1
紀元曆
The Jiyuan Calendar
2
步交會
Procedure for Lunar Nodes and Conjunctions
3
交終分：一十九萬八千三百七十七、秒八百八十。
Convergence termination parts: 198,377; seconds 880.
4
交終日：二十七、餘一千五百四十七、秒八百八十。
Convergence termination period: 27 days, remainder 1,547 parts, seconds 880.
5
交中日：一十三、餘四千四百一十八、秒五千四百四十。
Convergence middle period: 13 days, remainder 4,418 parts, seconds 5,440.
6
朔差日：二、餘二千三百二十、秒九千一百二十。
New-moon difference: 2 days, remainder 2,320 parts, seconds 9,120.
7
望策：一十四、餘五千五百七十九。
Full-moon stride: 14 days, remainder 5,579 parts.
8
已上秒母一萬。
For the above constants, the seconds denominator is 10,000.
9
交率：三百二十四。
The convergence ratio is 324.
10
交數：四千一百二十七。
The convergence number is 4,127.
11
交終度：三百六十三、約分七十九、秒四十四。
Convergence termination arc: 363°79′44″ (approximate parts).
12
交中度：一百八十一、約分八十九、秒七十二。
Convergence middle arc: 181°89′72″ (approximate parts).
13
交象度：九十、約分九十四、秒八十六。
Convergence image arc: 90°94′86″ (approximate parts).
14
半交象度：四十五、約分四十七、秒四十三。
Half convergence image arc: 45°47′43″ (approximate parts).
15
日食陽曆限：三千四百，定法三百四十。
Solar eclipse yang-calendar limit: 3,400; fixing divisor 340.
16
陰曆限：四千三百，定法四百三十。
Lunar eclipse yin-calendar limit: 4,300; fixing divisor 430.
17
月食限：六千八百，定法四百四十。
Lunar eclipse limit: 6,800; fixing divisor 440.
18
已上分秒母各同一百。
For the limits above, the parts-and-seconds denominators are each 100.
19
推天正十一月經朔加時入交：置天正十一月經朔加時積分，以交終分及秒去之，不盡，滿日法為日，不滿為餘秒，即天正十一月經朔加時入交泛日及餘秒。
To find the eleventh civil month's mean new-moon hour of entry into convergence: set the accumulated parts for that syzygy's hour, cast out full convergence termination parts and seconds, convert the remainder through the day divisor into days and remainder-seconds, and obtain the general convergence day and remainder-seconds for the eleventh month's mean new moon.
20
求次朔及望入交：置天正十一月經朔加時入交泛日及餘秒，求次朔，以朔差加之； 求望，以望策加之：滿交終日及餘秒去之，即各得次朔及望加時入交泛日及餘秒。 若以經朔、望小餘減之，各得朔、望夜半入交泛日及餘秒。
To find the next new and full moon entry into convergence: set the eleventh-month mean new-moon hour general convergence day and remainder-seconds; for the next new moon, add the new-moon difference; for the full moon add the full-moon stride; cast out full convergence termination days and remainder-seconds to obtain each next new or full moon hour general convergence day and remainder-seconds. Subtracting each mean syzygy's minor remainder yields the midnight general convergence day and remainder-seconds for that new or full moon.
21
求定朔望夜半入交：因經朔、望夜半入交泛日及餘秒，視定朔、望日辰有進退者，亦進退交日，否則因經為定，各得所求。
To find fixed syzygy midnight convergence entry: start from mean midnight general convergence day and remainder-seconds; if the fixed syzygy's day-period advances or retreats, adjust the convergence day likewise, otherwise take the mean value as fixed.
22
求次定朔夜半入交：各因定朔夜半入交泛日及餘秒，大月加二日，小月加一日，餘皆加五千七百四十二、秒九千一百二十，即次朔夜半入交； 若求次日，累加一日：滿交終日及餘秒皆去之，即每日夜半入交泛日及餘秒。
For the next fixed new-moon midnight entry: from each fixed new-moon midnight general convergence day and remainder-seconds, add two days in a long month and one in a short month, and always add 5,742 seconds 9,120 to the remainder for the next new moon's midnight entry; to advance day by day, add one day cumulatively, cast out full convergence termination days and remainder-seconds, and obtain each midnight general convergence day and remainder-seconds.
23
求定朔望加時入交：置經朔、望加時入交泛日及餘秒，以入氣、入轉朏朒定數朏減朒加之，即得定朔、望加時入交泛日及餘秒。
To find fixed syzygy hour-of-addition convergence entry: set mean syzygy hour general convergence day and remainder-seconds, apply entry-into-qi and entry-into-rotation tiao–chuo fixed numbers (tiao subtract, chuo add), and obtain fixed syzygy hour general convergence day and remainder-seconds.
24
求定朔望加時月行入交積度：置定朔、望加時入交泛日及餘秒，以日法通日，內餘，進一位，如五千四百五十三而一為度，不滿，退除為分，即定朔、望加時月行入交積度及分。 每日夜半，准此求之。
To find fixed syzygy hour moon convergence accumulated degrees: convert the general convergence day through the day divisor with the remainder, shift one place, divide by 5,453 for degrees, reduce the remainder to parts, and obtain the hour moon's convergence accumulated degrees and parts. For each midnight, follow this procedure.
25
求定朔望加時月行入交定積度：置定朔、望加時月行入交積度及分，以定朔、望加時入轉遲疾度遲減疾加之，滿與不足，進退交終度及分。
To find fixed syzygy hour moon fixed convergence accumulated degrees: set hour convergence accumulated degrees and parts, apply hour entry-into-rotation slow-fast degree (slow subtract, fast add), and advance or retreat convergence termination degrees and parts as needed.
26
即定朔、望加時月行入交定積度及分。 每日夜半，准此求之。
That yields the fixed syzygy hour moon's fixed convergence accumulated degrees and parts. For each midnight, follow this procedure.
27
求定朔望加時月行入陰陽曆積度：置定朔、望加時月行入交定積度及分，如在交中度及分已下為入陽曆積度； 已上者去之，餘為入陰曆積度。 每日夜半，准此求之。
To find fixed syzygy hour yang/yin calendar accumulated degrees: set fixed hour moon fixed convergence accumulated degrees and parts; at or below convergence middle counts as yang-calendar accumulated degrees; if above, subtract convergence middle; the remainder is yin-calendar accumulated degrees. For each midnight, follow this procedure.
28
求定朔望加時月去黃道度：視月入陰陽曆積度及分，如交象已下為在少象； 已上，覆減交中度，餘為入老象。 置所入老、少象度及分於上，列交象度於下，以上減下，餘以乘上，五百而一，所得，用減所入老、少象度及分，餘，列交中度於下，以上減下，餘以乘上，滿一千三百七十五而一，所得為度，不滿，退除為分，即為定朔、望加時月去黃道度及分。 每日夜半，准此求之。
To find fixed syzygy hour lunar ecliptic latitude: inspect entered yin- or yang-calendar accumulated degrees and parts; at or below the convergence image counts as the young image; if above, cover and subtract convergence middle; the remainder is entry into the old image. Set entered young or old image degrees above and convergence image degrees below, subtract and multiply (divide by 500), subtract from entered image degrees, then subtract and multiply against convergence middle (divide by 1,375), reduce to parts, and obtain fixed syzygy hour lunar ecliptic latitude in degrees and parts. For each midnight, follow this procedure.
29
求朔望加時入交常日：置其月經朔、望加時入交泛日及餘秒，以其月入氣朏朒定數朏減朒加之，滿與不足，進退其日，即得朔、望加時入交常日及餘秒。 近交初為交初，在二十六日、二十七日為交初； 近交中為交中，在十三日、十四日為交中。
To find syzygy hour general convergence day: set the month's mean syzygy hour general convergence day and remainder-seconds, apply entry-into-qi tiao–chuo fixed number (tiao subtract, chuo add), adjust the day as needed, and obtain syzygy hour general convergence day and remainder-seconds. Near the start of convergence counts as convergence beginning; days twenty-six or twenty-seven are convergence beginning; near convergence middle counts as convergence middle; days thirteen or fourteen are convergence middle.
30
求日月食甚定數：以其朔望入氣、入轉朏朒定數，同名相從，異名相消，副置之； 以定朔、望加時入轉算外損益率乘之，如日法而一，其定朔、望如算外在四七日者，視其餘在初數已下，初率乘之，初數而一； 初數已上，以末率乘之，末數而一。
To find eclipse greatest fixed number: combine that syzygy's entry-into-qi and entry-into-rotation tiao–chuo fixed numbers (same name together, different name cancel) and set aside as auxiliary; multiply auxiliary by fixed syzygy hour entry-into-rotation count-outside damage-benefit rate and divide by the day divisor; if fixed syzygy is count-outside on days four–seven and remainder is at or below the initial number, multiply by the initial rate and divide by the initial number; if above the initial number, multiply by the final rate and divide by the final number.
31
所得，視入轉，應朒者依其損益，應朏者益減損加其副； 以朏減朒加經朔望小餘，為泛餘。 滿與不足，進退大餘。
Apply the result according to entry-into-rotation: for chuo follow damage-benefit; for tiao benefit-subtract and damage-add to the auxiliary; apply tiao subtract and chuo add to the mean syzygy minor remainder to obtain the general remainder. Advance or retreat the major remainder when the sum overflows or falls short.
32
日食者視泛餘，如半法已下，為中前； 列半法於下，以上減下，餘以乘上，如一萬九百三十五而一，所得，為差； 以減泛餘，為食甚定餘； 用減半法，為午前分。 如泛餘在半法已上，減去半法，為中後； 列半法於下。 以上減下，餘以乘上，如日法而一，所得，為差； 以加泛餘，為食甚定餘； 乃減去半法，為午後分。 月食者視泛餘，如半法已上減去半法，餘在一千八百二十二半已下自相乘，已上者，覆減半法，餘亦自相乘，如三萬而一，所得，以減泛餘，為食甚定餘； 如泛餘不滿半法，在日出分三分之二已下，列於上位，已上者，用減日出分，餘倍之，亦列於上位，乃四因三約日出分，列之於下，以上減下，餘以乘上，如一萬五千而一，所得，以加泛餘，為食甚定餘。
For solar eclipse, if the general remainder is at or below the half method, it is before middle; array the half method below, subtract and multiply (divide by 10,935), and obtain the difference; subtract the difference from the general remainder to obtain the eclipse-greatest fixed remainder; subtracting the half method yields the before-noon fraction. if the general remainder is at or above the half method, subtract the half method for after middle; Place the half method in the lower row of the array. subtract below from above, multiply and divide by the day divisor, and obtain the difference; add the difference to the general remainder to obtain the eclipse-greatest fixed remainder; then subtract the half method to obtain the after-noon fraction. For lunar eclipse, if general remainder is at or above half method subtract half method; square remainder if ≤1,822.5, otherwise cover-subtract half method and square, divide by 30,000, and subtract from general remainder for eclipse-greatest fixed remainder; If general remainder is below half method and within two-thirds of sunrise part, array above; if above, adjust against sunrise part, double, array, reduce sunrise part by 4/3, subtract and multiply (divide by 15,000), and add to general remainder for eclipse-greatest fixed remainder.
33
求日月食甚辰刻：倍食甚定餘，以辰法除之為辰數，不盡，五因之，滿刻法除之為刻，不滿為分。 命辰數起子正，算外，即食甚辰刻及分。 若加半辰，命起子初。
To find eclipse-greatest watch and quarter: double the fixed remainder, divide by watch divisor for watches, convert remainder to quarters (multiply by five, divide by quarter divisor), and obtain parts. Count watches from zi-zheng, outside the count, to obtain eclipse-greatest watch, quarter, and parts. If adding half a watch, count from zi-initial.
34
求日月食甚入氣：食甚大、小餘及食定小餘，並定朔、望大餘，以此與經朔望大、小餘相減。
To find eclipse-greatest entry into qi: combine eclipse-greatest major and minor remainders with eclipse-fixed minor remainder and fixed syzygy major remainder, and compare with mean syzygy major and minor remainders.
35
置其朔望食甚大、小餘，與經朔望大、小餘相減之，餘以加減經朔望入氣日餘，經朔望少即加之，多即減之。
Set syzygy eclipse-greatest major and minor remainders, subtract mean syzygy remainders, and add or subtract mean syzygy entry-into-qi day remainder according to whether mean syzygy is less or more.
36
為日、月食甚入氣日及餘秒。 各置食甚入氣及餘秒，加其氣中積，其餘，以日法退除為分，即為日、月食甚中積及分。
That yields solar or lunar eclipse-greatest entry-into-qi day and remainder-seconds. For each eclipse-greatest entry into qi and remainder-seconds, add qi middle accumulation, convert remainder through day divisor into parts, and obtain eclipse-greatest middle accumulation and parts.
37
求日月食甚日行積度：置食甚入氣餘，以所入氣日盈縮分乘之，日法而一，加減其日先後數，至後加，分後減。
To find eclipse-greatest daily motion accumulated degrees: multiply eclipse-greatest entry-into-qi remainder by that qi day's excess-deficit parts (divide by day divisor), add or subtract day's before-after number (add after day, subtract after parts).
38
先加後減日、月食甚中積，即為日、月食甚日行積度及分。
Apply add-then-subtract to eclipse-greatest middle accumulation to obtain eclipse-greatest daily-motion accumulated degrees and parts.
39
求氣差：置日食甚日行積度及分，滿二至限去之，餘在象限已下為在初； 已上，覆減二至限，餘為在末。 皆自相乘，進二位，滿三百四十三而一，所得，用減二千四百三十，餘為氣差； 以午前、後分乘之。 如半晝分而一，以減氣差，為氣差定數。 在冬至後末限、夏至後初限，交初以減，交中以加。
To find qi difference: set solar eclipse-greatest daily-motion accumulated degrees and parts, cast out two solstice limit, and if remainder is at or below image limit it is initial; if above, cover and subtract two solstice limit; the remainder is final. Square each case, shift two places, divide by 343, subtract from 2,430, and obtain qi difference; multiply qi difference by the before- or after-noon part. Divide by half day-part, subtract from qi difference, and obtain qi-difference fixed number. After winter solstice in final limit and after summer solstice in initial limit: subtract at convergence beginning, add at convergence middle.
40
夏至後末限、冬至後初限，交初以加，交中以減。
After summer solstice in final limit and after winter solstice in initial limit: add at convergence beginning, subtract at convergence middle.
41
如半晝分而一，所得，在氣差已上者，即以氣差覆減之，余應加者為減，減者為加。
Divide by half day-part; if result meets or exceeds qi difference, cover-subtract qi difference and invert add/subtract as required.
42
求刻差：置日食甚日行積度及分，滿二至限去之，餘列二至限於下，以上減下，餘以乘上，進二位，滿三百四十三而一，所得為刻差。 以午前、後分乘而倍之，如半法而一，為刻差定數。 冬至後食甚在午前，夏至後食甚在午後，交初以加，交中以減。
To find quarter difference: set solar eclipse-greatest daily-motion accumulated degrees and parts, cast out two solstice limit, subtract and multiply against limit (divide by 343), and obtain quarter difference. Multiply quarter difference by before- or after-noon part, double, divide by half method, and obtain quarter-difference fixed number. After winter solstice with eclipse-greatest before noon, or after summer solstice after noon: add at convergence beginning, subtract at convergence middle.
43
冬至後食甚在午後，夏至後食甚在午前，交初以減，交中以加。
After winter solstice with eclipse-greatest after noon, or after summer solstice before noon: subtract at convergence beginning, add at convergence middle.
44
如半法而一，所得在刻差已上者，即倍刻差，以所得之數減之，餘為刻差定數，依其加減。
Divide by half method; if result meets or exceeds quarter difference, double quarter difference and subtract accordingly; apply add/subtract to obtain quarter-difference fixed number.
45
求朔入交定日：置朔入交常日及餘秒，以氣、刻差定數各加減之，交初加三千一百，交中減三千，為朔入交定日及餘秒。
To find fixed new-moon convergence fixed day: set new-moon general convergence day and remainder-seconds, apply qi- and quarter-difference fixed numbers, add 3,100 at convergence beginning and subtract 3,000 at convergence middle, and obtain new-moon fixed convergence day and remainder-seconds.
46
求望入交定日：置望入轉朏朒定數，以交率乘之，如交數而一，所得，以朏減朒加入交常日之余，滿與不足，進退其日，即望入交定日及餘秒。
To find fixed full-moon convergence fixed day: set full-moon entry-into-rotation tiao–chuo fixed number, multiply by convergence ratio (divide by convergence number), apply tiao subtract and chuo add to general convergence remainder, adjust day as needed, and obtain full-moon fixed convergence day and remainder-seconds.
47
求月行入陰陽曆：視其朔、望入交定日及餘秒，如在中日及餘秒已下為月在陽曆； 如中日及餘秒已上，減去中日，為月在陰曆。
To find moon yang/yin calendar entry: if fixed syzygy convergence fixed day and remainder-seconds are at or below middle day and remainder-seconds, the moon is in the yang calendar; if at or above middle day and remainder-seconds, subtract middle day; the moon is in the yin calendar.
48
求入食限交前後分：視其朔、望月行入陰陽曆，不滿日者為交後分； 在十三日上下者覆減交中日，為交前分； 視交前、後分各在食限已下者為入食限。
To find eclipse-limit before- and after-convergence parts: inspect fixed syzygy moon yang/yin calendar entry; remainder below one day is after-convergence part; if within about thirteen days, cover and subtract convergence middle day for before-convergence part; if each before- and after-convergence part is at or below the eclipse limit, the syzygy has entered the eclipse limit.
49
求日食分：以交前、後分各減陰陽曆食限，餘如定法而一，為日食之大分； 不盡，退除為小分。 命大分以十為限，即得日食之分。 其食不及大分者，行勢稍近交道，光氣微有映蔽，其日或食或不食。
To find solar eclipse magnitude: subtract yin/yang calendar eclipse limit from each before- and after-convergence part, divide remainder by fixed divisor, and obtain solar eclipse major part; reduce any remainder to minor parts. Take the major part on a scale of ten to obtain solar eclipse magnitude. If magnitude falls short of a major part, the path is near the node and light is only faintly dimmed; an eclipse may or may not occur that day.
50
求月食分：視其望交前、後分，如二千四百已下者，食既； 已上，用減食限，餘如定法而一，為月食之大分； 不盡，退除為小分。 命大分以十為限，得月食之分。
To find lunar eclipse magnitude: if full-moon before- and after-convergence parts are at or below 2,400, the eclipse is total; if above, subtract from eclipse limit, divide remainder by fixed divisor, and obtain lunar eclipse major part; reduce any remainder to minor parts. Take the major part on a scale of ten to obtain lunar eclipse magnitude.
51
求日食泛用分：置交前、後分，自相乘，退二位，陽曆一百九十八而一，陰曆三百一十七而一，所得，用減五百八十三，餘為日食泛用分。
To find solar eclipse general use-parts: square before- and after-convergence parts (shift two places), divide by 198 (yang) or 317 (yin), subtract from 583, and obtain solar eclipse general use-part.
52
求月食泛用分：置交前、後分，自相乘，退二位，如七百四而一，所得，用減六百五十六，餘為月食泛用分。
To find lunar eclipse general use-parts: square before- and after-convergence parts (shift two places), divide by 704, subtract from 656, and obtain lunar eclipse general use-part.
53
求日月食定用分：置日、月食泛用分，副之，以食甚加時入轉算外損益率乘之，如日法而一，如算外在四、七日者，依食定餘求之。
To find eclipse fixed use-parts: set general use-part as auxiliary, multiply by eclipse-greatest hour entry-into-rotation count-outside damage-benefit rate (divide by day divisor), and on days four or seven apply eclipse-fixed remainder rule.
54
所得，應朒者依其損益，應朏者益減損加其副，即為日月食定用分。
Apply result according to entry-into-rotation (chuo: follow damage-benefit; tiao: benefit-subtract, damage-add to auxiliary) to obtain solar and lunar eclipse fixed use-part.
55
求月食既內外分：置月食交前、後分，自相乘，退二位，如二百四十九而一，所得，用減二百三十一，餘以定用分乘之，如泛用分而一，為月食既內分； 用減定用分，餘為既外分。
To find lunar eclipse total inner and outer parts: square lunar eclipse before- and after-convergence parts (shift two places), divide by 249, subtract from 231, multiply remainder by fixed use-part (divide by general use-part), and obtain lunar eclipse total inner part; Subtract the total inner part from the fixed use-part to obtain the total outer part.
56
求日月食虧初復滿小餘：置日、月食甚小餘，各以定用分減之，為虧初； 加之，為復滿； 其月食既者，以既內分減之，為初既； 加之，為生光：即各得所求小餘。 如求時刻，依食甚術入之。
To find eclipse first diminishment, restoration fullness, and minor remainders: set sun and moon eclipse-greatest minor remainders, subtract the fixed use-part from each, and obtain first diminishment; Add the fixed use-part to obtain restoration fullness; For a total lunar eclipse, subtract the total inner part to obtain first totality; Add again to obtain light generation; each case then yields the required minor remainders. To find clock time, apply the eclipse-greatest procedure.
57
求月食更點法：置月食甚所入日晨分，倍之，減去七百二十九，餘五約之，為更法； 又五除之，為點法。
To find lunar eclipse watch and point: double the day's dawn parts at eclipse greatest, subtract 729, and five-reduce the remainder to obtain the watch method; Divide again by five to obtain the point method.
58
求月食入更點：置虧初、食甚、復末小餘，在晨分已下加晨分，昏分已上減去昏分，餘以更法除之為更數，不滿，以點法除之為點數。 其更數命初更，算外，即各得所入更、點。
To find lunar eclipse watch and point entry: set first diminishment, greatest eclipse, and restoration-end minor remainders; below dawn parts add dawn parts, above dusk parts subtract dusk parts; divide the remainder by the watch method for watch count, then by the point method for point count. Name from the first watch outside the count to obtain each entered watch and point.
59
求日食所起：日在陽曆，初起西南，甚于正南，復于東南； 日在陰曆，初起西北，甚於正北，復于東北。 其食八分已上，皆起正西，復於正東。 此據午地而論之。
To find solar eclipse direction of first contact: with the sun in the yang half, first contact is southwest, maximum due south, and recovery southeast; With the sun in the yin half, first contact is northwest, maximum due north, and recovery northeast. For eclipses of eight-tenths or greater, first contact is always due west and recovery due east. This assumes an observer on the local meridian (noon location).
60
求月食所起：月在陽曆，初起東北，甚於正北，復於西北； 月在陰曆，初起東南，甚于正南，復於西南。 其食八分已上，皆起正東，復於正西。 此亦據午地而論之。
To find lunar eclipse direction of first contact: with the moon in the yang half, first contact is northeast, maximum due north, and recovery northwest; With the moon in the yin half, first contact is southeast, maximum due south, and recovery southwest. For eclipses of eight-tenths or greater, first contact is always due east and recovery due west. This also assumes an observer on the local meridian.
61
求日月出入帶食所見分數：各以食甚小餘與日出、入分相減，餘為帶食差； 以乘所食之分，滿定用分而一，如月食既者，以既內分減帶食差，餘進一位，如既外分而一，所得，以減既分，即月帶食出入所見之分，不及減者，為帶食既出入。 以減所食分，即日月出、入帶食所見之分。 其食甚在晝，晨為漸進，昏為已退； 其食甚在夜，晨為已退，昏為漸進。
To find the visible fraction when an eclipse is caught at sunrise or sunset: subtract rise or set parts from eclipse-greatest minor remainder to obtain the horizon-partial difference; Multiply by the eclipsed fraction and divide by the fixed use-part; for total lunar eclipse, subtract the total inner part from the horizon-partial difference, advance the remainder one place, divide by the total outer part, subtract the result from the totality fraction to obtain the moon's visible fraction at rise or set; if subtraction cannot be completed, the eclipse is partially total at rise or set. Subtract the result from the eclipsed fraction to obtain the visible fraction at rise or set for sun or moon with horizon partial eclipse. If greatest eclipse is in daylight, at dawn the eclipse is still advancing and at dusk it has already retreated; If greatest eclipse is at night, at dawn it has already retreated and at dusk it is still advancing.
62
求日月食甚宿次：置食甚日行積度，望即更加半周天。
To find eclipse-greatest lodge position: set daily-motion accumulated degrees at eclipse greatest; at full moon add half a circuit of heaven.
63
以天正冬至加時黃道日度加而命之，即各得日、月食甚宿度及分。
Add to the winter solstice ecliptic solar degree at hour-of-addition, name the lodges, and obtain sun and moon eclipse-greatest lodge degrees and parts.
64
步五星
Step Five Planets
65
木星周率：二百九十萬七千八百七十九、秒六十四。
Jupiter circuit rate: 2,907,879, 64 seconds.
66
周差：二十四萬五千二百五十三、秒六十四。
Circuit difference: 245,253, 64 seconds.
67
曆率：二百六十六萬二千六百三十六、秒二十二。
Calendar rate: 2,662,636, 22 seconds.
68
周日：三百九十八、約分八十八、秒六十。
Circuit day: 398, 88 approximate parts, 60 seconds.
69
曆度：三百六十五、約分二十四、秒五十。
Calendar degree: 365, 24 approximate parts, 50 seconds.
70
曆中度：一百八十五、約分六十二、秒二十五。
Calendar mid-degree: 185, 62 approximate parts, 25 seconds.
71
曆策度：一十五、約分二十一、秒八十五。
Calendar stride degree: 15, 21 approximate parts, 85 seconds.
72
伏見度：一十三。
Heliacal setting visibility: 13 degrees.
73
木星盈縮曆
Jupiter excess-deficit calendar
74
火星周率：五百六十八萬五千六百八十七、秒六十四。
Mars circuit rate: 5,685,687, 64 seconds.
75
周差：三十六萬四百一十四、秒四十四。
Circuit difference: 360,414, 44 seconds.
76
曆率：二百六十六萬二千六百四十七、秒二十。
Calendar rate: 2,662,647, 20 seconds.
77
周日：七百七十九、約分九十二、秒九十七。
Circuit day: 779, 92 approximate parts, 97 seconds.
78
曆度：三百六十五、約分二十四、秒六十五。
Calendar degree: 365, 24 approximate parts, 65 seconds.
79
曆中度：一百八十二、約分六十二、秒三十二半。
Calendar mid-degree: 182, 62 approximate parts, 32½ seconds.
80
曆策度：二十五、約分二十一、秒八十六。
Calendar stride degree: 25, 21 approximate parts, 86 seconds.
81
伏見度：一十九。
Heliacal setting visibility: 19 degrees.
82
火星盈縮曆
Mars excess-deficit calendar
83
土星周率：二百七十五萬六千二百八十八、秒七十八。
Saturn circuit rate: 2,756,288, 78 seconds.
84
周差：九萬三千六百六十二、秒七十八。
Circuit difference: 93,662, 78 seconds.
85
曆率：二百六十六萬九千九百二十五、秒九十。
Calendar rate: 2,669,925, 90 seconds.
86
周日：三百七十八、約分九、秒一十七。
Circuit day: 378, 9 approximate parts, 17 seconds.
87
曆度：三百六十六、約分二十四、秒四十九。
Calendar degree: 366, 24 approximate parts, 49 seconds.
88
曆中度：一百八十三、約分一十二、秒二十四半。
Calendar mid-degree: 183, 12 approximate parts, 24½ seconds.
89
曆策度：一十五、約分二十六、秒二。
Calendar stride degree: 15, 26 approximate parts, 2 seconds.
90
伏見度：一十七。
Heliacal setting visibility: 17 degrees.
91
土星盈縮曆
Saturn excess-deficit calendar
92
金星周率：四百二十五萬六千六百五十一、秒四十三半。
Venus circuit rate: 4,256,651, 43½ seconds.
93
合日：二百九十一、約分九十五、秒一十四。
Conjunction day: 291, 95 approximate parts, 14 seconds.
94
曆率：二百六十六萬二千六百九十六、秒一十六。
Calendar rate: 2,662,696, 16 seconds.
95
周日：五百八十三、約分九十、秒二十八。
Circuit day: 583, 90 approximate parts, 28 seconds.
96
曆度：三百六十五、約分二十五、秒三十二。
Calendar degree: 365, 25 approximate parts, 32 seconds.
97
曆中度：一百八十二、約分六十二、秒六十六。
Calendar mid-degree: 182, 62 approximate parts, 66 seconds.
98
曆策度：一十五、約分二十一、秒八十九。
Calendar stride degree: 15, 21 approximate parts, 89 seconds.
99
伏見度：一十半。
Heliacal setting visibility: 10½ degrees.
100
金星盈縮曆
Venus excess-deficit calendar
101
HT5」SS〗水星周率：八十四萬四千七百三十八、秒五。
Mercury circuit rate: 844,738, 5 seconds.
102
合日：五十七、約分九十三、秒八十一。
Conjunction day: 57, 93 approximate parts, 81 seconds.
103
曆率：二百六十六萬二千七百九十四、秒九十五。
Calendar rate: 2,662,794, 95 seconds.
104
周日：一百一十五、約分八十七、秒六十二。
Circuit day: 115, 87 approximate parts, 62 seconds.
105
曆度：三百六十五、約分二十六、秒六十八。
Calendar degree: 365, 26 approximate parts, 68 seconds.
106
曆中度：一百八十二、約分六十三、秒三十四。
Calendar mid-degree: 182, 63 approximate parts, 34 seconds.
107
曆策度：一十五、約分二十一、秒九十四半。
Calendar stride degree: 15, 21 approximate parts, 94½ seconds.
108
晨伏夕見：一十四。
Morning hidden, evening visible: 14 degrees.
109
夕伏晨見：一十九。
Evening hidden, morning visible: 19 degrees.
110
水星盈縮曆
Mercury excess-deficit calendar
111
推五星天正冬至後平合及諸段中積中星：置氣積分，各以其星周率除之，所得周數。 不盡者，為前合。 以減周率，余滿日法為日，不滿，退除為分、秒，即其星天正冬至後平合中積； 命之為平合中星，以諸段常日、常度累加之，即諸段中積、中星。 其段退行者，以常度減之，即其段中星。
To find each planet's mean conjunction after civil new-year winter solstice and the central accumulation and central star for every segment: set the qi accumulated parts, divide each by that star's circuit rate, and obtain the circuit count. The remainder is the anterior conjunction. Subtract from the circuit rate; convert the remainder to days by the day divisor, reduce the remainder to parts and seconds, and obtain that star's mean-conjunction central accumulation after civil new-year winter solstice; Name the result the mean-conjunction central star; add each segment's standard days and degrees cumulatively to obtain every segment's central accumulation and central star. For retrograde segments, subtract the standard degree to obtain the segment's central star.
112
求木火土三星平合諸段入曆：置其星周數，求冬至後合，皆加一數置之。
To find mean-conjunction entry into the calendar cycle for Jupiter, Mars, and Saturn and each segment: set that star's circuit count, and for post-winter-solstice conjunction add one to the count.
113
以周差乘之，滿其星曆率去之，不盡，滿日法為度，不滿，退除為分、秒，即為其星平合入曆度及分、秒。 以其段限度依次累加之，即得諸段入曆。
Multiply by the circuit difference, cast out full calendar rates, convert the remainder to degrees by the day divisor, reduce the remainder to parts and seconds, and obtain the star's mean-conjunction entry-into-calendar degree and parts and seconds. Add each segment's limit degree cumulatively in order to obtain every segment's entry into the calendar cycle.
114
求金水二星平合及諸段入曆：置氣積分，各以其星曆率去之，不盡，滿日法除之為度，不滿，退除為分、秒，以加平合中星，即為其星天正冬至後平合入曆度及分、秒； 以其星其段限度依次累加之，即得諸段入曆。
To find Venus and Mercury mean conjunction and each segment's entry into the calendar cycle: set the qi accumulated parts, cast out full calendar rates for each star, convert the remainder to degrees and parts and seconds, add to the mean-conjunction central star, and obtain mean-conjunction entry-into-calendar degree and parts and seconds after civil new-year winter solstice; Add that star's segment limit degrees cumulatively in order to obtain every segment's entry into the calendar cycle.
115
求五星平合及諸段盈縮定差：各置其星其段入曆度及分，如曆中已下為在盈； 已上減去曆中，餘為在縮； 以其星曆策除之為策數，不盡，為入策度及分； 命策數，算外，以其策損益率乘之，如曆策而一為分，分滿百為度； 以損益其下盈縮積，即其星其段盈縮定差。
To find each planet's mean conjunction and every segment's excess-and-deficit fixed difference: set each star and segment's entry-into-calendar degree and parts; at or below the calendar mid counts as excess; If above the calendar mid, subtract it; the remainder is in deficit. Divide by that star's calendar stride to obtain the stride count; the remainder is entry-into-stride degree and parts; Name from outside the stride count, multiply the tabulated loss-and-gain rate for that stride, divide by the calendar stride for parts, and convert parts filling 100 to degrees; Apply it to the underlying excess-and-deficit accumulation to obtain that star and segment's excess-and-deficit fixed difference.
116
求五星平合及諸段定積：各置其星其段中積，以其段盈縮定差盈加縮減之，即其段定積日及分； 以天正冬至大餘及約分加之，即為定日及分； 盈紀法六十去之，不盡，命己卯，算外，即得日辰。
To find each planet's mean conjunction and every segment's fixed accumulation: set each star and segment's central accumulation, add or subtract the segment's excess-and-deficit fixed difference according to excess or deficit, and obtain the segment's fixed-accumulation days and parts; Add the civil new-year winter solstice greater remainder and approximate parts to obtain the fixed day and parts; Cast out full era rules of 60, name from jimao outside the count, and obtain the day and double-hour.
117
求五星平合諸段所在月日：各置其段定積，以天正閏日及約分加之，滿朔策及約分除之為月數，不盡，為入月已來日數及分。 其月數命天正十一月，算外，即其星其段入其月經朔日數及分，乃以日辰相距為定朔月、日。
To find the month and day for each planet's mean-conjunction segments: set each segment's fixed accumulation, add civil new-year intercalary days and approximate parts, divide by the new-moon policy and approximate parts for month count, and take the remainder as days and parts elapsed within the month. Name the month count from civil new-year month eleven outside the count to obtain the segment's days and parts since mean new moon, and take the day-and-double-hour interval as the fixed new-moon month and day.
118
求五星平合及諸段加時定星：各置其段中星，以其段盈縮定差盈加縮減之，金星倍之水星三之，乃可加減。
To find each planet's mean conjunction and every segment's hour-of-addition fixed star: set each segment's central star, apply the segment's excess-and-deficit fixed difference (double for Venus, triple for Mercury), then add or subtract according to excess or deficit.
119
即五星諸段定星； 以天正冬至加時黃道日度加而命之，即其星其段加時所在宿度及分秒。 五星皆因前留為前段初日定星，後留為後段初日定星，餘依術算。
That yields each planet's segment fixed star; Add to the winter solstice ecliptic solar degree at hour-of-addition, name the lodges, and obtain that star and segment's hour-of-addition lodge degree, parts, and seconds. For all five planets, the anterior station fixes the first-day star of the anterior segment and the posterior station the first-day star of the posterior segment; compute the remainder by the procedure.
120
求五星諸段初日晨前夜半定星：各以其段初行率乘其段加時分，百約之，乃以順減退加其日加時定星，即為其段初日晨前夜半定星； 加命如前，即得所求。
To find each planet's first-day before-dawn midnight fixed star for every segment: multiply the segment's initial motion rate by its hour-of-addition parts and hundred-reduce, then subtract for direct motion or add for retrograde from that day's hour-of-addition fixed star to obtain the segment's first-day before-dawn midnight fixed star; Add and name as before to obtain the result.
121
求諸段日率度率：各以其段日辰距至後段日辰，為其段日率； 以其段夜半定星與後段夜半定星相減，為其段度率及分秒。
To find each segment's day rate and degree rate: for each take the day-and-double-hour interval from that segment to the next segment as the segment's day rate; Subtract that segment's midnight fixed star from the next segment's midnight fixed star to obtain the segment's degree rate, parts, and seconds.
122
求諸段平行度：各置其段度率及分秒，以其段日率除之，為其段平行度及分秒。
To find each segment's parallel degree: set each segment's degree rate, parts, and seconds, divide by its day rate, and obtain the segment's parallel degree, parts, and seconds.
123
求諸段總差：各以其段平行分與後段平行分相減，餘為泛差； 並前段泛差，四因，退一位，為總差。 若前段無平行分相減為泛差者，因後段初日行分與其段平行分相減，餘為半總差； 倍之，為總差。 若後段無平行分相減為泛差者，因前段末日行分與其段平行分相減，餘為半總差，倍之，為總差。 晨遲末段，視段無平行分，因前初段末日行分與晨遲末段平行分相減，為半總差； 其退行者，各置本段平行分，十四乘之，十五而一，為總差。 內金星依順段術入之，即得所求。 夕遲初段，視前段無平行分，因後末段初日行分與夕遲初段平行分相減，為半總差。
To find each segment's total difference: subtract each segment's parallel parts from the next segment's parallel parts; the remainder is the general difference; Add the previous segment's general difference, multiply by four, and shift one decimal place to obtain the total difference. When the prior segment lacks parallel parts for subtraction, subtract that segment's parallel parts from the next segment's first-day motion parts; the remainder is half the total difference; Double it to obtain the total difference. When the following segment lacks parallel parts for subtraction, subtract that segment's parallel parts from the previous segment's last-day motion parts; double the half-total difference to obtain the total difference. For the morning-slow final segment, when parallel parts are absent, subtract its parallel parts from the previous initial segment's last-day motion parts to obtain half the total difference; For retrograde segments, set each segment's parallel parts, multiply by fourteen and divide by fifteen to obtain the total difference. For Mercury, apply the direct-segment procedure to obtain the result. For the evening-slow initial segment, when the prior segment lacks parallel parts, subtract its parallel parts from the next final segment's first-day motion parts to obtain half the total difference.
124
求諸段初末日行分：各半其段總差，加減其段平行分，後段平行分多者，減之為初，加之為末； 後段平行分少者，加之為初，減之為末。 其在退行者，前減之為初，加之為末； 後加之為初，減之為末。
To find each segment's first- and last-day motion parts: halve the segment's total difference and add or subtract its parallel parts; when the next segment's parallel parts are greater, subtract for the first day and add for the last; When the next segment's parallel parts are fewer, add for the first day and subtract for the last. In retrograde motion, subtract for the first day and add for the last; for the following segment, add for the first day and subtract for the last.
125
各為其星其段初、末日行度及分秒。 如前後段平行分俱多、俱少者，平注之； 本段總差不滿大分者，亦平注之。
Each yields that star's first- and last-day motion degrees, parts, and seconds for the segment. When both adjacent segments' parallel parts are greater or both are fewer, distribute evenly; When a segment's total difference is less than one major part, distribute evenly as well.
126
求每日晨前夜半星行宿次：置其段總差，減日率一以除之，為日差； 累損益初日行分，後行分少，損之； 後行分多，益之。
To find the lodge position at each day's before-dawn midnight: set the segment's total difference, divide by the day rate minus one, and obtain the day difference; Adjust the first-day motion parts cumulatively; when later motion parts decrease, subtract; when later motion parts increase, add.
127
為每日行度及分秒； 乃順加退減其段初日晨前夜半宿次命之，即每日晨前夜半星行所在宿次。
This yields each day's motion in degrees, parts, and seconds; For direct motion add and for retrograde subtract from the segment's first-day lodge at before-dawn midnight, naming each in turn to obtain the lodge at every day's before-dawn midnight.
128
徑求其日宿次：置所求日，減一，半之，以日差乘而加減初行日分，後行分少，減之； 後行分多，加之。
To find a given day's lodge directly: take the requested day minus one, halve it, multiply by the day difference and add or subtract from the initial motion parts; subtract when later motion parts are fewer; add when later motion parts are greater.
129
以所求日乘之，為積度； 乃順加退減其段初日宿次，即得所求日宿次。
Multiply by the requested day to obtain accumulated degrees; Add for direct motion or subtract for retrograde from the segment's first-day lodge to obtain the lodge for the requested day.
130
求五星平合及見伏入氣：置定積，以氣策及約分除之為氣數，不盡，為入氣已來日數及分秒。 其氣數命天正冬至，算外，即五星平合及見、伏入氣日及分秒。 其定積滿歲周日及分，去之，餘，在來年冬至後。
To find each planet's mean conjunction and appearance or occultation entry into a qi period: set the fixed accumulation, divide by the qi policy and approximate parts for the qi count, and take the remainder as days, parts, and seconds since qi entry. Count the qi from civil new-year winter solstice to obtain each planet's mean conjunction and appearance or hiding qi-entry day, parts, and seconds. When the fixed accumulation exceeds a full year's circuit, discard the excess; the remainder falls after the following year's winter solstice.
131
求五星合見伏行差：木、火、土三星，以其段初日星行分減太陽行分，餘為行差。 金、水二星順行者，以其段初日太陽行分減星行分，餘為行差。 金、水二星退行者，以其段初日星行分並太陽行分，為行差。
To find motion difference at conjunction, appearance, or occultation: for Jupiter, Mars, and Saturn, subtract the sun's motion parts from the segment's first-day planetary motion parts. For Venus and Mercury in direct motion, subtract the planet's motion parts from the sun's motion parts on the segment's first day. For Venus and Mercury in retrograde motion, add the planet's and sun's motion parts on the segment's first day.
132
求五星定合及見伏泛積：木、火、土三星，各以平合晨疾、夕伏定積，便為定合定見、定伏泛積。 金、水二星，各置其段盈縮定差，內水星倍之，以其段行差除之為日，不滿，退除為分秒，在平合夕疾、晨伏者，乃盈減縮加定積，為定合定見、定伏泛積； 在退合夕伏、晨見者，用盈加縮減定積，為定合定見、定伏泛積。
To find fixed conjunction and appearance or occultation general accumulations: for Jupiter, Mars, and Saturn, use the mean-conjunction morning-swift and evening-hidden fixed accumulations directly as the fixed conjunction, appearance, and occultation general accumulations. For Venus and Mercury, set each segment's excess-and-deficit fixed difference (double for Mercury), divide by the segment's motion difference for days and convert the remainder to parts and seconds; at mean conjunction evening-swift and morning-hidden, subtract excess or add deficit to the fixed accumulation; At retrograde conjunction evening-hidden and morning-appearance, add excess or subtract deficit from the fixed accumulation for the fixed conjunction, appearance, and occultation general accumulations.
133
求五星定合定積定星：木、火、土三星，以平合行差除其日先後數，為距合差日； 以先後數減之，為距合差度； 以差日、差度後加先減其星定合泛積，為其星定合日定積、定星。 金、水二星順合者，以平合行差除其日先後數，為距合差日； 以先後數加之，為距合差度； 以差日、差度先加後減其星定合泛積，為其星定合日定積、定星。 金、水二星退合者，以退合行差除其日先後數，為距合差日； 以減先後數，為距合差度； 以差日先減後加，以差度先加後減再定合泛積，為其星再定合積星。 各以冬至大餘及約分加定積，滿紀法去之，命己卯，算外，即得定合日辰； 以冬至加時黃道日度加定星，依宿次去之，即得定合所在宿次。
To find fixed conjunction accumulation and fixed star: for Jupiter, Mars, and Saturn, divide the day-prior-and-posterior count by mean-conjunction motion difference to obtain the conjunction-interval difference in days; subtract the prior-and-posterior count to obtain the conjunction-interval difference in degrees; Add the difference day and degree posteriorly and subtract anteriorly from the planet's fixed-conjunction general accumulation to obtain fixed-conjunction day, accumulation, and star. For Venus and Mercury in direct conjunction, divide the day-prior-and-posterior count by mean-conjunction motion difference for conjunction-interval difference days; add the prior-and-posterior count for conjunction-interval difference degrees; Add difference day and degree anteriorly and subtract posteriorly from the planet's fixed-conjunction general accumulation. For Venus and Mercury in retrograde conjunction, divide the day-prior-and-posterior count by retrograde-conjunction motion difference; subtract the prior-and-posterior count for conjunction-interval difference degrees; Subtract then add the difference day and add then subtract the difference degree from the re-fixed-conjunction general accumulation to obtain re-fixed conjunction accumulation and star. Add winter solstice great remainder and approximate parts to each fixed accumulation, cast out full era cycles, count from jimao, and obtain the fixed-conjunction day and double-hour; Add the fixed star to the winter solstice ecliptic solar degree at hour-of-addition and name the lodges to obtain the fixed conjunction's lodge position.
134
求木火土三星定見伏定積日：各置其星定見、伏泛積，晨加夕減象限日及分秒，如二至限已下自相乘，已上，覆減歲周，餘亦自相乘，百約為分，以其星伏見度乘之，十五除之，為差； 其差如其段行差而一為日，不滿，退除為分、秒，見加伏減泛積，為定積； 如前加命，即得日辰。
To find fixed appearance and occultation days for Jupiter, Mars, and Saturn: set each planet's general accumulation, add for morning appearances and subtract for evening occultations according to the image limit, apply the two-solstice limit formula, multiply by visibility degree and divide by fifteen for the difference; Convert the difference to days at the segment's motion difference rate, remainder to parts and seconds, add for appearance and subtract for occultation from the general accumulation; Name as before to obtain the day and double-hour.
135
求金水二星定見伏定日：夕見、晨伏，以行差除其日先後數，為日； 先加後減泛用積，為常用積。 晨見、夕伏，以行差除其日先後數，為日； 先減後加泛用積，為常用積。 如常用積在二至限已下為冬至後； 已上去之，餘為夏至後。 其二至後日及分在象限已下自相乘，已上，用減二至限，餘亦自相乘，如法而一，所得為分； 冬至後晨，夏至後夕，以十八為法； 冬至後夕、夏至後晨，以七十五為法。
To find fixed appearance and occultation days for Venus and Mercury: for evening appearance and morning occultation, divide the day-prior-and-posterior count by motion difference; add anteriorly and subtract posteriorly from the general-use accumulation for the regular-use accumulation. for morning appearance and evening occultation, divide by motion difference for days; subtract anteriorly and add posteriorly from the general-use accumulation. If the regular-use accumulation is at or below the two-solstice limit, count it as after winter solstice; if above, subtract the limit; the remainder is after summer solstice. For days and parts after a solstice, self-multiply if at or below the image limit, otherwise subtract from the two-solstice limit and self-multiply the remainder; divide per the method for parts; After winter solstice for morning, after summer solstice for evening, use eighteen as divisor; After winter solstice for evening, after summer solstice for morning, use seventy-five as divisor.
136
以伏見度乘之、十五除之，為差； 滿行差而一為日，不滿，退除為分秒，加減常用積，為定用積； 加命如前，即得定見、伏日辰。 冬至後，晨見、夕伏加之，夕見、晨伏減之； 夏至後，晨見、夕伏減之，夕見、晨伏加之。
multiply by the occultation-appearance degree and divide by fifteen for the difference; When full divide by motion difference for days, remainder to parts and seconds, add or subtract from regular-use accumulation for fixed-use accumulation; Name as before to obtain fixed appearance and occultation day and double-hour. After winter solstice, add for morning appearance and evening occultation, subtract for evening appearance and morning occultation; After summer solstice, subtract for morning appearance and evening occultation, add for evening appearance and morning occultation.
137
其水星，夕疾在大暑氣初日至立冬氣九日三十五分已下者，不見； 晨留在大寒氣初日至立夏氣九日三十五分已下者，春不晨見，秋不夕見。
For Mercury, when evening-swift falls from the start of Great Heat qi through day nine and thirty-five parts of Lesser Snow qi, it is invisible; When morning-station falls from the start of Great Cold qi through day nine and thirty-five parts of Start of Summer qi, Mercury does not appear in the morning in spring or in the evening in autumn.
138
熙寧六年六月，提舉司天監陳繹言：「渾儀尺度與《法要》不合，二極、赤道四分不均，規、環左右距度不對，遊儀重澀難運，黃道映蔽橫簫，遊規璺裂，黃道不合天體，天樞內極星不見。 天文院渾儀尺度及二極、赤道四分各不均，黃道、天常環、月道映蔽橫簫，及月道不與天合，天常環相攻難轉，天樞內極星不見。 皆當因舊修整，新定渾儀，改用古尺，均賦辰度，規、環輕利，黃赤道、天常環並側置，以北際當天度，省去月道，令不蔽橫蕭，增天樞為二度半，以納極星，規、環、二極，各設環樞，以便遊運。」 詔依新式製造，置於司天監測驗，以較疏密。 七年六月，司天監呈新制渾儀、浮漏于迎陽門，帝召輔臣觀之，數問同提舉官沈括，具對所以改更之理。 尋又言：「准詔，集監官較其疏密，無可比較。」 詔置於翰林天文院。 七月，以括為右正言，司天秋官正皇甫愈等賞有差。 初，括上《渾儀》、《浮漏》、《景表》三議，見《天文志》。 朝延用其說，令改造法物、曆書。 至是，渾儀、浮漏成，故賞之。
In the sixth month of the sixth year of Xining, Chen Yi, supervisor of the Directorate of Astronomy, reported: "The armillary sphere's measurements do not agree with the Essentials of Methods. The two poles and the equator's four quarters are uneven, the distance graduations on the rings do not match left and right, the sighting arm is too heavy to turn smoothly, the ecliptic band is blocked by the crossbeam, the touring sight-ring is cracked, the ecliptic fails to align with the heavens, and the pole star within the celestial pivot cannot be seen. The Astronomical Bureau's armillary sphere likewise has uneven scales on the two poles and equator, while the ecliptic, celestial constant ring, and lunar path block the crossbeam; the lunar path fails to match the heavens, the celestial constant ring binds and turns with difficulty, and the pole star within the celestial pivot remains unseen. Everything should be refurbished from the old instrument and recast as a new armillary sphere using ancient foot-lengths, with lodge degrees evenly marked, rings made light and easy to move, the ecliptic and equator and celestial constant ring set on edge with the northern rim aligned to celestial degree, the lunar path removed so it no longer blocks the crossbeam, the celestial pivot widened to two and a half degrees to take in the pole star, and pivots added to the rings and poles so the sighting arm may travel freely. An edict ordered manufacture in the new design and placement at the Directorate of Astronomy for trial, to compare coarse and fine readings. In the sixth month of the seventh year, the Directorate presented the new armillary sphere and clepsydra at the Gate Facing the Sun. The emperor summoned his chief ministers to inspect them and questioned Shen Kuo, co-supervisor of the project, again and again; Kuo explained in full the rationale behind every change. They soon reported further: "As the edict directed, directorate officers were gathered to compare the old and new instruments for accuracy, but no valid comparison could be made." An edict then ordered the instruments installed in the Hanlin Astronomical Bureau. In the seventh month Shen Kuo was appointed Right Remonstrator, and Huangfu Yu, Autumn Office Director of the Directorate, among others received graded rewards. Earlier Kuo had submitted three memorials on the armillary sphere, clepsydra, and gnomon table, recorded in the Astronomical Treatise. The court adopted his proposals and ordered the refashioning of ritual instruments and calendrical books. Now that the armillary sphere and clepsydra were finished, rewards were granted accordingly.
139
元豐五年正月，翰林學士王安禮言：「詳定渾儀官歐陽發所上渾儀、浮漏木樣，具新器之宜，變舊器之失，臣等竊詳司天監浮漏，疏謬不可用，請依新式改造。 其至道、皇祐渾儀、景表亦各差舛，請如法條奏修正。」 從之。 元祐四年三月，翰林學士許將等言：「詳定元祐渾天儀象所先奉詔製造水運渾儀木樣，如試驗候天不差，即別造銅器，今校驗皆與天合。」 詔以銅造，仍以元祐渾天儀象為名。 將等又言：「前所謂渾天儀者，其外形圓，可遍佈星度； 其內有璣、有衡，可仰窺天象。 今所建渾儀象，別為二器，而渾儀占測天度之真數，又以渾象置之密室，自為天運，與儀參合。 若並為一器，即象為儀，以同正天度，則渾天儀象兩得之矣。 請更作渾天儀。」 從之，七年四月，詔尚書左丞蘇頌撰《渾天儀象銘》。 六月，元祐渾天儀象成，詔三省、樞密院官閱之。 紹聖元年十月，詔禮部、秘書省，即詳定制造渾天儀象所，以新舊渾儀集局官同測驗，擇其精密可用者以聞。
In the first month of the fifth year of Yuanfeng, Academician-Expositor Wang Anli reported: "Ouyang Fa, officer charged with determining the armillary sphere, has submitted wooden models of the armillary sphere and clepsydra that show what the new instruments should do and remedy what the old ones got wrong. We find the Directorate's clepsydra coarse and unreliable and ask that it be rebuilt to the new design. The Zhidao and Huangyou armillary spheres and gnomon tables are likewise defective; we ask that they be corrected and reported to the throne as the regulations provide." The court approved. In the third month of the fourth year of Yuanyou, Academician-Expositor Xu Jiang and others reported: "The office determining the Yuanyou armillary sphere and celestial globe, having first been ordered to build a water-driven wooden model — with bronze to follow if observations showed no error — has now verified that every reading agrees with the sky." An edict ordered casting in bronze, retaining the name Yuanyou Armillary Sphere and Celestial Globe. Jiang and his colleagues added: "The instrument once called the armillary sphere has a round outer shell on which star degrees may be marked all around; inside are the ji and heng sighting components by which one peers up at the heavens. The instruments now built split these into two: the armillary sphere measures true celestial degrees, while a celestial globe is placed in a sealed chamber and driven to revolve like the sky, coordinated with the sphere. If combined in one, the globe would serve as the sphere and share the same corrected celestial degrees — and the armillary sphere and celestial globe would be won in a single instrument. We ask that a unified armillary-heaven instrument be made anew." The court agreed; in the fourth month of the seventh year an edict ordered Left Vice Director of the Secretariat Su Song to compose the inscription for the Armillary Sphere and Celestial Globe. In the sixth month the Yuanyou Armillary Sphere and Celestial Globe were completed, and officials of the Three Departments and Privy Council were summoned to inspect them. In the tenth month of the first year of Shaosheng, an edict ordered the Ministry of Rites and the Secretariat, at the office manufacturing the armillary sphere and celestial globe, to have bureau officers test old and new armillary spheres together and report whichever proved precise and fit for use.
140
宣和六年七月，宰臣王黼言：
In the seventh month of the sixth year of Xuanhe, Chief Councillor Wang Fu reported:
141
臣崇甯元年邂逅方外之士于京師，自雲王其姓，面出素書一，道璣衡之制甚詳。 比嘗請令應奉司造小樣驗之，逾二月，乃成璿璣，其圓如丸，具三百六十五度四分度之一，置南北極、昆侖山及黃、赤二道，列二十四氣、七十二候、六十四卦、十幹、十二支、晝夜百刻，列二十八宿、並內外三垣、周天星。 日月循黃道天行，每天左旋一周，日右旋一度，冬至南出赤道二十四度，夏至北入赤道二十四度，春秋二分黃、赤道交而出卯入酉。 月行十三度有餘，生明於西，其形如鉤，下環，西見半規，及望而圓； 既望，西缺下環，東見半規，及晦而隱。 某星始見，某星已中，某星將入，或左或右，或遲或速，皆與天象吻合，無纖毫差。 玉衡植於屏外，持扼樞鬥，注水激輪，其下為機輪四十有三，鉤鍵交錯相持，次第運轉，不假人力，多者日行二千九百二十八齒，少者五日行一齒，疾徐相遠如此，而同發於一機，其密殆與造物者侔焉。 自余悉如唐一行之制。
In the first year of Chongning I met by chance at the capital a man from beyond the ordinary world who gave his surname as Wang and handed me an unadorned book describing the mechanism of the ji and heng in meticulous detail. I had the Service-for-the-Emperor Office build a small model to test it; after more than two months the xuanji was finished, round as a pellet and marked with three hundred sixty-five and a quarter degrees, bearing the south and north poles, Mount Kunlun, and the yellow and red paths, with the twenty-four qi, seventy-two hou, sixty-four hexagrams, ten stems, twelve branches, the hundred day-and-night marks, the twenty-eight lodges, and the inner and outer three enclosures with every star of the circuit of heaven. The sun and moon travel along the ecliptic's daily course: each day the sky wheels left one full turn while the sun wheels right one degree; at winter solstice it emerges twenty-four degrees south of the equator, at summer solstice it enters twenty-four degrees north, and at the spring and autumn equinoxes the yellow and red paths cross so it emerges at mao and sets at you. The moon moves more than thirteen degrees a day; its light is born in the west, first shaped like a hook with the lower arc — a half-disk seen in the west — until at full it is round; after full the western arc is cut away in the lower ring while the eastern half-disk appears, until at last it vanishes. When a star first appears, when it culminates, when it is about to set — whether to left or right, slow or fast — everything matches the sky without the slightest error. The jade balance stands outside the screen, gripping the pivot's dipper; water is poured to drive the wheel beneath, where forty-three gear trains mesh by interlocking keys and turn in sequence without human hand — the fastest wheel runs two thousand nine hundred twenty-eight teeth a day, the slowest one tooth in five days; so wide the spread of speeds, yet all born of a single engine, its fineness nearly that of the Maker himself. Everything else follows the system of Tang Yixing.
142
然一行舊制機關，皆用銅鐵為之，澀即不能自運，今制改以堅木若美玉之類。 舊制外絡二輪，以綴日月，而二輪蔽虧星度，仰視躔次不審，今制日月皆附黃道，如蟻行磑上。 舊制雖有合望，而月體常圓，上下弦無辨，今以機轉之，使圓缺隱見悉合天象。 舊制止有候刻辰鐘鼓，晝夜短長與日出入更籌之度，皆不能辨，今制為司辰壽星，運十二時輪，所至時刻，以手指之，又為燭龍，承以銅荷，時正吐珠振荷，迴圈自運。 其制皆出一行之外。 即其器觀之，全象天體者，璿璣也； 運用水鬥者，玉衡也。 昔人或謂璣衡為渾天儀，或謂有璣而無衡者為渾天象，或謂渾儀望筒為衡：皆非也。 甚者莫知璣衡為何器。 唯鄭康成以運轉者為璣，持正者為衡，以今制考之，其說最近。
Yet Yixing's old mechanism relied entirely on copper and iron, which when stiff would not turn of themselves; the present design substitutes hard wood or fine jade and the like. The old design bound two outer wheels to carry sun and moon, and those wheels blocked and eclipsed star degrees so that looking up at the lodges' sequence was unclear; the new design fixes both sun and moon to the ecliptic, like ants crawling on a millstone rim. The old mechanism could mark conjunction and full moon, but the moon always appeared round and could not show the first and third quarters; the new design turns it by gears so that every phase of waxing, waning, and visibility matches the sky. The old device only struck bells and drums for clepsydra marks and double-hours; it could not show how day and night lengthen and shorten or how sunrise, sunset, watches, and night watches shift. The new design adds a Time-Keeper figure that drives a twelve-hour wheel and points to each moment, and a Candle Dragon on a bronze lotus that, at the proper hour, releases a pearl, shakes the lotus, and sets the ring turning on its own. Every feature of the design goes beyond what Yi Xing had accomplished. Among its parts, the component that fully represents the celestial sphere is the armillary sphere itself, the Xuanji; and the part that drives the water scoop is the Yuheng. Earlier scholars sometimes identified the entire Jiji-Yuheng with the armillary sphere, or called a globe with Ji but no Heng a celestial model, or took the sighting tube of the armillary instrument for the Heng — all of which is mistaken. Some did not even know what sort of instrument the Jiji and Yuheng were meant to be. Only Zheng Xuan identified the rotating component as Ji and the stabilizing component as Heng; compared with the present mechanism, his account comes nearest the truth.
143
又月之晦明，自昔弗燭厥理，獨揚雄云：「月未望則載魄於西，既望則終魄於東，其溯於日乎？」 京房云：「月有形無光，日照之乃光。」 始知月本無光，溯日以為光。 本朝沈括用彈況月，粉塗其半，以象對日之光，正側視之，始盡圓缺之形。 今制與三者之說若合符節。 宜命有司置局如樣制，相址於明堂或合台之內，築台陳之，以測上象。 又別制三器，一納禦府，一置鐘鼓院，一備車駕行幸所用。 仍著為成書，以詔萬世。
As for the moon's phases, antiquity never fully explained them. Yang Xiong alone wrote, "Before the full moon its spirit is borne in the west; after the full moon its spirit ends in the east — does it follow the sun?" Jing Fang said, "The moon has form but no light of its own; only sunlight makes it bright." Thus it was understood that the moon has no inherent light and shines only by facing the sun. In our own dynasty Shen Kuo modeled the moon with a clay ball, whitened half its surface to represent sunlight, and viewed it head-on and from the side, thereby showing every shape from full disk to crescent. The present mechanism agrees with all three accounts as if their seals matched exactly. The responsible offices should be ordered to establish a workshop from the model, choose a site within the Bright Hall or Joint Terrace, erect a platform, and install the instrument there to observe the heavens. Three additional instruments should be built: one for the Imperial Storehouse, one for the Bell and Drum Court, and one kept ready for the emperor's travels. A full book should also be compiled so that the design may instruct generations to come.
144
詔以討論製造璣衡所為名，命黼總領，內侍梁師成副之。
An edict established the Bureau for Discussing and Manufacturing the Jiji and Yuheng, placed Fu in overall charge, and made the inner attendant Liang Shicheng his deputy.