二十零度八四三六六三零六六四零六二五
20.84366306640625°
306
第六段
Sixth segment
307
二十三度四三一三三六二四一二五
23.43133624125°
308
第七段
Seventh segment
309
二十五度零九二四三五二八三四六八七五
25.09243528346875°
310
第八段
Eighth segment
311
二十五度六一八三七四七二
25.61837472°
312
凡平差
General mean difference
313
第一段
First segment
314
二十九分七一三一二六九三七五
29.7131269375′
315
第二段
Second segment
316
二十九分八四五七七五二五
29.84577525′
317
第三段
Third segment
318
二十九分五七八三五五零六二五
29.5783550625′
319
第四段
Fourth segment
320
二十八分六五四零六四
28.654064′
321
第五段
Fifth segment
322
二十七分三三三九五一五六二五
27.3339515625°
323
第六段
Sixth segment
324
二十五分六一八零一七七五
25.61801775°
325
第七段
Seventh segment
326
二十三分五零六二六二五六二五
23.5062625625°
327
第八段
Eighth segment
328
二十零分九九八六八六
20.998686°
329
凡平較
General mean comparison
330
凡立較
General upright comparison
331
第一段一十三秒二六四八三一二五
First segment: 13.26483125″
332
一十三秒五七六九七七五
13.5769775″
333
第二段二十六秒八四一八零八七五
Second segment: 26.84180875″
334
六十五秒五八七二九七五
65.5872975″
335
第三段九十二秒四二九一零六二五
Third segment: 92.42910625″
336
三十九秒五八二一三七五
39.5821375″
337
第四段一分三二零一一二四三七五
Fourth segment: 1′32.01124375
338
三十九秒五八二一三七五
39.5821375″
339
第五段一分七一五九三三八一二五
Fifth segment: 1′71.59338125
340
三十九秒五八二一三七五
39.5821375″
341
第六段二分一一一七五五一八七五
Sixth segment: 2′11.17551875
342
三十九秒五八二一三七五
39.5821375″
343
第七段二分五零七五七六二五
Seventh segment: 2′50.757625
344
取凡立較停者，三十九秒五八二一三七五，以較一段下凡平較一十三秒二六四八三一二五，餘二十六秒三一七三零六二五爲較較，以加一段下凡平差二十九分七一三一二六九三七五，得二十九分九十七秒六十三微，爲定差。 置較較二十六秒三一七三零六二五，以段日一十五日二十五刻而一，得一秒七二五七二五。 再置凡立較之半一十九秒七九一零六八七五，以段日而一，得一秒二九七七七五。 兩數並得三秒零二微三十五纖爲平差。 置凡立較之半一十九秒七九一零六八七五，以段日一十五日二五爲法除二次，得八微五十一纖，爲立差。
Take the general upright comparison at its stop point, 39.5821375″; compare with the first segment's lower general mean comparison of 13.26483125″; the remainder, 26.31730625″, is the comparison-of-comparisons; adding this to the first segment's lower general mean difference of 29.7131269375′ gives 29′97″63 wei as the fixed difference. Set the comparison-of-comparisons at 26.31730625″; dividing once by the segment length of 15 days 25 quarters gives 1.725725″. Take half the general upright comparison, 19.79106875″; dividing once by the segment days gives 1.297775″. Combining both values yields 3.02 wei 35 xian as the mean difference. Take half the general upright comparison, 19.79106875″; dividing twice by the segment length of 15 days 25 quarters yields 8 wei 51 xian as the upright difference.
345
已上爲火星平立定三差之原。
The above constitutes the origin of Mars's mean, upright, and fixed three differences.
346
▲土星盈歷立差加，平差減。
▲ Saturn surplus cycle: add the upright difference and subtract the mean difference.
347
第一段
First segment
348
一十一日五十刻
11 days 50 quarters
349
一度六八三二四五八二八七五
1.683245828875°
350
第二段
Second segment
351
二十三日
23 days
352
三度二三二一六四零一
3.23216401°
353
第三段
Third segment
354
三十四日五十刻
34 days 50 quarters
355
四度六二零九三零零八六二五
4.62093008625°
356
第四段
Fourth segment
357
四十六日
46 days
358
五度八二三七一九六
5.8237196°
359
第五段
Fifth segment
360
五十七日五十刻
57 days 50 quarters
361
六度八一四七零八六六八七五
6.81470866875°
362
第六段
Sixth segment
363
六十九日
69 days
364
七度五六八零七一一一
7.56807111°
365
第七段
Seventh segment
366
八十零日五十刻
80 days 50 quarters
367
八度零五七九八四一九一二五
8.05798419125°
368
第八段
Eighth segment
369
九十二日
92 days
370
八度二五八六二二八八
8.25862288°
371
凡平差
General mean difference
372
凡平較
General mean comparison
373
凡立較第一段一十四分六三六九二零二五五十八秒四零三三二五七秒四八五三五第二段一十四分零五二八八七
General upright comparison. First segment: 14′63692025; 58″403325; 7″48535. Second segment: 14′052887
374
六十五秒八八八六七五七秒四八五三五第三段一十三分三九四零零零二五七十三秒三七四零二五七秒四八五三五第四段一十二分六六零二六
65″888675; 7″48535. Third segment: 13′39400025; 73″374025; 7″48535. Fourth segment: 12′66026
375
八十零秒八五九三七五七秒四八五三五第五段一十一分八五一六六六二五八十八秒三四四七二五七秒四八五三五第六段一十一分九六八二一九
80″859375; 7″48535. Fifth segment: 11′85166625; 88″344725; 7″48535. Sixth segment: 11′968219
376
九十五秒八三零零七五七秒四八五三五第七段一十零分零零九九一八二五一分零三秒三一五四二五第八段八分九七六七六四
95″830075; 7″48535. Seventh segment: 10′00991825; 1′03″315425. Eighth segment: 8′976764
377
置第一段下凡平較，內減其下凡立較，餘五十零秒九一七九七五，爲平立較。 以平立較，加本段凡平差，得一十五分一十四秒六十一微，爲定差。 置平立較，內減凡立較之半，三秒七四二六七五，餘四十七秒一七五三，以段日十一日五十刻而一，得四秒一十零微二十二纖，爲平差。 置凡立較之半，以段日除二次，得二微八十三纖，爲立差。
Take the first segment's lower general mean comparison and subtract its lower general upright comparison; the remainder, 50.917975″, is the mean-upright comparison. Add the mean-upright comparison to this segment's general mean difference to obtain 15′14″61 wei as the fixed difference. Take the mean-upright comparison and subtract half the general upright comparison, 3.742675″; the remainder, 47.1753″, divided by the segment length of 11 days 50 quarters gives 4.10 wei 22 xian as the mean difference. Take half the general upright comparison; dividing twice by the segment days yields 2 wei 83 xian as the upright difference.
378
▲土星縮歷立差加，平差減。
▲ Saturn deficit cycle: add the upright difference and subtract the mean difference.
379
第一段
First segment
380
一十一日五十刻
11 days 50 quarters
381
一度二四一九七四二六八七五
1.24197426875°
382
第二段
Second segment
383
二十三日
23 days
384
二度四一三七三五六九
2.41373569°
385
第三段
Third segment
386
三十四日五十刻
34 days 50 quarters
387
三度四八五零七九六八六二五
3.48507968625°
388
第四段
Fourth segment
389
四十六日
46 days
390
四度四二五八零一六八
4.42580168°
391
第五段
Fifth segment
392
五十七日五十刻
57 days 50 quarters
393
五度二零五六九七零九三七五
5.20569709375°
394
第六段
Sixth segment
395
六十九日
69 days
396
五度七九四五六一三五
5.79456135°
397
第七段
Seventh segment
398
八十零日五十刻
80 days 50 quarters
399
六度一六二四一一零零四七五
6.16241100475°
400
第八段
Eighth segment
401
九十二日
92 days
402
六度二七八三七八零八
6.27837808°
403
凡平差
General mean difference
404
凡平較
General mean comparison
405
凡立較第一段一十分七九九七七六二五三十零秒五二七三二五八秒七五四九五第二段一十分四九四五零三
General upright comparison. First segment: 10′79977625; 30″527325; 8″75495. Second segment: 10′494503
406
三十九秒二八二二七五八秒七五四九五第三段一十分一零一六八零二五四十八秒零三七二二五八秒七五四九五第四段九分六二一三零八五十六秒七九二一七五八秒七五四九五第五段九分零五三三八六二五
39″282275; 8″75495. Third segment: 10′10168025; 48″037225; 8″75495. Fourth segment: 9′621308; 56″792175; 8″75495. Fifth segment: 9′05338625
407
六十五秒五四七一二五八秒七五四九五第六段八分三九七九一五
65″547125; 8″75495. Sixth segment: 8′397915
408
七十四秒三零三零七五八秒七五四九五第七段七分六五四八九四二五
74″303075; 8″75495. Seventh segment: 7′65489425
409
八十三秒零五七零七五第八段六分八二四三二四
83″057075. Eighth segment: 6′824324
410
置一段凡平較，內減其下凡立較，餘二十一秒七七二三七五，爲平立較。 以平立較加入本段凡平差，得一十一分零一秒七十五微，爲定差。 置平立較，內減凡立較之半，四秒三七七四七五，餘一十七秒三九四九，以段日一十一日五十刻爲法除之，得一秒五十一微二十六纖，爲平差。 置凡立較之半，以段日爲法除二次，得三微三十一纖爲立差。
Take the first segment's general mean comparison and subtract its lower general upright comparison; the remainder, 21.772375″, is the mean-upright comparison. Add the mean-upright comparison to this segment's general mean difference to obtain 11′1″75 wei as the fixed difference. Take the mean-upright comparison and subtract half the general upright comparison, 4.377475″; the remainder, 17.3949″, divided by the segment length of 11 days 50 quarters gives 1.51 wei 26 xian as the mean difference. Take half the general upright comparison; dividing twice by the segment days yields 3 wei 31 xian as the upright difference.
411
已上爲土星平定三差之原。
The above constitutes the origin of Saturn's mean, upright, and fixed three differences.
412
金星立差加，平差減。
Venus: the upright difference is added and the mean difference subtracted.
413
第一段
First segment
414
一十一日五十刻
11 days 50 quarters
415
空度四零二一三四零九八七五
0.40213409875°
416
第二段
Second segment
417
二十三日
23 days
418
空度七九一三九三六六
0.79139366°
419
第三段
Third segment
420
三十四日五十刻
34 days 50 quarters
421
一度一五四九一二零八一二五
1.15491208125°
422
第四段
Fourth segment
423
四十六日
46 days
424
一度七四九八二二七六
1.74982276°
425
第五段
Fifth segment
426
五十七日五十刻
57 days 50 quarters
427
一度七五三二五九零九三七五
1.75325909375°
428
第六段
Sixth segment
429
六十九日
69 days
430
一度九六二三五四四八
1.96235448°
431
第七段
Seventh segment
432
八十零日五十刻
80 days 50 quarters
433
二度零九四二四二三一六二五
2.09424231625°
434
第八段
Eighth segment
435
九十二日
92 days
436
二度一三六零五六
2.136056°
437
凡平差
General mean difference
438
凡平較
General mean comparison
439
凡立較第一段
General upright comparison, first segment
440
三分四九六八一八二五五秒五九七六二五
3.49681825′; 5.597625″
441
三秒七二九四五第二段
3.72945″; second segment
442
三分四四零八四二零零九秒三二七零七五
3.44084200′; 9.327075″
443
三秒七二九四五第三段三分三四七五七一二五一十三秒零六五五二五三秒七二九四五第四段
3.72945″; third segment 3.34757125′; 13.065525″; 3.72945″; fourth segment
444
三分二一七零零六
3.217006′
445
一十六秒七八五九七五三秒七二九四五第五段
16.785975″; 3.72945″; fifth segment
446
三分零四九一四六二五二十零秒五一五四二五三秒七二九四五第六段
3.04914625′; 20.515425″; 3.72945″; sixth segment
447
二分八四三九九二
2.843992′
448
二十四秒二四四八七五三秒七二九四五第七段二分六零一五四三二五二十七秒九七四三二五第八段
24.244875″; 3.72945″; seventh segment 2.60154325′; 27.974325″; eighth segment
449
二分三二一八
2.3218′
450
置一段下凡平較，與其凡立較相減，餘一秒八六一七五爲平立較，以加凡平差，得三分五十一秒五十五微，爲定差。 置平立較，與凡立較之半，一秒八六四七二五相減，餘三十四纖，以段日一十一日五十刻爲法除之，得三纖，爲平差。 置凡立較之半，以段日爲爲法除二次，得一微四十一纖，爲立差。
Take the first segment's lower general mean comparison and subtract its lower general upright comparison; the remainder, 1″86175, is the mean-upright comparison; adding it to the general mean difference gives 3′51″55 wei as the fixed difference. Take the mean-upright comparison and subtract half the general upright comparison, 1″864725; the remainder, 34 xian, divided by the segment length of 11 days 50 quarters gives 3 xian as the mean difference. Take half the general upright comparison; dividing twice by the segment days yields 1 wei 41 xian as the upright difference.
451
已上爲金星平立定三差之原。
The above constitutes the origin of Venus's mean, upright, and fixed three differences.
452
▲水星立差加，平差減。
▲ Mercury: the upright difference is added and the mean difference subtracted.
453
第一段
First segment
454
一十一日五十刻
11 days 50 quarters
455
空度四四零八四七三五三七五
0.44084735375°
456
第二段
Second segment
457
二十三日
23 days
458
空度八六三一零一六八
0.86310168°
459
第三段
Third segment
460
三十四日五十刻
34 days 50 quarters
461
一度二五三八九六三七六二五
1.25389637625°
462
第四段
Fourth segment
463
四十六日
46 days
464
一度六零零三六四八四
1.60036484°
465
第五段
Fifth segment
466
五十七日五十刻
57 days 50 quarters
467
一度八八九六三一零四三七五
1.88963104375°
468
第六段
Sixth segment
469
六十九日
69 days
470
二度一零八八六六六
2.0888666°
471
第七段
Seventh segment
472
八十零日五十刻
80 days 50 quarters
473
二度二四五二九二一一三七五
2.24529211375°
474
第八段
Eighth segment
475
九十二日
92 days
476
二度二八五六四四三二
2.28564432°
477
凡平差
General mean difference
478
凡平較
General mean comparison
479
凡立較
General upright comparison
480
第一段三分八三三四五五二五八秒零八三九二五
First segment: 3.83345525′; 8.083925″
481
三秒七二九四五
3.72945″
482
第二段三分七五二六一六
Second segment: 3.752616′
483
一十一秒八一三三七五三秒七二九四五
11.813375″; 3.72945″
484
第三段三分六三四四八二二五一十五秒五四二八二五三秒七二九四五
Third segment: 3.63448225′; 15.542825″; 3.72945″
485
第四段三分四七九零五四
Fourth segment: 3.479054′
486
一十九秒二七二二七五三秒七二九四五
19.272275″; 3.72945″
487
第五段三分二八六三三一二五二十三秒零零一七二五三秒七二九四五
Fifth segment: 3.28633125′; 23.001725″; 3.72945″
488
第六段三分零五六三一四
Sixth segment: 3.056314′
489
二十六秒七三二一七五三秒七二九四五
26.732175″; 3.72945″
490
第七段二分七八九零零二二五三十零秒四六零六二五
Seventh segment: 2.78900225′; 30.460625″
491
第八段二分四八四三九六
Eighth segment: 2.484396′
492
術同金星，求得定差三分八十七秒九十微，平差二十一微六十五纖，立差一微四十一纖。
The procedure matches Venus, yielding a fixed difference of 3′87″90 wei, a mean difference of 21 wei 65 xian, and an upright difference of 1 wei 41 xian.
493
已上爲水星平立定三差之原。
The above constitutes the origin of Mercury's mean, upright, and fixed three differences.
494
在五星，皆以立差爲秒，平差爲本，定差爲總。 五星各以段次因秒，木土金水四星並本，惟火星較本，各以積日而積，五星皆較總，又各以積日乘之，得各實測之度分。
For all five planets, the upright difference counts as seconds, the mean difference as the base, and the fixed difference as the total. Each planet applies the seconds by segment order; Jupiter, Saturn, Venus, and Mercury all add to the base, while only Mars compares against the base; each accumulates by accumulated days; all five then compare to the total and multiply again by accumulated days to obtain each planet's observed degrees and parts.
495
五星積日，皆本度率，除週日得三百六十五度二十五分太。 各以四分之一爲象限，惟火星用象限三之一，減象限爲盈初縮末限，加象限爲縮初盈末限。 其命度爲日者，爲各取盈縮歷乘除之便，其實積日之數，即積度也。
Each planet's accumulated days derives from the degree rate; dividing by the circuit days yields 365°25′ surplus. Each planet takes one-quarter of its cycle as the quadrant; Mars alone uses one-third; subtracting the quadrant gives the surplus-at-beginning and deficit-at-end limit, and adding it gives the deficit-at-beginning and surplus-at-end limit. Naming degrees as days is only a convenience for multiplying and dividing in each planet's surplus-deficit table; in reality accumulated days and accumulated degrees are the same number.
496
▲裏差刻漏
▲ Latitude difference and clepsydra divisions
497
求二至差股及出入差。 術曰：置所測北極出地四十度九十五分爲半弧背，以前割圓弧矢法，推得出地半弧弦三十九度二十六分，爲大三斜中股。 置測到二至黃赤道內外度二十三度九十分爲半弧背，以前法推得內外半弧弦二十三度七十一分。 又爲黃赤道大句，又爲小三斜弦。 置內外半弧弦自之爲句冪，半徑自之爲弦冪，二冪相減，開方得股，以股轉減半徑，餘四度八十一分爲二至出入矢，即黃赤道內外矢。 夏至日，南至地平七十四度二十六分半爲半弧背，求得日下至地半弧弦五十八度四十五分。 半徑六十零度八十七分半，爲大三斜中弦。 置大三斜中股三十九度二十六分，以二至內外半弧弦二十三度七十一分乘之爲實，以半徑六十零度八十七分半爲法除之，得一十五度二十九分，爲小三斜中股又爲小股。 置小三斜中股一十五度二十九分，去減日下至地半弧弦五十八度中十一分，餘四十三度一十六分，爲大股。 以出入矢四度八十一分，去減半徑六十零度八十七分半，餘五十六度零六分半，爲大股弦。 置大股弦，以小股一十五度二九乘之爲實，大股四十三度一六爲法除之，得一十九度八十七分爲小弦，即爲二至出入差半弧弦。 置二至出入差半弧弦，依法求到二至出入差半弧背一十九度九十六分一十四秒。 置二至出入差半弧背一十九度九十六一四秒，置二至出入半弧背一十九度九六一四，以二至黃赤道內外半弧弦二十三度七十一分除之，得八十四分一十九秒，爲度差分。
To find the solstitial-difference legs and the ingress-egress difference. Method: Take the observed north-pole altitude of 40°95′ as the half-arc back; by the prior circle-cutting arc-sagitta method, derive the altitude half-arc chord of 39°26′ as the large third triangle's middle leg. Take the measured solstitial ecliptic-equator inner-outer arc of 23°90′ as the half-arc back; by the same method derive the inner-outer half-arc chord of 23°71′. This is also the ecliptic-equator's large leg and the small third triangle's hypotenuse. Square the inner-outer half-arc chord for the leg power and the half-diameter for the chord power; subtract, take the square root for the leg, and subtract that leg from the half-diameter; the remainder, 4°81′, is the solstitial ingress-egress sagitta—the ecliptic-equator inner-outer sagitta. At the summer solstice, take the sun's southern altitude of 74°26½ as the half-arc back; the half-arc chord from the sun down to the earth is 58°45′. The half-diameter of 60°87½ serves as the large third triangle's middle hypotenuse. Take the large third triangle's middle leg of 39°26′, multiply by the solstitial inner-outer half-arc chord of 23°71′ as dividend, and divide by the half-diameter of 60°87½ to obtain 15°29′ as the small third triangle's middle leg—the small leg. Take the small third triangle's middle leg of 15°29′ and subtract it from the sun-to-earth half-arc chord of 58°45′; the remainder, 43°16′, is the large leg. Subtract the ingress-egress sagitta of 4°81′ from the half-diameter of 60°87½; the remainder, 56°06½, is the large leg's hypotenuse. Set the great-leg chord; multiply by the small leg of 15°29′ for the dividend and divide by the great leg of 43°16′ as divisor; the quotient, 19°87′, is the small chord—that is, the solstitial entry-and-exit difference half-arc chord. Set the solstitial entry-and-exit difference half-arc chord; by the prescribed method obtain the solstitial entry-and-exit difference half-arc back of 19°96′14″. Set the solstitial entry-and-exit difference half-arc back at 19°9614″ and the solstitial entry-and-exit half-arc back at 19°9614; dividing by the solstitial ecliptic-equator inner-and-outer half-arc chord of 23°71′ yields 84′19″ as the degree-difference parts.
498
求黃道每度書夜刻。 術曰：置所求每度黃赤道內外半弧弦，以二至出入差半弧背乘之爲實，二至黃赤道內外半弧弦爲法除之，爲每度出入差半弧背。 又術：置黃赤道內外半弧弦，以度差八十四分一十九秒乘之，亦得出入差半弧背。 置半徑內減黃赤道內外矢，即赤道二絃差，見前條立成。 餘數倍之，又三因之，得數加一度，爲日行百刻度。 又術：以黃赤道內外矢倍之，以減全徑餘數，三因加一度，爲日行百刻度，亦同。 置每度出入半弧背，以百刻乘之爲實，日行百刻爲法除之，得數爲出入差刻。 置二十五刻，以出入差刻視黃道，在赤道內加之，在赤道外減之，得數爲半晝刻，倍之爲晝刻，以減百刻，爲夜刻。
Procedure to find the day-and-night quarter-hour graduations for each degree of the ecliptic. The procedure says: Take the inner-and-outer half-arc chord of the ecliptic-equator for the sought degree, multiply by the solstitial entry-and-exit difference half-arc back as dividend, and divide by the solstitial ecliptic-equator inner-and-outer half-arc chord as divisor to obtain the per-degree entry-and-exit difference half-arc back. Alternate procedure: Take the ecliptic-equator inner-and-outer half-arc chord and multiply by the degree difference of 84′19″; the entry-and-exit difference half-arc back is obtained likewise. Subtract the ecliptic-equator inner-and-outer versine from the semidiameter; the remainder is the equatorial two-chord difference—see the ready reckoners in the preceding section. Double the remainder, multiply by three, and add one degree; the result is the daily motion in hundred-ke graduations. Alternate procedure: Double the ecliptic-equator inner-and-outer versine and subtract from the full diameter; triple the remainder and add one degree to obtain the daily motion in hundred-ke graduations—the result is the same. Take the per-degree entry-and-exit half-arc back, multiply by 100 ke as dividend, and divide by the daily motion in hundred-ke graduations as divisor; the quotient is the entry-and-exit difference in ke. Set 25 ke; according to whether the ecliptic lies within or outside the equator, add or subtract the entry-and-exit difference ke to obtain half-daytime ke; double it for daytime ke and subtract from 100 ke for nighttime ke.
499
如求冬至後四度晝刻。 術曰：置冬至後四十四度黃赤道內外半弧一十七度二十五分六十九秒，又爲黃赤道小弧弦，前立成中取之。 以二至出入差半弧背一十九度九十六分一十四秒乘之爲實，以二至黃赤道內外半弧弦二十三度七十一分爲法除之，得一十四度五十二分八十五秒，爲出入半弧背。 又法：置黃赤道內外半弧弦一十七度二五六九，以度差零度八四一九乘之，亦得一十四度五二八五，爲出入半弧背。 置半徑六十零度八七五，以四十四度黃赤道內外矢二度五十一分八十一秒又爲赤道二絃差，前立成中取之。 減之，餘五十八度三十五分六十九秒，即赤道小弦。 倍之，得一百一十六度七十一分三十八秒，三因之，加一度，得三百五十一度一十四分一十四秒，爲日行百刻度。 又術：倍黃赤道內外矢得五度零三分六十二秒，以減全徑一百二十一度七十五分，亦得一百一十六度七十一分三十八秒，三因加一度，爲日行百刻度，亦同。 置出入半弧背一十四度五十二分八十五秒，以百刻乘之爲實，以日行百刻度三百五十一度一十四分一十四秒爲法除之，得四刻一十三分七十五秒，爲出入差刻。 置二十五刻，以出入差刻四刻一十三分七十五秒減之，因冬至後四十四度，黃道在赤道外，故減。 餘二十零刻八十六分二十五秒，爲半晝刻。 倍之得四十一刻七十二分半，爲晝刻。 以晝刻減百刻，餘五十八刻二十七分半，爲夜刻。 又術：置出入差刻四刻一十三分七十五秒，倍之，得八刻二十七分半，以減春秋分晝夜五十刻，得四十一刻七十二分半，爲晝刻。 以倍刻加五十刻，得五十八刻二十七分半，爲夜刻。 晝減故廢加，餘仿此。
Example: finding the daytime ke four degrees after the winter solstice. The procedure says: Take the ecliptic-equator inner-and-outer half-arc at 44° after the winter solstice, 17°25′69″—which also serves as the ecliptic-equator small arc-chord—from the ready reckoners above. Multiply by the solstitial entry-and-exit difference half-arc back of 19°96′14″ as dividend and divide by the solstitial ecliptic-equator inner-and-outer half-arc chord of 23°71′ as divisor; the quotient, 14°52′85″, is the entry-and-exit half-arc back. Alternate method: Take the inner-and-outer half-arc chord of 17°2569 and multiply by the degree difference of 0°8419; the entry-and-exit half-arc back of 14°5285 is obtained likewise. Set the semidiameter at 60°875 and subtract the ecliptic-equator inner-and-outer versine at 44° after the winter solstice, 2°51′81″—also taken as the equatorial two-chord difference from the ready reckoners above. The remainder, 58°35′69″, is the equatorial small chord. Double it to obtain 116°71′38″; triple the result and add one degree to obtain 351°14′14″ as the daily motion in hundred-ke graduations. Alternate procedure: Double the versine to obtain 5°03′62″ and subtract from the full diameter of 121°75′; the remainder is again 116°71′38″; triple it and add one degree for the daily motion in hundred-ke graduations—the result is the same. Take the entry-and-exit half-arc back of 14°52′85″, multiply by 100 ke as dividend, and divide by the daily motion of 351°14′14″ as divisor; the quotient is 4 ke 13 parts 75 seconds as the entry-and-exit difference ke. Set 25 ke and subtract the entry-and-exit difference ke of 4 ke 13 parts 75 seconds; at 44° after the winter solstice the ecliptic lies outside the equator, so one subtracts. The remainder, 20 ke 86 parts 25 seconds, is half-daytime ke. Double it to obtain 41 ke 72 parts and a half as daytime ke. Subtract the daytime ke from 100 ke; the remainder, 58 ke 27 parts and a half, is nighttime ke. Alternate procedure: Take the entry-and-exit difference ke of 4 ke 13 parts 75 seconds and double it to obtain 8 ke 27 parts and a half; subtract from the spring-and-autumn equinox day-and-night of 50 ke each to obtain 41 ke 72 parts and a half as daytime ke. Add the doubled ke to 50 ke to obtain 58 ke 27 parts and a half as nighttime ke. Where daytime is found by subtraction, the addition method is set aside; apply the rest by analogy.
500
表格略
[Table omitted.]
501
右《歷草》所載晝夜刻分，乃大都即燕京晷漏也。 夏晝、冬夜極長，六十一刻八十四分，冬晝、夏夜極短，三十八刻一十六分。 明既遷都於燕，不知遵用。 惟正統己巳奏準頒歷用六十一刻，而羣然非之。 景泰初仍復用南京晷刻，終明之世未能改正也。
The day-and-night quarter-hour graduations recorded in the Calendar Draft above are those of Dadu—that is, the clepsydra graduations of Yanjing (Beijing). Summer daytime and winter night reach their maximum at 61 ke 84 parts; winter daytime and summer night their minimum at 38 ke 16 parts. After the Ming moved the capital to Yan, they did not adhere to these graduations. Only in the Zhengtong jisi year (1449) was approval granted to issue the calendar using 61 ke—and there was widespread objection. At the beginning of the Jingtai reign they reverted to Nanjing clepsydra graduations, and through the entire Ming the error was never corrected.