← Back to 後漢書

第三 律曆下 曆法

Volume 93: Rhythm and the Calendar Part Three

Chapter 104 of 後漢書 · Book of Later Han
← Previous Chapter
Chapter 104
Next Chapter →
1
後漢書志第三律歷下曆法
Book of Later Han, Treatise 3: “Pitchpipes and the Calendar, Part 3 — Calendrical Methods”
2
昔者聖人之作歷也，觀琁璣之運，三光之行，道之發斂，景之長短，斗綱 (之) [所]建，青龍所躔，參伍以變，錯綜其數，而制術焉。
When the sages of old devised the calendar, they watched the turning armillary sphere, the paths of Sun, Moon, and stars, how the ecliptic widens and narrows, how shadows lengthen and shorten, and how the Dipper’s (of) handle is aligned, where the Azure Dragon stands along the sky, matched permutations and interlaced the reckoning until they could set down the algorithms.
3
天之動也，一晝一夜而運過周，星從天而西，日違天而東。 日之所行與運周，在天成度，在歷成日。 居以列宿，終於四七，受以甲乙，終於六旬。 日月相推，日舒月速，當其同[所]，謂之合朔。 舒先速後，近一遠三，謂之弦。 相與為衡，分天之中，謂之望。 以速及舒，光盡體伏，謂之晦。 晦朔合離，斗建移辰，謂之[月]。 日月之 (術) [行]，則有冬有夏； 冬夏之閒，則有春有秋。 是故日行北陸謂之冬，西陸謂之春，南陸謂之夏，東陸謂之秋。 日道發南，去極彌遠，其景彌長，遠長乃極，冬乃至焉。 日道斂北，去極彌近，其景彌短，近短乃極，夏乃至焉。 二至之中，道齊景正，春秋分焉。
The sky turns once between dusk and dawn; the stars ride westward with it, but the Sun slips eastward against that flow. The Sun’s path around the circuit becomes celestial degrees in the sky and counted days on the calendar. It is pinned to the twenty-eight lodges (four times seven) and tallied through the sixty-day cycle in ten-day units. Sun and Moon chase one another—the Sun lags, the Moon runs ahead—and when they meet in the same longitude we call it new moon. When the slow phase leads and the fast follows—near on one limb, farther by three—it is the first or last quarter. When they face each other across the sky and split the heavens down the middle, we have full moon. When the faster overtakes the slower and the Moon’s light is gone and her disk concealed, that is month’s end, the dark night. New moon and month-end alternate, the Dipper’s handle advances the celestial hour—thus the lunar month is defined. In the Sun and Moon’s (methods) courses, and so we get winter and summer; between those extremes lie spring and autumn. So the Sun in the northern mansions means winter, in the west spring, in the south summer, in the east autumn. As the ecliptic tilts south the Sun recedes from the pole, shadows lengthen, and at their greatest length midwinter comes. As the path swings north the Sun approaches the pole, shadows shorten, and at their shortest midsummer arrives. Between the solstices the solar path levels and noon shadows match—those are the spring and autumn equinoxes.
4
日周於天，一寒一暑，四時備成，萬物畢改，攝提遷次，青龍移辰，謂之歲。
One circuit of the Sun brings cold and heat in turn, rounds out the four seasons, renews the myriad things, advances Sheti’s pointer and the Azure Dragon’s station—what we call a year.
5
歲首至也，月首朔也。 至朔同日謂之章，同在日首謂之蔀，蔀終六旬謂之紀，歲朔又復謂之元。 是故日以實之，月以閏之，時以分之，歲以周之，章以明之，蔀以部之，紀以記之，元以原之。 然後雖有變化萬殊，贏朒無方，莫不結繫於此而業正焉。
The civil year opens at winter solstice; the lunar month opens at new moon. Solstice and new moon on one day make a zhang; both starting the same day make a bu; six decadal cycles close a bu and form a ji; when year-start and new moon realign you have a yuan. Thus the Sun completes the reckoning, intercalary months adjust it, seasons slice it, the year encloses it, the zhang defines cycles, the bu groups them, the ji logs them, the yuan anchors their beginning. Whatever the endless variety of calendrical drift, every correction still hooks to these principles and sets the system right.
6
極建其中，道營於外，琁衡追日，以察[發]斂，光道生焉。 孔壺為漏，浮箭為刻，下漏數刻，以考中星，□明生焉。 日有光道，月有九行，九行出入而交生焉。 朔會望衡，鄰於所交，虧薄生焉。 月有晦朔，星有合見，月有弦望，星有留逆，其歸一也，步術生焉。 金、水承陽，先後日下，速則先日，□而後留，留而後逆，逆與日違，違而後速，速與日競，競又先日，□速順逆，晨夕生焉。
The pole stands at the hub, the ecliptic girdles it, and the armillary instrument tracks the Sun’s advance and retreat—whence the ecliptic itself is derived. The pierced pot is the water-clock, the floating rod marks the clepsydra divisions; counting drips against the meridian stars fixes the hours and defines twilight. The Sun follows the ecliptic; the Moon has nine paths; where those paths cross, eclipses and nodes appear. New moon is their conjunction, full moon their opposition; near the nodes they line up, and partial or total eclipse follows. Months have dark and new moons, stars have heliacal rise and visibility, months have quarters and full, stars have stationary points and retrograde loops—all reduce to one method of computation. Venus and Mercury ride the sunlight, hugging the Sun inside its orbit: moving fast they outpace it, moving slowly they fall back and halt, halt and then retrograde, slip westward then accelerate again—hence their alternating appearances as morning and evening stars.
7
日、月、五緯各有終原，而七元生焉。 見伏有日，留行有度，而率數生焉。 參差齊之，多少均之，會終生焉。 引而伸之，觸而長之，探賾索隱，鉤深致遠，無幽辟潛伏，而不以其精者然。 故陰陽有分，寒暑有節，天地貞觀，日月貞明。
Sun, Moon, and five planets each have a fundamental period, and from them the seven calendrical “elements” are derived. Heliacal rising and setting take definite days; motion and pause take measured arcs—so mean motions are computed. Unequal motions are averaged, discrepancies reconciled, and the epoch when cycles close is found. Extend the pattern, press every case, sound the hidden depths—nothing in the sky’s recesses escapes this reckoning. Thus Yin and Yang part cleanly, cold and heat keep their beats, Heaven and Earth hold a fixed measure, Sun and Moon a steady light.
8
若夫佑術開業，淳耀天光，重黎其上也[1]。 承聖帝之命若昊天，典歷像三辰，以授民事，立閏定時，以成歲功，羲和其隆也[2]。 取象金火，革命創製，治歷明時，應天順民，湯、武其盛也[3]。 及王德之衰也，無道之君亂之於上，頑愚之史失之於下。 夏後之時，羲和淫湎，廢時亂日，胤乃征之。 紂作淫虐，喪其甲子，武王誅之。 夫能貞而明之者，其興也勃焉； 回而敗之者，其亡也忽焉。 巍巍乎若道天地之綱紀，帝王之壯事，是以聖人寶焉，君子勤之。
Those who first framed these arts and let celestial splendor shine clear were Chong and Li above all[1]. Commissioned like a mandate from Heaven, they aligned the calendar with the three lights to guide the people’s labor, set intercalary rules and nailed the seasons—Xi and He’s glory was in that service[2]. They read the omens in the Venus and Mars years, seized the mandate, reformed the system, fixed the calendar to the seasons, matching Heaven and winning the people—Tang of Shang and Wu of Zhou stand highest here[3]. When royal virtue failed, reckless sovereigns corrupted the system aloft and dull scribes bungled it below. Under the later Xia, the Xi and He clans drowned in wine, abandoned the calendar and scrambled the days, until Yin’s punitive expedition was launched. The last king of Shang gave himself to cruelty and lost even his day-count; King Wu overthrew him. States that honor true calendrical order rise with sudden force; those that twist or break it collapse overnight. So august is this art—the very fabric of cosmos and the crown work of kings—that sages cherish it and gentlemen toil at it without rest.
9
夫歷有聖人之德六焉：以本氣者尚其體，以綜數者尚其文，以考類者尚其象，以作事者尚其時，以占往者尚其源，以知來者尚其流。 大業載之，吉凶生焉，是以君子將有興焉，咨焉而以從事，受命而莫之違也。 若夫用天因地，揆時施教，頒諸明堂，以為民極者，莫大乎月令。 帝王之大司備矣，天下之能事畢矣。
Calendrical science wears six virtues of the sage: root it in vital qi for its body, braid the numbers for its pattern, compare types for its imagery, time undertakings by it, trace antiquity to its source, and read the future in its drift. Great undertakings rest on it; fortune and misfortune flow from it—so the gentleman consults these reckonings before acting and, once charged, does not defy Heaven’s measure. Nothing matches the Monthly Ordinances for aligning Heaven and Earth, timing the edicts, and publishing them in the Bright Hall as the people’s standard. With that, the sovereign’s great calendar is whole and every task under Heaven is covered.
10
過此而往，腢忌苟禁，君子未之或知也。
Past that lie minor divinatory taboos and arbitrary bans—topics a serious scholar does not bother with.
11
斗之二十一度，去極至遠也，日在焉而冬至，腢物於是乎生。 故律首黃鐘，歷始冬至，月先建子，時平夜半。 當漢高皇帝受命四十有五歲，陽在上章，陰在執徐，冬十有一月甲子夜半朔旦冬至，日月閏積之數皆自此始，立元正朔，謂之漢歷。 又上兩元，而月食五星之元，並發端焉。
Where the Dipper’s circle stands twenty-one degrees from the pole—winter solstice—the Sun lies farthest south and the myriad things begin their yearly renewal. Hence the pitch sequence starts with Yellow Bell, the civil year at winter solstice, the month series with jianzi, the day-count from midnight. Forty-five years after Gaozu founded the Han, on a jiazi midnight in the eleventh winter month—new moon and solstice together—the grand conjunction of intercalation and planetary epochs was fixed; they set the epoch and standard month, and that is the Han calendar. Push two higher epochs back from that point and the eclipse cycle and five-planet epochs all start there too.
12
歷數之生也，乃立儀、表，以校日景。 景長則日遠，天度之端也。 日發其端，周而為歲，然其景不復，四周千四百六十一日，而景復初，是則日行之終。 以周除日，得三百六十五四分度之一，為歲之日數。 日日行一度，亦為天度。 察日月俱發度端[4]，日行十九周，月行二百五十四周，復會於端，是則月行之終也。 以日周除月周，得一歲周天之數。
To derive the basic numbers you erect a gnomon and shadow-table and read the noon shadow. A longer shadow means a more distant Sun—that marks the limit of the solar degree. The Sun leaves that point and tours the sky for a year, but the shadow does not repeat until four such cycles—1,461 days—when the gnomon reading matches the start: one complete solar period. Divide those days by four circuits and you get 365¼ days for the tropical year. The Sun moves one degree a day—that is both a solar day and a sky degree. From a common starting longitude the Sun completes nineteen revolutions while the Moon makes 254 and they conjoin again—closing the Moon’s cycle. Divide the Moon’s revolutions by the Sun’s and you have how many sidereal months fit a year.
13
以日一週減之，余十二十九分之七，則月行過周及日行之數也，為一歲之月。
Subtract one solar revolution and the remainder is 10 7/29—the Moon’s yearly advance over the Sun—which yields the synodic months in a year.
14
以除一歲日，為一月之數。 月之餘分積滿其法，得一月，月成則其歲[大]。
Divide that into the days of a year to get the mean length of a month. When leftover parts of a month add up to a full divisor you gain a month; when the months line up, the year is reckoned “long.”
15
月 (大) 四時推移，故置十二中以定月位。 有朔而無中者為閏月。 中之始 (日) [曰]節，與中為二十四氣。 以除一歲日，為一氣之日數也。 其分積而成日為沒，並歲氣之分，如法為一歲沒。 沒分於終中，中終於冬至，冬至之分積如其法得一日，四歲而終。 月分成閏，閏七而盡，其歲十九，名之曰章。 章首分盡，四之俱終，名之曰蔀。 以一歲日乘之，為蔀之日數也。 以甲子命之，二十而復其初，是以二十蔀為紀。 紀歲青龍未終，三終歲後復青龍為元。
Month (long) As the seasons turn, twelve “central” qi are inserted to anchor each lunar month. If a month contains a new moon but never receives its mid-qi, that month is intercalated. The start of a mid- (segment) qi segment is a jie; paired with the zhong qi you have all twenty-four solar terms. Spread the year’s days across them and you get the mean length of one qi. Leftover fractions pile into “submergence” days; summed against the rule they give the yearly mo total. Mo parts run out at the last zhong qi, which closes at winter solstice; when solstice remainders stack to a full day, the cycle closes every four years. Month remainders force intercalation; seven leap months in nineteen years balance the books—that is a zhang. When a zhang opens clean and all four seasonal markers align, you close a bu. Multiply a year’s days through that fold to get the bu in days. Counting with the sexagenary cycle, twenty bu bring the stem-branch back to the start—so twenty bu form a ji. Before the ji’s Azure Dragon cycle finishes, three ji later the Dragon station returns and opens a new yuan.
16
元法，四千五百六十[5]。
Yuan divisor: 4,560[5].
17
紀法，千五百二十[6]。
Ji divisor: 1,520[6].
18
紀月，萬八千八百。
Months in a ji: 18,800.
19
蔀法，七十六[7]。
Bu divisor: 76[7].
20
蔀月，九百四十。
Months in a bu: 940.
21
章法，十九。
Zhang divisor: 19.
22
章月，二百三十五[8]。
Months in a zhang: 235[8].
23
周天，千四百六十一。
Celestial circumference divisor: 1,461.
24
日法，四。
Day divisor: 4.
25
蔀日，二萬七千七百五十九。
Days in a bu: 27,759.
26
沒數，二十一，為章閏。
Mo count: 21—this sets intercalation within a zhang.
27
通法，四百八十七。
Common factor: 487.
28
沒法，七，因為章閏。
Mo divisor: 7—it builds the zhang intercalary rule.
29
日餘，百六十八。
Day remainder: 168.
30
中法， (四) [三]十二。
Zhong-qi divisor, (four) thirty-two.
31
大周，三十四萬三千三百三十五。
Grand cycle: 343,335.
32
月周千一十六。
Lunar cycle constant: 1,016.
33
月食數之生也，乃記月食之既者。 率二十三食而復既，其月 (食) 百三十五，率之相除，得五 (百) [月]二十三之二十而一食。 以除一歲之月，得歲有再食五百一十三分之五十[五]也。 分終其法，因以與蔀相約，得四與二十七，互之，會二千五十二，二十而與元會。
Eclipse theory begins by recording total lunar eclipses. The rule runs: after twenty-three eclipses totality repeats; the month The gloss marks the word ‘eclipse.’ After 135 months, cross-dividing the eclipse rates yields five. The gloss supplies ‘hundred.’ That is, twenty eclipses every twenty-three months—one eclipse each time that ratio completes. Spread across the months of a year, the mean is two eclipses per year, leaving a remainder of 55/513. Clear the fractional rule, reduce against the bu cycle, pair 4 with 27, cross them to a rendezvous of 2,052, and twenty such steps align with the grand epoch.
34
元會，四萬一千四十。
Grand-epoch conjunction: 41,040.
35
蔀會， (三) [二]千五十 (三) [二]。
Bu conjunction, Gloss: three. Two thousand fifty. Gloss: three. Two.
36
歲數，五百一十三。
Annual eclipse constant: 513.
37
食數，千八十一。
Eclipse-cycle constant: 1,081.
38
月數，百 (二) [三]十五。
Month constant: one hundred Gloss: two. Thirty-five.
39
食法，二十 (二) [三]。
Eclipse divisor: twenty Gloss: two. Three.
40
推入蔀術曰：以元法除去上元，其餘以紀法除之，所得數從天紀，筭外則所入紀也。 不滿紀法者，入紀年數也。 以蔀法除之，所得數從甲子蔀起，筭外，所入紀歲名命之，筭上，即所求年太歲所在。
To find your place in the calendrical cycle: remove whole yuan periods, divide what is left by the cycle length, count forward from the Celestial cycle, and the overflow tells which cycle you are in. Any remainder short of a full ji counts as the years elapsed inside that ji. Divide next by the bu length, count bu from the jiazi bu, read the stem-branch name for the year inside the ji, and the top of the count gives Tai-sui’s station for the target year.
41
推月食所入蔀會年，以元會除去上元，其餘以蔀會除之，所得以 (七) [二]十 (二) [七]乘之，滿六十除去之，余以二十除所得數，從天紀，筭 (之起) 外，所 (以) 入紀，不滿二十者，數從甲子蔀起，筭外，所入蔀會也。 其初不滿蔀會者，入蔀會年數也，各以 (不) [所]入紀歲名命之，筭上，即所求年 (蔀) [太歲所在]。
To locate a lunar eclipse within the bu-hui: strip out yuan hui periods, divide the remainder by the bu-hui length, take the quotient, and Fragment (with following glosses) completing seventy-two. …twenty… …two… multiply that quotient by seventy-two (the glosses spell out the digits), drop multiples of sixty, divide what remains by twenty, count from the Celestial ji, and This gloss marks the start of the rod-count. beyond the rods, what by which enters the ji; if the tally is under twenty, begin from the jiazi bu; the overflow names the bu hui you stand in. If the opening figure is short of a full bu hui, that counts years inside the bu hui; then, with The gloss negates the following graph. name the stem-branch for the ji-cycle you entered; above the rods lies the year you seek The gloss marks the bu cycle. [where Grand Year stands].
42
天紀歲名地紀歲名人紀歲名蔀首庚辰庚子庚申甲子一丙申丙辰丙子癸卯二壬子壬申壬辰壬午三戊辰戊子戊申辛酉四甲申甲辰甲子庚子五庚子庚申庚辰己卯六丙辰丙子丙申戊午七壬申壬辰壬子丁酉八戊子戊申戊辰丙子九甲辰甲子甲申乙卯十庚申庚辰庚子甲午十一丙子丙申丙辰癸酉十二壬辰壬 (午) [子]壬申壬子十三戊申戊辰戊子辛卯十四甲子甲申甲辰庚午十五庚辰庚子庚申 (乙) [己]酉十六丙申丙辰丙子戊子十七壬子壬申壬辰丁卯十八戊辰戊子戊申丙午十九甲申甲辰甲子乙酉二十推天正術，置入蔀年減一，以章月乘之，滿章法得一，名為積月，不滿為閏余，十二以上，其歲有閏。
Celestial-ji year-names, Terrestrial-ji year-names, Human-ji year-names, bu-head: gengchen, gengzi, gengshen, jiazi—one; bingshen, bingchen, bingzi, guimao—two; renzi, renshen, renchen, renwu—three; wuchen, wuzi, wushen, xinyou—four; jiashen, jiachen, jiazi, gengzi—five; gengzi, gengshen, gengchen, jimao—six; bingchen, bingzi, bingshen, wuwu—seven; renshen, renchen, renzi, dingyou—eight; wuzi, wushen, wuchen, bingzi—nine; jiachen, jiazi, jiashen, yimao—ten; gengshen, gengchen, gengzi, jiawu—eleven; bingzi, bingshen, bingchen, guiyou—twelve; renchen, ren The gloss reads the branch as wu (Horse). [zi] renshen, renzi—thirteen; wushen, wuchen, wuzi, xinmao—fourteen; jiazi, jiashen, jiachen, gengwu—fifteen; gengchen, gengzi, gengshen The gloss reads the stem as yi. [ji]you—sixteen; bingshen, bingchen, bingzi, wuzi—seventeen; renzi, renshen, renchen, dingmao—eighteen; wuchen, wuzi, wushen, bingwu—nineteen; jiashen, jiachen, jiazi, yiyou—twenty. Method for reckoning the celestial-standard month: set the years entered in the bu, subtract one, multiply by the zhang months; when it fills the zhang rule, that is one—call it accumulated months; what does not fill is the intercalary remainder; if it is twelve or more, that year has an intercalary month.
43
推天正朔日，置入蔀積月，以蔀日乘之，滿蔀月得一，名為積日，不滿為小余，積日以六十除去之，其餘為大余，以所入蔀名命之，筭盡之外，則前年天正十一月朔日也。 小余四百四十一以上，其月大。 求後月朔，加大余二十九，小余四百九十[九]，小余滿蔀月得一，上加大余，命之如前。
For new moon of month XI (winter solstice month): multiply bu accumulated months by bu days, fold in bu months to get accumulated days, keep the minor remainder, reduce the major remainder modulo sixty, read against the bu name, and the exhausted count gives the prior year’s first-month new moon. A minor remainder of 441 or above marks a long month. Step each new moon by +29 major, +499 minor; carry a full bu month from the minor into the major; name stems and branches as above.
44
一術，以大周乘年，周天乘[閏余]減之，余滿蔀 (日) [月]，則天正朔日也。
Alternate rule: year × grand cycle minus intercalary remainder × celestial circumference; when the residue fills the bu (days) month divisor, you have the civil new moon day.
45
推二十四氣術曰：置入蔀年減一，以 (月) [日]余乘之，滿中法得一，名曰大余，不滿為小余，大余滿六十除去之，其餘以蔀名命之，筭盡之外，則前年冬至之日也。
For the twenty-four solar terms: take bu years minus one, multiply by The gloss supplies ‘month’ in the compound. …day remainder; each full mid-qi divisor gives a major remainder; the shortfall is minor remainder; reduce modulo sixty and read against the bu name to get the previous winter solstice.
46
求次氣，加大余十五，小余七，除命之如前，小寒日也。
Each following qi: +15 major, +7 minor, reduce and name as before—next is Minor Cold.
47
推閏月所在，以閏余減章法，余以十二乘之，滿章閏數得一，滿四以上亦得一筭之數，從前年十一月起，筭盡之外，閏月也。 或進退，以中氣定之。
Intercalary placement: zhang rule minus leap surplus, ×12, divide by zhang intercalation; advance from month XI until the tally clears—that month is intercalated. Adjust ±1 month as needed so the mid-qi alignment holds.
48
推弦、望日，因其月朔大小余之數，皆加大余七，小余三百五十九四分三，小余滿蔀月得一，加大余，大余命如法，得上弦。 又加得望，次下弦，又後月朔。
Quarters and full moon: from the month’s new-moon fractions add +7 major, +359¾ minor, carry into bu months, then name—you get first quarter. Repeat the step for full moon, last quarter, and the following new moon.
49
其弦、望小余二百六十以下，每以百刻乘之，滿蔀月得一刻，不滿其 (數) [所]近節氣夜漏之半者，以筭上為日。
If quarter/full minor remainder ≤260, multiply by 100 ke; each bu month of product is one ke; the remainder—its (fraction) …proximity to half the night drip of the governing term fixes whether the event falls on the counted day.
50
推沒滅術，置入蔀年減一，以沒數乘之，滿日法得一，名為積沒，不盡為沒余。
Mo/mie algorithm: (bu years −1) × mo number; each full day rule adds one accumulated mo; leftover is mo remainder.
51
以通法乘積沒，滿沒法得一，名為大余，不盡為小余。 大余滿六十除去之，其餘以蔀名命之，筭盡之外，前年冬至前沒日也。 求後沒，加大余六十九，小余四，小余滿沒法，從大余，命之如前，無分為滅。
Accumulated mo × common divisor; fold by mo divisor into major and minor remainders. Reduce the major remainder modulo sixty, read against the bu name—the exhausted count is the last mo day before the prior solstice. Next mo: +69 major, +4 minor, carry by mo divisor; no remainder means mie (extinction).
52
一術，以 (為) [十]五乘冬至小余，以減通法，余滿沒法得一，則天正後沒也。
Alternate rule: (as) Multiply solstice minor remainder by fifteen, subtract from the common divisor; each mo divisor in the result is the post–new-year mo.
53
推合朔所在度，置入蔀積 (月) [日]以 (日) [蔀月]乘之，滿大周除去之，其余滿蔀月得一，名為積度，不盡為余分。 積度加斗二十一度，加二百三十五分，以宿次除之，不滿宿，則日月合朔所在星度也。 求後合朔，加度二十九，加分四百九十九，分滿蔀月得一度，經斗除二百三十五分。
To find conjunction’s ecliptic longitude: take bu accumulated (months) [days] and take (daily) …× bu months, drop whole great cycles, fold remainders into accumulated degrees and fractional parts. Add 21°35′ of Dipper measure, step through lodges; the residue is conjunction’s sidereal place. Next conjunction: +29° +499 parts, carry a degree per bu month, subtract 235 parts when crossing the Dipper field.
54
一術，以閏余乘周天，以減大周余，滿蔀月得一，合以斗二十一度四分一，則天正合朔日月所在度。
Shortcut: great-cycle residue minus intercalary remainder × celestial circumference, folded by bu months, plus 21¼° Dipper—gives conjunction longitude for civil new moon.
55
推日所在度，置入蔀積日之數，以蔀法乘之，滿蔀日除去之，其餘滿蔀法得一，為積度，不盡為余分。 積度加斗二十一度，加十九分，以宿次除去之，則夜半日所在宿度也。
Solar longitude: accumulated bu days × bu rule, modulo bu days and bu rule into degrees and fractions. Add 21°19′ Dipper, reduce by lodges—the midnight Sun position.
56
求次日，加一度。 求次月，大加三十度，小加二十九度，經斗除十[九]分。
For the next night: add 1°. Next month: +30° if long, +29° if short; subtract 19 parts when threading the Dipper.
57
一術，以朔小余減合[朔]度分，即日夜半所在。 其分 (三) [二]百 (二) [三]十五約之，十九乘之。
Alternate: new-moon minor remainder minus conjunction fraction gives midnight solar longitude. Take that fraction Gloss: three. two hundred Gloss: two. …235, reduce, then multiply by nineteen.
58
推月所在度，置入蔀積日之數，以月周乘之，滿蔀日除去之，其餘滿蔀法得一，為積度，不盡為余分。 積度加斗二十一十[九]分，除如上法，則所求之日夜半月所在宿度也。
Lunar longitude: bu accumulated days × lunar cycle, same reduction into degrees and parts. Add the Dipper offset of 21°19′, then lodge-reduce as before—that is midnight lunar longitude for the night in question.
59
求次日，加十三度二十八分。 求次月，大加三十五度六十一分，月小二十二度三十三分，分滿法得一度，經斗除十九分。 其冬下旬月在張、心署之，謂 (盡) [晝]漏分後盡漏盡也。
Next day: +13°28′. Next month: long +35°61′, short +22°33′, carry degrees, subtract 19 parts at the Dipper. Late winter nights when the Moon sits in Zhang and Heart are called (spent) “daylight-drip exhaustion”—when the day’s drip runs out.
60
一術，以蔀法除朔小余，所得以減日半度也。 余以減分，即月夜半所在度也。
Shortcut: new-moon minor remainder ÷ bu rule, subtracted from half a day. Subtract that from the fractional column to get midnight lunar longitude.
61
推日明所入度分術曰：置其月節氣夜漏之數，以蔀法乘之，二百除之，得一分，即夜半到明所行分也。 以增夜半日所在度分，為明所在度分也。
Dawn Sun: take the term’s night drip × bu rule ÷ 200 for degrees moved before sunrise. Add to midnight solar longitude for dawn position.
62
求□日所入度，以夜半到明日所行分 (分) 減蔀法，其餘即夜半到□所行分也。
For dusk’s solar longitude: take the motion from midnight to the following (fraction) subtract from the bu divisor; the remainder is midnight-to-dusk travel.
63
以加夜半所在度分，為□日所在度也。
Add to midnight longitude for the Sun’s position at dusk.
64
推月明所入度分術曰：置其節氣夜 (半) [漏]之數，以月周乘之，以二百除之，為積分。 積分滿蔀法得一，以增夜半度，即 (明) 月[明]所在度也。
Dawn Moon: set the term’s night The gloss supplies ‘half’ (of the night drip). Take the night-drip figure for the term, multiply by the lunar cycle, divide by 200 to get accumulated fractional parts. When those parts complete one bu divisor, add the result to midnight longitude—then The gloss marks ‘dawn’ or ‘brightness.’ you have the Moon’s longitude at dawn.
65
求□月所入度：以明積分減月周，其餘滿蔀法得一度，加夜半，則□月所在度也。
For the following month’s longitude: subtract dawn’s fractional advance from one lunar cycle; each full bu divisor adds one degree to midnight’s position—that gives the Moon’s place for that month.
66
推弦、望日所入星度術曰：置合朔度分之數，加七度三百五十九分四分 (之) 三，[以]宿次除之，即得上弦日所入宿度分也。
To find ecliptic longitudes at quarters and full moon: start from conjunction’s longitude and add 7°359′4″ The gloss completes the fraction ‘of’ the next unit. …then lodge-reduce to get the first quarter’s sidereal position.
67
求望、下弦，加除如前法，小分[滿]四從大分，[大分]滿蔀月從度。
For full and last quarters, repeat the same addition and reduction; carry small parts into large parts by fours, and large parts into degrees when they fill a bu month.
68
推弦、望月所入星度術曰：置月合朔度分之數，加度九十八，加分六百五十三半，以宿次除之，即上弦月所入宿度分也。
For the Moon at quarters: from conjunction longitude add 98°653½′, then lodge-reduce—that is first-quarter lunar longitude.
69
求望、下弦，加除如前分，滿蔀月從度。
Step to full and last quarters the same way, carrying a full bu month of parts into whole degrees.
70
推月食術曰：置入蔀會年數，減一，以食數乘之，滿歲數得一，名曰積食，不滿為食余。 以月數乘積[食]，滿食法得一，名為積月，不滿為月餘分。 積月以章月除去之，其餘為入章月數。 當先除入章閏，乃以十二除去之，不滿者命以十一月，筭盡之外，則前年十一月前食月也。 求入章閏者，置入章月，以章閏乘之，滿章月得一，則入章閏數也。 余分滿二百二十四以上至二百三十一，為食在閏月。 閏或進退，以朔日定之。 求後食，加五 (百) [月]二十分，滿法得一月數，命之如法，其分盡食筭上。
Eclipse reckoning: (bu-hui years − 1) × eclipse number, folded by the year divisor into accumulated eclipses and remainder. Multiply accumulated eclipses by the month constant; reduce by the eclipse divisor into whole months and leftover parts. Strip whole zhang-month periods from accumulated months; what remains counts months within the current zhang. First clear the zhang’s intercalary offset, then reduce by twelve; name from month XI; the exhausted count lands on the eclipse month before the prior year’s eleventh month. Intercalation within the zhang: months-in-zhang × zhang intercalation, divided by zhang months. Remainders from 224 through 231 put the eclipse in a leap month. Adjust the leap month ±1 as needed to match the new-moon reckoning. For the next eclipse, add five The gloss supplies ‘hundred’ in the numeral. …hundred twenty fractional parts; each full divisor adds a month; name as usual; zero remainder means the eclipse sits on the named day.
71
推月食朔日術曰：置食積月之數，以二十九乘之，為積日。 又以四百九十[九]乘積月，滿蔀月得一，以並積日，以六十除之，其餘以所會蔀名命之，筭盡之外，則前年天正前食月朔日也。
Eclipse new moon: accumulated eclipse months × 29 gives accumulated days. Add 499 × accumulated months with bu-month carry into days, reduce modulo sixty, read against the bu name—the exhausted count is the new moon before the eclipse month in the prior civil year.
72
求食日，加大余十四，小余七百一十九半，小余滿蔀月為大余，大余命如前，則食日也。
Eclipse day: +14 major, +719½ minor; carry bu months into major; name stems and branches—that is eclipse day.
73
求後食朔及日，皆加大余二十七，小余六百一十五。 其月餘分不滿二十者，又加大余二十九，小余四百九十九。 其食小餘者，當以漏刻課之，夜漏未盡，以筭上為日。
Each following eclipse: +27 major, +615 minor (with standard carries). If month remainder < 20, add another +29 major and +499 minor. Fractional eclipse times need clepsydra conversion; if night drip has not run out, the event still belongs to the previous civil day.
74
一術，以歲數去上元，余以為積月，以百一十二乘之，滿月數去之，余滿食法得一，則天正後食。
Shortcut: high-yuan residue → ×112 → reduce by month number → fold by eclipse divisor for the first post–new-year eclipse.
75
推諸加時，以十二乘小余，先減如法之半，得一時，其餘乃以法除之，所得筭之數從夜半子起，筭盡之外，則所加時也。
Convert fractional day to the twelve double-hours: 12 × minor remainder, adjust by half the rule, divide for the hour count from midnight zi.
76
推諸上水漏刻：以百乘其小余，滿其法得一刻； 不滿法 (法) 什之，滿法得一分。 積刻先減所入節氣夜漏之半，其餘為晝上水之數。 過晝漏去之，余為夜上水數。 其刻不滿夜漏半者，乃減之，余為昨夜未 (晝) [盡]，其弦望其日。
Clepsydra marks after midnight: minor remainder × 100 ÷ rule = whole ke. The shortfall below one full divisor The gloss identifies the divisor in the rule. …split the remainder into tenths of a ke; ten tenths make one fen when they fill the divisor. Subtract half the term’s night drip from the accumulated marks to get daytime clepsydra after sunrise. Overflow past day drip rolls into the night drip tally. If the count falls short of half the night drip, subtract from that half—the residue is time still belonging to the previous The gloss contrasts ‘day’ with night. …day’s drip; for quarters and full moons, assign the event to that civil day.
77
五星數之生也，各記於日，與周天度相約而為率。 以章法乘周率為 (用) [月]法，章月乘日率，如月法，為積月月餘。 以月之 (月) [日]乘積[月]，為朔大小余。 乘為入月日餘。 以日法乘周率為日度法，以[周]率去日率，余以乘周天，如日度法，為[積]度 (之) [度]余也。 日率相約取之，得二千九百九十萬一千六百二十一億五十八萬二千三百，而五星終，如蔀之數，與元通。
Planetary epochs are keyed to days and reduced against the celestial circumference to yield each planet’s rate. Zhang rule × circumferential rate gives The gloss reads ‘use’ in the compound. …the month divisor; zhang months × day rate, compared to the month rule, yields accumulated months and remainder. Using the month’s The gloss marks the second ‘month’ in the phrase. …day factor × accumulated months gives conjunction major and minor fractions. The product also gives days and parts into the synodic month. Day rule × circumferential rate defines the degree divisor; (circumference rate − day rate) × celestial circumference ÷ that divisor gives accumulated longitude The gloss links ‘accumulated’ to ‘degrees.’ …and its fractional remainder. Mutual reduction of the planetary rates yields 2,990,162,100,582,300, after which the five wanderers finish their common period and align with the bu and yuan epochs.
78
木，周率，四千三百二十七。 日率，四千七百二十五。 合積月，十三。 月餘，四萬一千六百六。 月法，八萬二千二百一十三。 大余，二十三。 小余，八百四十七。 虛分，九十三。 入月日，十五。 日餘，萬四千六百四十 (七) [一]。
Jupiter: circumferential rate 4,327. Day rate: 4,725. Synodic months per cycle: 13. Month remainder: 41,606. Month divisor: 82,213. Major remainder: 23. Minor remainder: 847. Empty fraction: 93. Days into the synodic month: 15. Day remainder: 14,640 Gloss digit seven. …to finish the day-remainder figure 14,647.
79
日度法，萬七千三百八。 積度，三十三。 度余，萬三百一十四。
Degree divisor: 17,308. Accumulated longitude: 33°. Degree fraction: 10,314.
80
火，周率，八百七十九。 日率，千八百七十六。 合積月，二十六。 月餘，六千六百三十四。 月法，萬六千七百一。 大余，四十七。 小余，七百五十四。 虛分，一百八十六。 入月日，十 (一) [二]。 日餘，千八百七十二。 日度法，三千五百一十六。 積度，四十九。 度余，一百一十四。
Mars: circumferential rate 879. Day rate: 1,876. Synodic months per cycle: 26. Month remainder: 6,634. Month divisor: 16,701. Major remainder: 47. Minor remainder: 754. Empty fraction: 186. Days into the synodic month: ten Gloss digit one. …forming twelve days. Day remainder: 1,872. Degree divisor: 3,516. Accumulated longitude: 49°. Degree fraction: 114.
81
土，周率，九千九十六。 日率，九千四百一十五。 合積月，十二。 月餘，十三萬八千六百三十七。 月法，十七萬二千八百二十四。 大余，五十四。 小余，三百四十八。 虛分，五百九十二。 入月日，二十 (三) [四]。 日餘，二千一百六十三。 日度法，三萬六千三百八十四。 積度，十二。 度余，二萬九千四百五十一。
Saturn: circumferential rate 9,096. Day rate: 9,415. Synodic months per cycle: 12. Month remainder: 138,637. Month divisor: 172,824. Major remainder: 54. Minor remainder: 348. Empty fraction: 592. Days into the synodic month: twenty Gloss digit three. …forming twenty-four days. Day remainder: 2,163. Degree divisor: 36,384. Accumulated longitude: 12°. Degree fraction: 29,451.
82
金，周率，五千八百三十。 日率，四千六百六十一。 合積月，九。 月餘，九萬八千四百五。 月法，十[一]萬七百七十。 大余，二十五。 小余，七百三十一。
Venus: circumferential rate 5,830. Day rate: 4,661. Synodic months per cycle: 9. Month remainder: 98,405. Month divisor: 110,770. Major remainder: 25. Minor remainder: 731.
83
虛分，二百九。 入月日，二十六。 日餘，二百八十一。 日度法，二萬三千三百二十。 積度，二百九十二。 度余，二百八十一。
Empty fraction: 209. Days into the synodic month: 26. Day remainder: 281. Degree divisor: 23,320. Accumulated longitude: 292°. Degree fraction: 281.
84
水，周率，萬一千九百八。 日率，千八百八十九。 合積月，一。 月餘，二十一萬七千六百六十[三]。 月法，二十二萬六千二百五十二。 大余，二十九。 小余，四百九十九。 虛分，四百四十 (九) [一]。 入月日，二十 (七) [八]。
Mercury: circumferential rate 11,108. Day rate: 1,889. Synodic months per cycle: 1. Month remainder: 217,663. Month divisor: 226,252. Major remainder: 29. Minor remainder: 499. Empty fraction: 440 Gloss digit nine. …completing 441 for the void fraction. Days into the synodic month: twenty Gloss digit seven. …forming twenty-eight days.
85
日餘，四萬四千八百五。 日度法，四萬七千六百三十 (一) [二]。 積度，五十七。 度余四萬四千八百五。
Day remainder: 44,805. Degree divisor: 47,631 Gloss digit one. …completing 47,632. Accumulated longitude: 57°. Degree fraction: 44,805.
86
推五星術，置上元以來，盡所求年，以周率乘之，滿日率得一，名為積合； 不盡名[為]合余。 [合]余以周率除之，不得焉退歲； 無所得，星合其年，得一合前年，二合前二年。 金、水積合奇為晨，偶為夕。 其不滿周率者反減之，余為度分。
For planetary conjunctions: years since high yuan × circumferential rate ÷ day rate gives accumulated conjunctions; the undivided part is conjunction remainder. Divide conjunction remainder by the circumferential rate; if it fails to divide, step back one year; no quotient means conjunction this year, one means last year, two means two years prior. Venus and Mercury: odd accumulated conjunction counts as a morning apparition, even as an evening one. Take the shortfall below one circumferential rate, subtract it from the rate; the residue is ecliptic longitude in degrees.
87
推星合月，以合積月乘積合為小積，又以月餘乘積合，滿其月法得一，從小積[為積月，不盡]為月餘。 積月滿紀月去之，余為入紀月。 每以章閏乘之，滿章月得一為閏； 不盡為閏余。 以閏減入紀月，其餘以十二去之，余為入歲月數，從天正十一月起，筭外，星合所在之月也。 其閏[余]滿二百二十四以上至二百三十一星合閏月。 閏或進退，以朔制之。
Conjunction month: conjunction count × synodic months plus cross-terms with month remainder, reduced by the month divisor into accumulated months and fractional remainder. Modulo ji months gives position within the ji cycle. Multiply by zhang intercalation; each full zhang month adds an intercalary slot; the leftover is intercalary fractional remainder. Strip leap from ji months, reduce by twelve, count from month XI—where the tally ends is conjunction’s civil month. Intercalary remainders 224–231 place conjunction in a leap month. Adjust the leap month ±1 to match new-moon placement.
88
推朔日，以蔀日乘 (之) 入紀月，滿蔀月得一為積日，不盡為小余。 積日滿六十去之，余為大余，命以甲子，筭外，星合月朔日。
New moon day: multiply bu days by The gloss completes the object of ‘multiply.’ …months-in-ji; each bu month folds into accumulated days with minor remainder. Reduce days modulo sixty, name with sexagenary cycle—the exhausted count is conjunction new moon.
89
推入月日，以蔀日乘月餘，以其月法乘朔小余，從之，以四千四百六十五約之，所得 (得) 滿日度法得一，為入月日，不盡為日餘。 以朔命入月日，筭外，星合日也。
Days into synodic month: (bu days × month remainder + month method × new-moon minor) ÷ 4465 yields The gloss repeats ‘obtain’ from the rule. …quotient folded by the degree-day divisor into whole days and day remainder. Read against new moon to name the day-within-month of conjunction.
90
推合度，以周天乘度分，滿日度法得一為積度，不盡為度余。 以斗二十一四分之一命度，筭外，星合所在度也。
Conjunction longitude: celestial circumference × fractional parts ÷ day-degree method. Reduce from 21¼° in the Dipper field to get conjunction ecliptic longitude.
91
一術，加退歲一，以減上元，滿八十除去之，余以沒數乘之，滿日法得一，為大余，不盡為小余。 以甲子命大余，則星合歲天正冬至日也。 以周率[乘]小余，並度余，余滿日度法從度，即 (正) [至]後星合日數也，命以冬至。 求後合月，加合積月於入歲月，加月餘於月餘，滿其月法得一，從入歲月。 入歲月滿十二去之，有閏計焉，余命如前，筭外，後合月也。 (余一) [金、水]加晨得夕，加夕得晨。
Alternate: (high yuan − retreat year − 1) mod 80 × mo number, reduced to major and minor remainders. Sexagenary major remainder names conjunction-year winter solstice in the civil calendar. Circumferential rate × minor remainder + degree remainder, carried by the degree divisor—then The gloss reads ‘correct’ for the alignment. …you have days after solstice to conjunction; count from winter solstice. Step conjunction month: add synodic months and remainders with carries into the civil year count. Reduce by twelve with leap adjustment; name as before for the following conjunction month. Note the single-remainder carry in the rule. For Venus and Mercury, toggling morning and evening apparitions each cycle.
92
求朔日，以大小余加今所得，其月餘得一月者，又[加大]余二十九，[小余四百九十九，]小飾滿蔀月得一， (如) [加]大余，大余命如前。
New-moon step: add current major/minor; if a full month accrues, add +29/+499 and carry bu months into major; The gloss reads ‘as’ in the instruction. …then add to major remainder and name stems and branches as before.
93
求入月日，以入月日[日]余加今所得，余滿日度法得一，從日。 其前合月朔小余 (不) 滿其虛分者，空加一日。 日滿月先去二十九，其後合月朔小余不滿四百九十九，又減一日，其餘命如前。
Update days-in-month by adding day parts with carry through the degree-day divisor. If the prior conjunction’s new-moon minor remainder The gloss negates the verb. fails to reach its empty fraction, insert an extra day in the count. On month rollover subtract 29 first; if new moon minor <499, subtract another day; then name as usual.
94
求合度，以積度度余加今所得，余滿日度法得一從度，命如前，經斗除如周率矣。
Advance longitude by accumulated degrees and parts, carry through the divisor, Dipper-subtract like the circumferential step.
95
木，晨伏，十六日七千 (二) [三]百二十分半，行二度萬三千八百一十一分，在日後十三度有奇，而見東方。 見順，日行五十八分度之十一，五十八日行十一度，微□，日行九分，五十八日行九度。 留不行，二十五日。 旋逆，日行七分度之一，八十四日 (進) [退]十二度。 復留，二十五日。 復順，五十八日行九度，又五十八日行十一度，在日前十三度有奇，而夕伏西方。 除伏逆，一見三百六十六日，行二十八度。 伏復十六日七千 (二) [三]百二十分半，行二度萬三千八百一十一分，而與日合。 凡一終，三百九十八日有萬四千六百四十一分，行星三十 (二) [三]度與萬三百一十四分，通率日行四千七百二十五分之三百九十八。
Jupiter morning apparition: 16 days plus 7,000 Gloss digit two. …723½ fractional parts; motion 2°13,811′; about 13° east of the Sun at heliacal rise. Direct motion: 11/58° per day for 58 days (11°), then a slower 9/58° per day for another 58 days (9°). Stationary: 25 days. Retrograde: 1/7° per day; in 84 days The gloss marks forward motion in the manuscript. …it retreats 12° (text: advance/retreat). Second station: 25 days. Second direct leg mirrors the first, ending ~13° west of the Sun at heliacal set. One apparition, omitting dusk and retrograde, lasts 366 days and spans 28°. Evening apparition ends with 16 days plus 7,000 Gloss two. …723½ parts and 2°13,811′ of motion to solar conjunction. One synodic period: 398 days 14,641 parts; the planet moves 30 Gloss two. …33° and 10,314 parts; mean daily motion 398/4725 of a degree.
96
火，晨伏，七十一日二千六百九十四分，行五十五度二千二百五十四分半，在日後十六度有奇，而見東方。 見順，日行二十三分度之十四，[百]八十四日行[百]一十二度。 微□，日行十二分，九十二日行四十八度。 留不行，十一日。 旋逆，日行六十二分度之十七，六十二日退十七度。 復留，十一日。 復順，九十二日，行四十八度，又百八十四日行百一十二度，在日前十六度有奇，而夕伏西方。
Mars morning: 71d 2694p; motion 55°2254½p; ~16° east of the Sun at appearance. Direct: 14/23° per day for 184 days covering 112°. Slower leg: 12 parts per day for 92 days (48°). Stationary: 11 days. Retrograde: 17/62° per day for 62 days (−17°). Second station: 11 days. Return direct mirrors the outbound leg; ends ~16° west of the Sun at disappearance.
97
除伏逆，一見六百三十六日，行[三]百三度。 伏復，七十一日二千六百九十四分，行五十五度二千二百五十四分半，而與日合。 凡一終，七百七十九日有千八百七十二分，行星四百一十四度與九百九十三分。 通率日行千八百七十六分之九百九十七。
One apparition (no dusk/retrograde): 636 days, 303° of travel. Concluding dusk: 71d 2694p and 55°2254½p motion to conjunction. Synodic period 779d 1872p; arc 414°993p. Mean daily motion 997/1876 of a degree.
98
土，晨伏，十九日千八十一分半，行三度萬四千七百二十五分半，在日後十五度有奇，而見東方。 見順，日行四十三分度之三，八十六日行六度。 留不行，三十三日。 旋逆，日行十七分度之一百二，日退六度。 復留，三十三日。 復順，八十六日，行六度，在日前十五度有奇，而夕伏西方。 除伏逆，[一]見三百四十日，行六度。 伏復，十九日千八十一分半，行三度萬四千七百二十五分半，與日合。 凡一終，三百七十八日有二千一百六十三分，行星十二度與二萬九千四百五十一分。 通率日行九千四百一十五分之三百一十九。
Saturn morning: 19d 1081½p; motion 3°14,725½p; ~15° east of the Sun. Direct: 3/43° per day for 86 days (6°). Stationary: 33 days. Retrograde leg: 102/17° per day, yielding 6° of retreat per day in the tabulation. Second stationary phase: 33 days. Return direct: 86 days for 6°, ending ~15° west of the Sun at heliacal set. One apparition (omitting dusk and retrograde): 340 days, 6° of net motion. Closing dusk: 19d 1081½p and 3°14,725½p to conjunction. Saturn’s synodic period: 378d 2163p; arc 12°29,451p. Mean daily motion: 319/9415 of a degree.
99
金，晨伏，五日，退四度，在日後九度，而見東方。 見逆，日行五分度之三，十日，退六度。 留不行，八日。 [旋]順，日行 (行) 四十六分度之三十三，四十六日行三十三度。 而[疾]，日行一度九十[一]分度之十五，九十一日行百六度。 益疾，日行一度二十二分，九十一日行百一十三度，在日後九度，而晨伏東方。 除伏逆，一見二百四十六日，行二百四十六度。 伏四十一日二百八十一分，行五十度二百八十一分，而與日合。 一合二百九十二日[二]百八十一分，行星如之。
Venus morning apparition: 5 days, −4°, ~9° east of the Sun at rise. Initial retrograde: 3/5° per day for 10 days (−6°). Stationary: 8 days. Then direct motion at a daily rate of The gloss marks the verb ‘travel.’ 33/46° per day for 46 days (33°). Faster leg: 15/91° per day for 91 days (106°). Peak speed: 1°22′ per day for 91 days (113°), ending ~9° east of the Sun at morning disappearance. One morning apparition (no dusk/retrograde): 246 days, 246°. Morning dusk: 41d 281p, motion 50°281p to inferior conjunction. Inferior conjunction to inferior conjunction: 292d 281p; angular travel matches.
100
金，夕伏，四十一日二百八十一分，行五十度二百八十一分，在日前九度，而見西方。 見順，疾，日行一度九十一分度之二十二，九十一日行百一十三度。
Venus evening apparition opens: 41d 281p, 50°281p motion, ~9° west of the Sun. Evening direct (fast): 22/91° per day for 91 days (113°).
101
微□，日行一度十五分，九十一日行百六度。 而 (進) [□]，日行四十六分度之三十三，四十六日行三十三度。 留不行，八日。 旋逆，日行五分度之三，十日退六度，在日前九度，而夕伏西方。 除伏逆，一見二百四十六日，行二百四十六度，伏五日，退四度而 (後) [復]合。 凡 (三) [再]合一終，五百八十四日有五百六十二分，行星如之。 通率日行一度。
Slower direct leg: 1°15′ per day for 91 days (106°). Then The gloss marks forward motion. …continuing at 33/46° per day for 46 days (33°). Stationary: 8 days. Evening retrograde: 3/5° per day for 10 days (−6°), ~9° west of the Sun at set. Evening apparition (no dusk arc): 246 days, 246°; then 5 days dusk, −4°, and The gloss reads ‘after’ or ‘behind.’ …inferior conjunction again. In total The gloss supplies the numeral three. Two inferior conjunctions complete one synodic cycle: 584d 562p with matching angular travel. Mean motion averages one degree per day over the cycle.
102
水，晨伏，九日，退七度，在日後十六度，而見東方。 見逆，一日退一度。 留不得，二日。 旋順，日行九分度之八，九日行八度。 而疾，日行一度四分度之一，二十日行二十五度，在日後十六度，而晨伏東方。 除伏逆，一見，三十二日，行三十二度，伏十六日四萬四千八百五分，行三十二度四萬四千八百五分，而與日合。 一合五十七日有四萬四千八百五分，行星如之。
Mercury morning: 9 days, −7°, ~16° east of the Sun. Retrograde: about 1° per day. No true station: 2-day transition. Direct: 8/9° per day for 9 days (8°). Fast direct: 1¼° per day for 20 days (25°), ~16° east of the Sun at disappearance. Morning apparition: 32d, 32°; dusk 16d 44,805p, motion 32°44,805p to conjunction. One synodic arc: 57d 44,805p with matching travel.
103
水，夕伏，十六日四萬四千八百五分，行三十二度四萬四千八百五分，在日前十六度，而見西方。 見順，疾，日行一度四分度之一，二十日行二十五度。 而□，日行九分度之八，九日行八度。 留不行，二日。 [旋]逆，一日退一度，在日前十六度，而夕伏西方。 除伏逆，一見三十二日，行三十[二]度，伏九日，退七度而復合。 凡再合一終，百一十五日有四萬一千九百七十八分，行星如之。 通率日行一度。
Evening apparition: 16d 44,805p, 32°44,805p, ~16° west of the Sun. Evening direct (fast): 1¼° per day for 20 days (25°). Then slower: 8/9° per day for 9 days (8°). Stationary: 2 days. Retrograde: ~1° per day, ~16° west of the Sun at evening set. Evening apparition: 32d, 32°; then 9d dusk, −7° to conjunction. Two conjunctions per cycle: 115d 41,978p with matching arc. Mean motion: one degree per day.
104
步術，以步法伏日度分， (如) [加]星合日度余，命之如前，得星見日度也。
To step planetary motion: take tabulated dusk days and fractions, as add conjunction’s longitude remainder and reduce as before for apparent position.
105
(術) [行]分母乘之，分 (日) 如[日]度法而一，分不盡如 (法) 半[法]以上，亦得一，而日加所行分，滿其母得一度。 逆順母不同，以當行之母乘故分，如故母，如一也。 留者承前，逆則減之，伏不書度。 經斗除如行母，四分具一。 其分有損益，前後相放。 其以赤道命度，進加退減之。 其步以黃道。
The gloss marks ‘method’ in the heading. Multiply by the motion’s fractional denominator; the parts column The gloss supplies ‘day’ in the compound. …is ratioed to the daily degree divisor; leftover parts compare to the divisor half-divisor or more carries one; add the day’s motion; a full denominator of parts makes one degree. When direct and retrograde use different denominators, cross-multiply fractions to a common base. Station inherits prior longitude; retrograde subtracts; dusk intervals omit degree entries. Crossing the Dipper field, reduce by the motion denominator (quarters to wholes). Round fractional drift forward and backward so the columns balance. For equatorial longitudes, add for direct motion and subtract for retrograde. These steps follow ecliptic (Yellow Path) longitudes.
106
(日) [月]名 (日) [月]名天正十一月十二月正月二月三月四月五月六月冬至大寒雨水春分穀雨小滿夏至大暑七月八月九月十月處暑秋分霜降小雪[9]
Column header fragment: day. Column: month name. Repeated column marker: day. Table heading pairing month names (civil months XI–XII and I–X) with the twenty-four qi nodes listed in order (Winter Solstice through Light Snow, with a footnote mark [9]).
107
斗二十六[10]牛八女十二[11]虛十[12]危十六[13]室十六[14]壁十[15][一]北方九十八度四分一奎十六婁十二 (進) [退]一胃十四 (進二) [退一]昴十一 (進) [退]二畢十六 (進) [退]三觜二退三參九退四西方八十度井三十三退三鬼四柳十五星七進一張十八進一翼十八進 (一) [二]軫十七進一南方百一十二度角十二亢九退一氐十五退二房五退三心五退三尾十八 (進) [退]三箕十一退三東方七十五度右赤道度周天三百六十五度四分一斗二十四 (進一) [四分一]牛七女十一虛十危十六室十八壁十北方九十六度四分一奎十七婁十二胃十五昴十二畢十六觜三參八西方八十三度井三十鬼四柳十四星七張十七翼十九軫十八南方百九度角十三亢十氐十六房五心五尾十八箕十東方七十七度右黃道度三百六十五四分一
Equatorial lodge breadths (first segment): Dipper 26° (fn.10), Ox 8°, Girl 12° (fn.11), Emptiness 10° (fn.12), Rooftop 16° (fn.13), Hall 16° (fn.14), Wall 10° (fn.15); Northern Quadrant subtotal 98¼°; then Kui 16°, Lou 12°. Manuscript gloss: advance. Retreat correction −1; Stomach 14°. Advance correction +2. Retreat −1; Pleiades 11°. Advance. Retreat −2; Net (Bi) 16°. Advance. Further equatorial entries with advance/retreat notations; Western Quadrant totals 80°; Well 33° (−3), Ghost 4°, Willow 15°, Star 7° (+1), Extended Net 18° (+1), Wing 18°, advance… Gloss: one. …Chariot 17° (+1); Southern Quadrator 112°; Horn 12°, Horn-Guard 9° (−1), Base 15° (−2), Chamber 5° (−3), Heart 5° (−3), Tail 18°. Advance. Retreat −3; Winnowing Basket 11° (−3); Eastern Quadrant 75°; right-hand column totals equator 365¼°; Dipper 24°. Advance +1. Yellow-path lodge table (second column): same quadrants with updated widths—Northern 96¼°, Western 83°, Southern 109°, Eastern 77°; circuit 365¼° on the ecliptic.
108
黃道去極，日景之生，據儀、表也。 漏刻之生，以去極遠近差乘節氣之差。 如遠近而差一刻，以相增損。 □明之生，以天度乘晝漏，夜漏減 (三) [之，二]百而一，為定度。 以減天度，余為明； 加定度一為□。 其餘四之，如法為少。 [二為半，三為太，]不盡，三之，如法為強，余半法以上以成強。 強三為少，少四為度，其強二為少弱也。 又以日度余為少強，而各加焉[16]。
Ecliptic polar distance and noon shadows come from the armillary ring and shadow-table. Seasonal clepsydra corrections scale term-to-term polar-distance differences. Adjust one ke up or down with latitude (near or far from the pole). Twilight length: scale daylight drip by sky degrees; subtract night drip Gloss: three. …divide by 200 for the fixed arc. Subtract that from the day-arc; the remainder is bright twilight; add one fixed degree for the dark interval. Take the leftover, divide in fourths; by rule these are ‘weak’ quarter-degrees. Traditional strength labels: two parts weak, three great; leftovers tripled per rule give ‘strong’ fractions, rounded up past half the divisor. Three strong quarters fold to one weak; four weak make a degree; two strong form a ‘weak-less’ notch. Also fold the Sun’s fractional degrees into lesser-strong units and add as the table requires [16].
109
二十四氣
Section title: the twenty-four solar terms.
110
冬至[17]
Subheading: Winter Solstice, with footnote [17].
111
日所在黃道去極晷景晝漏刻夜漏刻□中星[一]旦中星二十四氣日所在黃道去極晷景晝漏刻夜漏刻□中星[一]旦中星冬至[二]斗二十一度八分退二百一十五度丈三尺四十五五十五奎六弱亢二少強退一
Column headers for the solar-term almanac, then Winter Solstice row [two]: solar longitude in Dipper 21°8′ (retreat), polar distance 215, shadow 1 zhang 3 chi, day drip 45 ke, night 55 ke, culminating Kui 6 weak, dawn Horn-Guard 2 lesser-strong, with retreat notation −1.
112
女二度七分進一百一十三強丈二尺三寸四十五八分五十四二分婁六半強退一氐七少弱退二
Next almanac row (Great Cold segment per table): Girl 2°7′ advance; polar distance 113 greater-strong; shadow 1 zhang 2 chi 3 cun; drips 45 ke 8 fen / 54 ke 2 fen; Lou 6 half-strong (−1); Base 7 lesser-weak (−2).
113
虛五度十四分進二百一十大弱丈一尺四十六八分五十三八分胃十一半強退一心半退三
Rain Water row: Emptiness 5°14′; ecliptic polar distance 210 (da-weak); noon shadow 1 zhang 1 chi; day drip 46 ke 8 fen, night 53 ke 8 fen; Stomach 11 half-strong (−1); Heart half (−3).
114
危十度二十一分進二百六少強九尺六寸四十八六分五十一四分畢五少弱退三尾七半弱退三
Spring Equinox row: Rooftop 10°21′; polar distance 206 (shao-strong); noon shadow 9 chi 6 cun; day drip 48 ke 6 fen, night 51 ke 4 fen; Net (Bi) 5 shao-weak (−3); Tail 7 half-weak (−3).
115
室八度二十八分進三百一強七尺九寸五分五十八分四十九二分參六半弱退四箕大弱退三
Grain Rain almanac row: solar longitude in Hall 8°28′; ecliptic polar distance 301 (strong); noon shadow 7 chi 9 cun 5 fen; clepsydra 58:49 / 58:49; culminating Triaster 6 half-weak (−4); dawn Winnowing Basket great-weak (−3).
116
壁八度三分進一九十五強六尺五寸五十三三分四十六七分井十七少弱退三斗少退二
Pure Brightness row: Wall 8°3′; polar distance 195 strong; shadow 6 chi 5 cun; drips 53:3 / 46:7; Well 17 shao-weak (−3); Dipper minor (−2).
117
奎十四度十分八十九強五尺二寸五分五十五八分四十四二分鬼四斗十一弱退二
Continued almanac: Kui 14°10′; polar distance 89 strong; shadow 5 chi 2 cun 5 fen; drips 55:8 / 44:2; Ghost 4°; Dipper 11 weak (−2).
118
胃一度十七分退一八十三少弱四尺一寸五分五十八三分四十一七分星四大進一斗二十一半退二
Stomach 1°17′ (−1); polar distance 83 shao-weak; shadow 4 chi 1 cun 5 fen; drips 58:3 / 41:7; Star 4 great (+1); Dipper 21 half (−2).
119
昴二度二十四分退二七十七大強三尺二寸六十五分三十九五分張十七進一牛六半
Pleiades 2°24′ (−2); polar distance 77 great-strong; shadow 3 chi 2 cun; drips 65 / 39:5; Extended Net 17 (+1); Ox 6 half.
120
畢六度三十一分退三七十三少弱二尺五寸二分六十二四分三十七六分翼十七大進二女十少進一
Net (Bi) 6°31′ (−3); polar distance 73 shao-weak; shadow 2 chi 5 cun 2 fen; drips 62:4 / 37:6; Wing 17 great (+2); Girl 10 shao (+1).
121
參四度六分退四六十九大弱尺九寸八分六十三九分三十六一分角大弱危大弱進二
Triaster 4°6′ (−4); polar distance 69 great-weak; shadow 1 chi 9 cun 8 fen; drips 63:9 / 36:1; Horn great-weak; Rooftop great-weak (+2).
122
井十度十三分退三六十七少弱尺六寸八分六十四九分三十五一分亢五大退一危十四強進二
Well 10°13′ (−3); polar distance 67 shao-weak; shadow 1 chi 6 cun 8 fen; drips 64:9 / 35:1; Horn-Guard 5 great (−1); Rooftop 14 strong (+2).
123
夏至[18]
Subheading: Summer Solstice, footnote [18].
124
井二十五度二十分退三六十七強尺五寸六十五三十五氐十二少弱退二室十二少弱進三
Summer Solstice row: Well 25°20′ (−3); polar distance 67 strong; shadow 1 chi 5 cun; drips 65 / 35; Base 12 shao-weak (−2); Hall 12 shao-weak (+3).
125
柳三度二十七分六十七大強尺七寸六十四七分三十五三分尾一大強退三奎二大強
Willow 3°27′; polar distance 67 great-strong; shadow 1 chi 7 cun; drips 64:7 / 35:3; Tail 1 great-strong (−3); Kui 2 great-strong.
126
星四度二分進一七十二尺六十三八分三十六二分尾十五半弱退三婁三大退一
Star 4°2′ (+1); polar distance 72; shadow 2 chi; drips 63:8 / 36:2; Tail 15 half-weak (−3); Lou 3 great (−1).
127
張十二度九分進一七十三半強二尺五寸五分六十二三分三十七七分箕九大強退三胃九弱退一
Extended Net 12°9′ (+1); polar distance 73 half-strong; shadow 2 chi 5 cun 5 fen; drips 62:3 / 37:7; Winnowing Basket 9 great-strong (−3); Stomach 9 weak (−1).
128
翼九度十六分進二七十八半強三尺三寸三分六十二分三十九八分斗十少退二畢三大退三
Wing 9°16′ (+2); polar distance 78 half-strong; shadow 3 chi 3 cun 3 fen; drips 62 / 39:8; Dipper 10 shao (−2); Net 3 great (−3).
129
軫六度二十三分進一八十四少強四尺三寸五分五十七八分四十二二分斗二十一強退二參五半弱退四
Chariot 6°23′ (+1); polar distance 84 shao-strong; shadow 4 chi 3 cun 5 fen; drips 57:8 / 42:2; Dipper 21 strong (−2); Triaster 5 half-weak (−4).
130
角四度三十分九十半強五尺五寸五十五二分四十四八分牛五少井十六少強退三
Horn 4°30′; polar distance 90 half-strong; shadow 5 chi 5 cun; drips 55:2 / 44:8; Ox 5 shao; Well 16 shao-strong (−3).
131
亢八度五分退一九十六大強六尺八寸五分五十二六分四十七四分女七大進一鬼三少強
Horn-Guard 8°5′ (−1); polar distance 96 great-strong; shadow 6 chi 8 cun 5 fen; drips 52:6 / 47:4; Girl 7 great (+1); Ghost 3 shao-strong.
132
氐十四度十二分退二百二少強八尺四寸五十三分四十九七分虛六大進二星三大強進一
Base 14°12′ (−2); polar distance 202 shao-strong; shadow 8 chi 4 cun; drips 53 / 49:7; Emptiness 6 great (+2); Star 3 great-strong (+1).
133
尾四度十九分退三百七少強丈四十八二分五十一八分危八強進二張十五大強進一
Tail 4°19′ (−3); polar distance 307 shao-strong; shadow 1 zhang; drips 48:2 / 51:8; Rooftop 8 strong (+2); Extended Net 15 great-strong (+1).
134
箕一度二十六分退三百一十一弱丈一尺四寸四十六七分五十三三分室三半強進三翼十五大強進二
Winnowing Basket 1°26′ (−3); polar distance 311 weak; shadow 1 zhang 1 chi 4 cun; drips 46:7 / 53:3; Hall 3 half-strong (+3); Wing 15 great-strong (+2).
135
斗六度一分退二百一十三大強丈二尺五寸六[四]分四十五五分五十四五分壁半強進一軫十五弱進一
Dipper 6°1′ (−2); polar distance 213 great-strong; shadow 1 zhang 2 chi 5 cun 6 fen; drips 45:5 / 54:5; Wall half-strong (+1); Chariot 15 weak (+1).
136
注[一]月令章句曰：「中星當中而不中，日行□也。 未當中而中，日行疾也。」
Cai Yong's gloss: "if the meridian star should cross at mid-heaven yet does not, the Sun is moving too slowly. If it culminates before its due moment, the Sun is moving too fast."
137
注[四]易緯所稱晷景長短，不與相應，今列之於後，並至與不至各有所候，以參廣異同。 冬至，晷長一丈三尺。 當至不至，則旱，多溫病。 未當至而至，則多病暴逆心痛，應在夏至。 小寒，晷長一丈二尺四分。 當至不至，先小旱，後小水，丈夫多病喉痺。 未當至而至，多病身熱，來年麻不為耳。 大寒，晷長一丈一尺八分。 當至不至，則先大旱，後大水，麥不成，病厥逆。 未當至而至，多病上氣、嗌腫。 立春，晷長一丈一寸六分。 當至不至，兵起，麥不成，民疲瘵。 未當至而至，多病熛、疾疫。 雨水，晷長九尺一寸六分。 當至不至，早麥不成，多病心痛。 未當至而至，多病□。 驚蟄，晷長八尺二寸。 當至不至，則霧，稚禾不成，老人多病嚏。 未當至而至，多病漢疽、脛腫。 春分，晷長七尺二寸四分。 當至不至，先旱後水，歲惡，米不成，多病耳癢。 清明，晷長六尺二寸八分。 當至不至，菽豆不熟，多病嚏、振寒 (溫) 、[洞]洩。 未當至而至，多溫病、暴死。 穀雨，晷長五尺三寸六分。 當至不至，水物雜稻等不為，多病疾瘧、振寒、霍亂。 未當至而至，老人多病氣腫。 立夏，晷長四尺三寸六分。 當至不至，旱，五穀傷，牛畜疾。 未當至而至，多病頭痛、腫嗌、喉痺。 小滿，晷長三尺四寸。 當至不至，凶言，[國]有大喪，先水後旱，多病筋急、痺痛。 未當至而至，多熛、嗌腫。 芒種，晷長二尺四寸四分。
Note [4]: the Yijing apocryphon's gnomon figures disagree with the official calendar; the following lists each with omens for early or late arrival. Winter Solstice: noon shadow 1 zhang 3 chi. If the solstice is late: drought and epidemic fevers. If the solstice comes early: violent chest pains; the paired sign appears at summer solstice. Minor Cold: noon shadow 1 zhang 2 chi 4 fen. Late Minor Cold: brief drought then minor flooding; men suffer sore throats. Early Minor Cold: fevers; the next year's hemp crop fails. Great Cold: noon shadow 1 zhang 1 chi 8 fen. Late Great Cold: drought then flood, wheat fails, cold limbs and fainting. Early Great Cold: panting and swollen throat. Spring Begins: noon shadow 1 zhang 1 cun 6 fen. Late Spring Begins: war, failed wheat, wasting illness among the people. Early Spring Begins: heatstroke and epidemics. Rain Water: noon shadow 9 chi 1 cun 6 fen. Late Rain Water: winter wheat fails; cardiac pain spreads. Early Rain Water: many cases of an unrecorded malady (lacuna in text). Waking of Insects: noon shadow 8 chi 2 cun. Late Waking of Insects: fog, failed young crops, elders sneezing. Early Waking of Insects: leg ulcers and shin edema. Spring Equinox: noon shadow 7 chi 2 cun 4 fen. Late equinox: drought then flood, bad harvest, itching ears. Pure Brightness: noon shadow 6 chi 2 cun 8 fen. Late Pure Brightness: beans fail; sneezing and chills. The gloss marks warm fever alongside chill. and watery diarrhea. Early Pure Brightness: fevers and sudden deaths. Grain Rain: noon shadow 5 chi 3 cun 6 fen. Late Grain Rain: wet-field crops and rice fail; malaria, chills, cholera. Early Grain Rain: edema among the elderly. Summer Begins: noon shadow 4 chi 3 cun 6 fen. Late Summer Begins: drought, crop loss, sick cattle. Early Summer Begins: headache and throat ailments. Small Fullness: noon shadow 3 chi 4 cun. Late Small Fullness: ill omens, royal bereavement, flood then drought, cramps and numb pain. Early Small Fullness: heatstroke and throat swelling. Grain in Ear: noon shadow 2 chi 4 cun 4 fen.
138
當至不至，凶言，國有狂令。 未當至而至，多病厥眩、頭痛。 夏至，晷長一尺四寸八分。 當至不至，國有大殃，旱，陰陽並傷，草木夏落，有大寒。 未當至而至，病眉腫。 小暑，晷長二尺四寸四分。 當至不至，前小水，後小旱，有兵，多病洩注、腹痛。 未當至而至，病臚腫。 大暑，晷長三尺四寸。 當至不至，外兵作，來年饑，多病筋痺、匣痛。 未當至而至，多病脛痛、惡氣。 立秋，晷長四尺三寸六分。 當至不至，暴風為災，來年黍不為。 未當至而至，多病咳上氣、咽腫。 處暑，晷長五尺三寸二分。 當至不至，國多浮令，兵起，來年麥不為。
Late Grain in Ear: ill omens and lawless commands. Early Grain in Ear: fainting vertigo and headaches. Summer Solstice: noon shadow 1 chi 4 cun 8 fen. Late summer solstice: national disaster, drought, Yin-Yang imbalance, trees drop leaves in summer, untimely cold. Early summer solstice: swollen brows. Minor Heat: noon shadow 2 chi 4 cun 4 fen. Late Minor Heat: brief wet then dry spell, warfare, dysentery and bellyache. Early Minor Heat: abdominal swelling. Great Heat: noon shadow 3 chi 4 cun. Late Great Heat: border war, next year's famine, sinew paralysis and thoracic pain. Early Great Heat: shin pain and foul miasma. Autumn Begins: noon shadow 4 chi 3 cun 6 fen. Late Autumn Begins: wind damage, failed millet next year. Early Autumn Begins: cough, panting, throat swellings. End of Heat: noon shadow 5 chi 3 cun 2 fen. Late End of Heat: empty decrees, war, failed winter wheat.
139
未當至而至，病脹，耳熱不出行。 白靈，晷長六尺二寸八分。 當至不至，多病痤、疽、洩。 未當至而至，多病水、腹閉疝瘕。 秋分，晷長七尺二寸四分。 當至不至，草木復榮，多病溫，悲心痛。 未當至而至，多病匣鬲痛。 寒露，晷長八尺二寸。 當至不至，來年谷不成，六畜鳥獸被殃，多病疝瘕、□痛。 未當至而至，多病疢熱中。 霜降，晷長九尺一寸六分。 當至不至，萬物大耗，年多大風，人病□痛。 未當至而至，多病匣駿支滿。 立冬，晷長丈一寸二分。 當至不至，地氣不藏，來年立夏反寒，早旱，晚水，萬物不成。 未當至而至，多病臂掌痛。 小雪，晷長一丈一尺八分。 當至不至，來年蠶麥不成，多病恭腕痛。 未當至而至，亦為多肘腋痛。 大雪，晷長一丈二尺四分。 當至不至，溫氣洩，夏蝗蟲生，大水，多病少氣、五疸、水腫。 未當至而至，多病漢疽痛，應在芒種。
Early End of Heat: bloating, burning ears, people keep indoors. White Dew (written White Spirit in the manuscript): noon shadow 6 chi 2 cun 8 fen. Late White Dew: boils, abscesses, diarrhea. Early White Dew: edema, abdominal blockage, hernial masses. Autumn Equinox: noon shadow 7 chi 2 cun 4 fen. Late autumn equinox: untimely green growth, fevers, sorrowful chest pain. Early equinox: pain below the heart and across the diaphragm. Cold Dew: noon shadow 8 chi 2 cun. Late Cold Dew: crop failure, livestock plague, hernial lumps and rheumatic pain (character missing). Early Cold Dew: internal burning fevers. Frost's Descent: noon shadow 9 chi 1 cun 6 fen. Late Frost's Descent: heavy loss of growth, gales, bodily pains (lacuna). Early Frost's Descent: congested chest and swollen limbs (corrupt graphs in source). Winter Begins: noon shadow 1 zhang 1 cun 2 fen. Late Winter Begins: earth fails to close; next summer stays cold, drought then flood, crops fail. Early Winter Begins: sore arms and hands. Light Snow: noon shadow 1 zhang 1 chi 8 fen. Late Light Snow: silk and wheat fail next year; sore wrists. Early Light Snow: elbow and axilla pain. Great Snow: noon shadow 1 zhang 2 chi 4 fen. Late Great Snow: warmth escapes, summer locusts, floods, fatigue, jaundice, dropsy. Early Great Snow: ulcer pain answered at Grain in Ear.
140
月令章句曰：「周天三百六十五度四分度之一，分為十二次，日月之所躔也。 地有十二分，王侯之所國也。 每次三十 (二) 度三十 (三) [二]分之十四，日至其初為節，至其中為中氣。 自危十度至壁 (八) [九]度謂之豕韋之次，立春、驚蟄居之，衛之分野。 自壁 (八) [九]度至胃一度，謂之降婁之次，雨水、春分居之，魯之分野。 自胃一度至畢六度，謂之大梁之次，清明、穀雨居之，趙之分野。 自畢六度至井十度，謂之實沉之次，立夏、小滿居之，晉之分野。 自井十度至柳三度，謂之鶉首之次，芒種、夏至居之，秦之分野。 自柳三度至張十二度，謂之鶉火之次，小暑、大暑居之，周之分野。 自張十二度至軫六度，謂之鶉尾之次，立秋、處暑居之，楚之分野。 自軫六度至亢八度，謂之壽星之次，白露、秋分居之，鄭之分野。 自亢八度至尾四度，謂之大火之次，寒露、霜降居之，宋之分野。 自尾四度至斗六度，謂之析木之次，立冬、小雪居之，燕之分野。 自斗六度至須女二度，謂之星紀之次，大雪、冬至居之，越之分野。 自須女二度至危十度，謂之玄枵之次，小寒、大寒居之，齊之分野。」
Cai Yong: "the sky spans 365 degrees and one-quarter, split into twelve lunar stations along the ecliptic. The realm below is mapped into twelve regions matching those stations. Each celestial station spans thirty degrees of the circuit. Manuscript gloss: digit two in the fraction. …thirty degrees… Gloss: three. Each station spans 30°32′14″ (text split across glosses): the Sun’s entry marks a jie; its midpoint marks a zhong qi. From longitude Rooftop 10° through Wall Gloss: eight. …8°–9° of Wall—this is the Shiwei lodge; Spring Begins and Waking of Insects fall here; it maps to the region of Wei. From Wall Gloss: eight. …Wall 8°–9° to Stomach 1° is Jianglou; Rain Water and the spring equinox belong to it; the allotment is Lu. Stomach 1° to Net (Bi) 6° is Daliang; Pure Brightness and Grain Rain; allotment Zhao. Net 6° to Well 10° is Shichen; Summer Begins and Small Fullness; allotment Jin. Well 10° to Willow 3° is Chunshou; Grain in Ear and summer solstice; allotment Qin. Willow 3° to Extended Net 12° is Chunhuo; the two heats; allotment Zhou. Extended Net 12° to Chariot 6° is Chunwei; Autumn Begins and End of Heat; allotment Chu. Chariot 6° to Horn-Guard 8° is Shouxing; White Dew and autumn equinox; allotment Zheng. Horn-Guard 8° to Tail 4° is Dahuo; Cold Dew and Frost’s Descent; allotment Song. Tail 4° to Dipper 6° is Ximu; Winter Begins and Light Snow; allotment Yan. Dipper 6° to Girl 2° is Xingji; Great Snow and winter solstice; allotment Yue. Girl 2° to Rooftop 10° is Xuanxiao; Minor and Great Cold; allotment Qi—closing Cai Yong’s quotation.
141
蔡邕分星次度數與皇甫謐不同，兼明氣節所在，故載焉。 謐所列在郡國志。
Cai Yong’s celestial-field boundaries disagree with Huangfu Mi’s; since he ties in the solar terms, his version is included. Huangfu Mi’s table appears in the geography monograph.
142
中星以日所在為正，日行四歲乃終，置所求年二十四氣小余四之，如法為少、大，余不盡，三之，如法為強、弱，以減節氣□明中星，而各定矣。 強，正；
Meridian stars are keyed to solar longitude; the Sun’s cycle closes in four years—quadruple the year’s qi fractions, reduce by the calendrical rule to strong/weak quarter-degrees, and subtract from the tabulated culminations to update the almanac. A ‘strong’ label counts as positive correction.
143
弱， (直) [負]也。 其強弱相減，同名相去，異名從之。 從強進少為弱，從弱退少而強。 從上元太歲在庚辰以來，盡熹平三年，歲在甲寅，積九千四百五十五歲也[19]。
A ‘weak’ label The gloss reads ‘straight’ versus oblique. …counts as negative offset. Combine corrections: like signs subtract, unlike signs add. Carry overflows from strong into weak ranks and vice versa per the rod rules. From the epoch (Tai-sui at gengchen) to Xiping 3 (jiayin) is 9,455 years [19].
144
論曰：易有太極，是生兩儀。 兩儀之分尚矣，乃有皇犧。 皇犧之有天下也，未有書計。 歷載彌久，暨於黃帝，班示文章，重黎記注，像應著名，始終相驗，准度追元，乃立歷數。 天難諶斯，是以五、三迄於來今，各有改作，不通用。
The essay opens: the Great Ultimate begets Yin and Yang. Once Yin and Yang parted, culture begins with Fu Xi. Under Fu Xi there was neither script nor arithmetic. By the Yellow Emperor texts appeared, Chong and Li kept celestial logs, instruments matched heaven, and only then could the calendar be founded. Heaven’s measure is subtle, so every dynasty from the Five Thearchs onward has revised the calendar—no single system serves all ages.
145
贊曰：象因物生，數本杪曶。 律均前起，准調後發。 該核衡琁，檢會日月。
Envoi: phenomena yield emblems; numbers spring from the infinitesimal. The pitchpipes set the measure; the gnomon and ring respond. They brace the armillary sphere and reconcile Sun and Moon.
146
校勘記
Section title: textual collation.
147
三０五五頁四行斗綱 (之) [所]建集解引盧文弨說，謂「之」御覽作「所」。 按：與下「青龍所纏」相對成文，作「所」是，今據改。 又按：「綱」原斗「剛」，徑改正。
Collation: p. 3055 l.4, lemma ‘Dipper handle.’ Variant gloss ‘of.’ Lu Wenchao: "Yulan has the cited text for the cited text." Note: It pairs with ‘where the Azure Dragon treads’ below; writing ‘place’ is correct; now emended accordingly. Also note: ‘handle’ was miswritten ‘hard’ in the original; corrected directly.
148
三０五五頁七行當其同[所]集解引盧文弨說，謂「同」下脫「所」字，御覽有。 今據補。
Page 3055, line 7: when they share the same [place]: Collected Commentary cites Lu Wenchao, saying ‘same’ lacks ‘place’ below; the Yulan has it. Supplemented.
149
三０五五頁九行斗建移辰謂之[月]據集解引李銳說補。
Page 3055, line 9: Dipper handle shifts the hour, called [month]: supplied per Li Rui in the Collected Commentary.
150
三０五五頁九行日月之 (術) [行]據集解引李銳說改。 按：殿本作「行」。
Lemma: Sun and Moon’s… (methods) [motion]: emended per Li Rui in the Collected Commentary. Note: The palace edition reads ‘motion.’
151
三０五六頁五行以察[發]斂據集解引錢大昕說補。
Page 3056, line 5: to observe [expansion] contraction: supplied per Qian Daxin in the Collected Commentary.
152
三０五七頁一三行乃立儀表按：集解引李銳說，謂儀謂渾儀，表謂圭表。 今於儀表之閒加頓號。
Page 3057, line 13: then set up instrument and gnomon: Note: Collected Commentary cites Li Rui—‘instrument’ means the armillary sphere, ‘gnomon’ means the shadow-table. Editor adds caesura between the two words.
153
三０五八頁二行為一月之數按：依文義當云「為一月之日數」，疑脫「日」字。
Page 3058, line 2: makes the number for one month: Note: by the sense it should read ‘makes the day-count for one month’; suspect omission of the word ‘day.’
154
三０五八頁三行月成則其歲[大]月 (大) 四時推移集解引張文虎說，謂「月大」二字斗倒，「大」字絕句，「月」字當屬下。 此謂有閏之年為大歲也。 歲之餘分滿月法而置閏謂之大歲，與月之餘分滿日法而成日謂之大月正同。 然閏月四時推移或有進退，故置中氣以定之。 今據改。
Lemma: month complete / year great. (great) four seasons shift: Collected Commentary cites Zhang Wenhu, saying the two graphs ‘month great’ are inverted; ‘great’ ends the phrase, ‘month’ should go with what follows. ‘Great year’ means leap-year. Analogous to great month when day fractions fill. Leap months shift seasons—hence mid-qi placement. Emended per above.
155
三０五八頁四行中之始 (日) [曰]節據集解本改。
Lemma: start of mid-qi. (day) [is] called jie: emended per the Collected Commentary edition.
156
三０五八頁一三行故一 (共) [元]以四千五百六十為甲寅之終也據汲本改。
Lemma: therefore one… (altogether) [origin] uses four thousand five hundred sixty as the end of jiayin: emended per the Ji edition.
157
三０五九頁一一行沒數二十一 (為章閏) 據集解引李銳說刪。
Lemma: mo 21. Marginal gloss deleted. Li Rui: delete spurious gloss.
158
三０五九頁一五行中法 (四) [三]十二據集解引錢大昕說改。
Lemma: zhong qi divisor. Gloss four. 32 per Qian Daxin.
159
三０六０頁三行其月 (食) 百三十五據集解引錢大昕說刪。
Lemma: its month. (eclipse) Delete erroneous 135.
160
三０六０頁四行得五 (百) [月]二十三之二十而一食據集解引錢大昕說改。
Lemma: obtain five. Gloss hundred. Eclipse ratio lemma fixed.
161
三０六０頁四行得歲有再食五百一十三分之五十[五]也據集解引錢大昕說補。
Supplement 55/513.
162
三０六０頁五行得四與二十七互之會二千五十二按：「互」殿本作「五」。 集解引錢大昕說，謂「五之」兩字難解，閩本、汲古閣本作「互」，亦非是。 當云「名之曰蔀會」，傳寫脫斗耳。 又引李銳說，謂「互之」者互乘之也。 四為七十六約數，以乘五百一十三，得二千五十二； 二十七為五百一十三約數，以乘七十六，亦得二千五十二，為蔀會。
Page 3060, line 5: obtain four and twenty-seven, cross them, rendezvous two thousand fifty-two: Note: ‘cross’ in the palace edition reads ‘five.’ Collected Commentary cites Qian Daxin, saying the two words ‘five it’ are hard to interpret; the Min and Jigu editions read ‘cross,’ also not right. It should read ‘call it bu hui’; scribal omission of graphs. Also citing Li Rui: ‘cross them’ means cross-multiply. 4×513=2052 (factor of 76). 27×76=2052; defines bu hui.
163
三０六０頁七行蔀會 (三) [二]千五十 (三) [二]據集解引錢大昕說改。
Lemma: bu hui. Gloss three. Two thousand fifty. Gloss three. 2052/2053 fixed per Qian.
164
三０六０頁一０行月數百 (二) [三]十五據集解引錢大昕說改。
Lemma: month number. Gloss two. 135 month constant.
165
三０六０頁一一行食法二十 (二) [三]據集解引錢大昕說改。
Lemma: eclipse divisor. Gloss two. 223 per Qian.
166
三０六０頁一三行筭外所入紀歲名命之筭上即所求年太歲所在集解引李銳說，謂「筭外」下有脫文，當云「筭外，所入蔀也。 不滿蔀法者，入蔀年數也，各以所入紀歲名命之，筭上，即所求年太歲所在」。 按：如李說，則「筭外」下當補「所入蔀也不滿蔀法者入蔀年數也各以」十六字。
Li Rui notes that after the phrase 'beyond the rod-count' the text is defective; the missing clause should state which bu you have entered before naming the stem-branch year, matching the parallel instructions earlier in the eclipse algorithm. Full restored sentence for eclipse bu-hui algorithm." Note: according to Li's view, below 'beyond the count' one should add sixteen graphs: 'the bu entered; what does not fill the bu rule is the count of years within the bu; each with.'
167
三０六０頁一五行所得以 (七) [二]十 (二) [七]乘之據集解引李銳說改。
Page 3060, line 15: what you obtain, take Gloss digit seven. Completes seventy-two. Gloss two. 72× multiplier restored per Li Rui.
168
三０六一頁一行筭 (之起) 外所 (以) 入紀集解引錢大昕說，謂「之」「起」「以」三字皆衍文。 今據刪。
Lemma: rod-count. (its start) Fragment before editorial deletion. (by which) Qian Daxin: "delete three spurious characters." Removed per Qian.
169
三０六一頁二行各以 (不) [所]入紀歲名命之據集解引錢大昕說改。
Page 3061, line 2: each with Negative gloss. [the] entered ji year-name is named: emended per Qian Daxin in the Collected Commentary.
170
三０六一頁三行即所求年 (蔀) [太歲所在]據集解引李銳說刪補。
Lemma: year sought. (bu) [where Grand Year stands]: supplied and deleted per Li Rui in the Collected Commentary.
171
三０六一頁四行紀蔀表張文虎舒蓺室隨筆云：「案此表首行序題，各本誤以『天紀歲名』對蔀名『甲子』『癸卯』為第一列，『地紀歲名』對『庚辰』『丙申』為第二列，『人紀歲名』對『庚子』『丙辰』為第三列，『蔀首』二字對『庚申一』『丙子二』為第四列。 李尚之四分術注依錢少詹說更正，以天、地、人三紀序題各降一列，而以『蔀首』二字獨對一、二、三、四數目，今局中新刊本從之。 其實蔀名『甲子』、『癸卯』一列當移末列，與數目字相屬，王氏太歲考改如此。 或移蔀首數目為第一列，與蔀名相屬，庶為明白。」 今依張說移正。
Page 3061, line 4: ji bu table: Zhang Wenhu in Casual Notes from the Shuyi Studio says: 'Examining this table's header row, each edition wrongly pairs the column title "Celestial-ji year-name" with the bu names jiazi and guimao as the first column, "Terrestrial-ji year-name" with gengchen and bingshen as the second, "Human-ji year-name" with gengzi and bingchen as the third, and the two graphs "bu head" with gengshen one and bingzi two as the fourth. Li Rui and Qian reorder columns; the official reprint adopts their layout. The bu stem-branch pair belongs beside the round number, as Wang's Tai-sui study argues. Alternative layout: put serial numbers first next to bu names." Table transposed following Zhang Wenhu.
172
三０六二頁二行壬 (午) [子]據集解引盧文弨說改。
Lemma: stem ren. Gloss wu (Horse). Renzi for corrupt graph per Lu.
173
三０六二頁五行 (乙) [己]酉據集解引盧文弨說改。
Page reference line 5. Gloss yi stem. Jiyou corrected per Lu.
174
三０六二頁一四行小余四百九十[九]據集解引錢大昕、李銳說補。
499 restored in new-moon rule.
175
三０六三頁一行以大周乘年周天乘[閏余]減之餘滿蔀 (日) [月]則天正朔日也據集解引錢大昕說補改。
Page 3063, line 1: multiply the year by the great cycle; subtract the product of the celestial circumference and [intercalary remainder]; when the remainder fills bu (days) Closes the divisor phrase.
176
三０六三頁二行以 (月) [日]余乘之據集解引錢大昕說改。
Page 3063, line 2: take (month) [day] remainder multiply it: emended per Qian Daxin in the Collected Commentary.
177
三０六三頁九行不滿其 (數) [所]近節氣夜漏之半者集解引李銳說，謂「數」當作「所」，聲之鬥。 今據改。
Page 3063, line 9: what does not fill its (number) [what] approaches half the night drip of the nearest solar term: Collected Commentary cites Li Rui, saying 'number' should be 'place'—a phonetic slip. Adopted.
178
三０六三頁一四行以 (為) [十]五乘冬至小余據集解引錢大昕說改。
Lemma. (as) [ten] five multiply the winter-solstice minor remainder: emended per Qian Daxin in the Collected Commentary.
179
三０六三頁一五行置入蔀積 (月) [日]以 (日) [蔀月]乘之據集解引錢大昕說改。
Page 3063, line 15: set accumulated months within the bu (months) [days] take Gloss. [bu months] multiply it: emended per Qian Daxin in the Collected Commentary.
180
三０六四頁八行經斗除十[九]分據集解引錢大昕說補。
19 fen subtraction at Dipper.
181
三０六四頁九行以朔小余減合[朔]度分據集解引盧文弨說補。
Page 3064, line 9: subtract conjunction degree-parts from the new-moon minor remainder: supplied per Lu Wenchao in the Collected Commentary.
182
三０六四頁九行其分 (三) [二]百 (二) [三]十五約之據汲本、殿本改。
Page 3064, line 9: its fraction Gloss three. Two hundred. Gloss two. 235 reduction step.
183
三０六四頁一二行積度加斗二十一十[九]分據集解引錢大昕說補。
21°19′ Dipper offset.
184
三０六四頁一四行謂 (盡) [晝]漏分後盡漏盡也集解引李銳說，謂「謂盡漏」當作「謂晝漏」。 晝漏分後者，晝漏與夜漏分之後，謂自夜上水後至夜漏盡，月在張、心，則注於術。 今據改。
Page 3064, line 14: call it (exhausted) [daylight] drip parts afterward exhausted drip exhausted: Collected Commentary cites Li Rui, saying 'call exhausted drip' should read 'call daylight drip.' Li Rui defines the dusk interval when the Moon sits in Zhang and Heart. Accepted Li's fix.
185
三０六五頁三行以夜半到明日所行分 (分) 減蔀法據集解引李銳說刪。
Lemma midnight-to-dawn fraction. (parts) subtract bu rule: deleted per Li Rui in the Collected Commentary.
186
三０六五頁五行置其節氣夜 (半) [漏]之數據集解引錢大昕說改。
Lemma night drip. (half) [drip] count: emended per Qian Daxin in the Collected Commentary.
187
三０六五頁六行即 (明) 月[明]所在度也據集解引盧文弨說改。
Page 3065, line 6: then (bright) Moon [bright] degree where it stands: emended per Lu Wenchao in the Collected Commentary.
188
三０六五頁八行加七度三百五十九分四分 (之) 三據集解引盧文弨說刪。
Lemma +7°359′4. (of) three: deleted per Lu Wenchao in the Collected Commentary.
189
三０六五頁八行[以]宿次除之據集解引盧文弨說補。
Page 3065, line 8: [and] remove by lunar-lodge steps: supplied per Lu Wenchao in the Collected Commentary.
190
三０六五頁一０行小分[滿]四從大分[大分]滿蔀月從度據集解引李銳說補。
Carry rules for quarter moons.
191
三０六五頁一五行以月數乘積[食]據集解引錢大昕說補。
Page 3065, line 15: multiply accumulated [eclipse] by the month number: supplied per Qian Daxin in the Collected Commentary.
192
三０六六頁三行加五 (百) [月]二十分據集解引錢大昕說改。
Lemma +5. Gloss hundred. 520 fractional parts.
193
三０六六頁五行又以四百九十[九]乘積月據集解引錢大昕說補。
499× term added.
194
三０六六頁一一行余以為積月按：集解引李銳說，謂此省文也。 以術為之，當以章月乘餘年，滿章法得一為積月，不滿為閏余。
Li Rui: text abbreviated. Full algorithm for accumulated months spelled out.
195
三０六六頁一五行不滿法 (法) 什之據集解引錢大昕說刪。
Lemma. Gloss. take tenths: deleted per Qian Daxin in the Collected Commentary.
196
三０六七頁二行余為昨夜未 (晝) [盡]據集解引李銳說改。
Lemma night remainder. (daytime) [exhausted]: emended per Li Rui in the Collected Commentary.
197
三０六七頁三行以章法乘周率為 (用) [月]法據集解引錢大昕說改。
Page 3067, line 3: multiply the zhang rule by the circumferential rate to make (use) [month] method: emended per Qian Daxin in the Collected Commentary.
198
三０六七頁四行以月之 (月) [日]乘積[月]為朔大小余據集解引李銳說改。
Lemma. Gloss. Conjunction fraction rule fixed.
199
三０六七頁四行乘為入月日餘按：集解引錢大昕說，謂此處有脫鬥。 今以算術求之，當以蔀日乘積月，如蔀月而一，為積日，不盡為小余； 積日滿六十去之，余為大余也。 又以蔀日乘月餘，以月法乘朔小余，並之，以四千四百六十五約之，所得如日度法而一，為入月日，不盡為日餘也。 又引李銳說，謂以算求之，當以蔀日乘月餘，以月法乘朔小余，從之，章法乘章月，得數約之，如日度法，為入月日、日餘。
Page 3067, line 4: multiply to make days entered into the month and remainder: Note: Collected Commentary cites Qian Daxin, saying here there is a corrupted passage. Reconstruction step one: bu days × months. Step two: sexagenary major. Full formula for days into synodic month. Li Rui's alternate reduction path.
200
三０六七頁五行以[周]率去日率據集解引錢大昕說補。
Page 3067, line 5: subtract [circumferential] rate from day rate: supplied per Qian Daxin in the Collected Commentary.
201
三０六七頁五行如日度法為[積]度 (之) [度]余也集解引錢大昕說，謂「為度之餘」當云「為積度度余」。 又引李銳說，謂「如日度法，為度之餘也」，當云「如日度法為積度，不盡為度之餘也」。 今按：錢、李二氏之說皆合理，局本依錢說改，今從之。
Page 3067, line 5: as the day-degree method makes [accumulated] degrees (of) [degree] remainder: Collected Commentary cites Qian Daxin, saying 'as degree remainder' should read 'as accumulated degrees and degree remainder.' Also citing Li Rui: saying 'as day-degree method, as degree remainder' should read 'as day-degree method makes accumulated degrees; what is not exhausted is degree remainder.' Editor adopts Qian's reading for the bureau text.
202
三０六七頁一０行日餘萬四千六百四十 (七) [一]據集解引錢大昕說改。
Lemma: Jupiter day remainder. Gloss seven. Completes 14,647.
203
三０六七頁一四行入月日十 (一) [二]據集解引錢大昕說改。
Lemma: Mars days in month. Gloss one. Twelve days.
204
三０六八頁四行入月日二十 (三) [四]據集解引錢大昕說改。
Lemma: Saturn. Gloss three. Twenty-four days.
205
三０六八頁七行月法十[一]萬七百七十據汲本、殿本補。
110,770 divisor restored.
206
三０六八頁一０行月餘二十一萬七千六百六十[三]據集解引錢大昕說補。
Mercury month remainder 217,663.
207
三０六八頁一二行虛分四百四十 (九) [一]據集解引錢大昕說改。
Lemma: void fraction. Gloss nine. 441 void fraction.
208
三０六八頁一二行入月日二十 (七) [八]據集解引錢大昕說改。
Mercury days. Gloss seven. Twenty-eight days.
209
三０六八頁一三行日度法四萬七千六百三十 (一) [二]據集解引錢大昕說改。
Lemma: Mercury degree divisor. Gloss one. 47,632.
210
三０六八頁一四行不盡名[為]合余集解引惠棟說，謂「名」下干象歷有「為」字，應增入。 今據補。
Page 3068, line 14: what is not exhausted is named [as] conjunction remainder: Collected Commentary cites Hui Dong, saying below 'named' the Qianxiang calendar has 'as'; it should be added. Now supplied accordingly.
211
三０六八頁一五行[合]余以周率除之據集解引李銳說補。
Page 3068, line 15: [conjunction] remainder divided by circumferential rate: supplied per Li Rui in the Collected Commentary.
212
三０六九頁二行從小積[為積月不盡]為月餘據集解引李銳說補。
Li Rui restores bracketed phrase.
213
三０六九頁五行其閏[余]滿二百二十四以上據集解引李銳說補。
Page 3069, line 5: if intercalary [remainder] reaches two hundred twenty-four or more: supplied per Li Rui in the Collected Commentary.
214
三０六九頁六行以蔀日乘 (之) 入紀月據集解引錢大昕說刪。
Lemma. Gloss. months entered into the ji: deleted per Qian Daxin in the Collected Commentary.
215
三０六九頁八行所得 (得) 滿日度法得一據集解引錢大昕說刪。
Page 3069, line 8: what you obtain Duplicate gloss. Delete erroneous phrase.
216
三０六九頁一三行以周率[乘]小余據集解引盧文弨說補。
Page 3069, line 13: multiply circumferential rate by [the] minor remainder: supplied per Lu Wenchao in the Collected Commentary.
217
三０六九頁一四行即 (正) [至]後星合日數也據集解引李銳說改。
Page 3069, line 14: then (correct) [to] after the star conjunction day-count: emended per Li Rui in the Collected Commentary.
218
三０六九頁一五行 (余一) [金水]加晨得夕據集解引錢大昕說改。
Line break lemma. (remainder one) [metal, water] add morning to get evening: emended per Qian Daxin in the Collected Commentary.
219
三０七０頁二行又[加大]余二十九[小余四百九十九]集解引錢大昕說，謂「又」下疑有脫文，當云「加大余二十九，小余四百九十九」。 今據補。 按：此即上求後合月中所謂「加月餘於月餘，滿其月法得一」也，故應再加大余二十九，小余四百九十九。
Page 3070, line 2: again [add major] remainder twenty-nine [minor remainder four hundred ninety-nine]: Collected Commentary cites Qian Daxin, saying below 'again' there is suspected lacuna; it should read 'add major remainder twenty-nine, minor remainder four hundred ninety-nine.' Supplemented. Note: this is what the preceding 'seek next conjunction month' means by 'add month remainder to month remainder; when it fills month method, that is one'—therefore one should again add major twenty-nine, minor four hundred ninety-nine.
220
三０七０頁三行 (如) [加]大余據集解引錢大昕說改。
Page ref. Gloss. The collator restores the verb 'add' before 'major remainder' so the new-moon carry instruction reads as a complete step in the planetary algorithm.
221
三０七０頁四行以入月日[日]余加今所得據集解引盧文弨說補。
Page 3070, line 4: add day entered-into-month [and] day remainder to what you now obtain: supplied per Lu Wenchao in the Collected Commentary.
222
三０七０頁四行其前合月朔小余 (不) 滿其虛分者據集解引李銳說刪。
Lemma. Negation gloss. fills its void fraction: deleted per Li Rui in the Collected Commentary.
223
三０七０頁八行木晨伏十六日七千 (二) [三]百二十分半據集解引錢大昕說改。
Jupiter dusk lemma. Gloss. Qian Daxin restores the fractional tail to seven hundred twenty-three and one-half parts so Jupiter’s dusk interval matches the Quarter-day canon.
224
三０七０頁一０行八十四日 (進) [退]十二度據集解引錢大昕說改。
Lemma. Gloss. [retreat] twelve degrees: emended per Qian Daxin in the Collected Commentary.
225
三０七０頁一二行伏復十六日七千 (二) [三]百二十分半據集解引錢大昕說改。
Lemma. Gloss. 723½ repeat.
226
三０七０頁一四行行星三十 (二) [三]度與萬三百一十四分據集解引錢大昕說改。
Lemma. Gloss. 33°10,314p.
227
三０七一頁一行[百]八十四日行[百]一十二度據集解引錢大昕說補。
Page 3071, line 1: [one hundred] eighty-four days travels [one hundred] twelve degrees: supplied per Qian Daxin in the Collected Commentary.
228
三０七一頁四行行[三]百三度據集解引錢大昕說補。
The editor supplies the missing hundreds digit so Mars’ direct-motion arc is three hundred three degrees as required by the table.
229
三０七一頁六行通率日行千八百七十六分之九百九十七「九十七」原斗「九十六」，據張元濟校勘記謂「六」字原作「大」，影印上板時描改也。
Page 3071, line 6: uniform rate nine hundred ninety-seven parts per one thousand eight hundred seventy-sixths of a day—the 'nine-seven' was corrupted to 'nine-six' in the original; Zhang Yuanji's collation notes say the 'six' graph was originally 'great' and was traced into 'six' when the block was photographed.
230
三０七一頁一一行[一]見三百四十日據集解引盧文弨說補。
Page 3071, line 11: [one] appearance three hundred forty days: supplied per Lu Wenchao in the Collected Commentary.
231
三０七一頁一五行[旋]順按：依文義當脫一「旋」字，今補。
Page 3071, line 15: [then] direct: Note: by the sense a 'then' is missing; now supplied.
232
三０七一頁一五行日行 (行) 四十六分度之三十三據集解引錢大昕說刪。
Lemma. Gloss. Erroneous duplicate line removed.
233
三０七一頁一五行而[疾]日行一度九十[一]分度之十五據集解引錢大昕說補。
Page 3071, line 15: then [fast] one degree fifteen parts in ninety-one parts of a degree: supplied per Qian Daxin in the Collected Commentary.
234
三０七二頁三行一合二百九十二日[二]百八十一分據集解引錢大昕說補。
Qian Daxin restores the missing 'two hundred' fraction so Venus’ inferior-conjunction cycle reads two hundred ninety-two days and two hundred eighty-one parts.
235
三０七二頁七行而 (進) [□]據集解引錢大昕說改。
Lemma. Gloss. Corrupt graph fixed.
236
三０七二頁九行退四度而 (後) [復]合據集解引錢大昕說改。
Lemma. Gloss. The lemma is emended to 'again conjoin' so the line describes returning to inferior conjunction after the evening dusk arc.
237
三０七二頁九行凡 (三) [再]合一終據集解引錢大昕說改。
Lemma. Gloss. The phrase is corrected so the synodic cycle counts two inferior conjunctions closing one full Venus period.
238
三０七三頁二行[旋]逆據集解引錢大昕說補。
Page 3073, line 2: [then] retrograde: supplied per Qian Daxin in the Collected Commentary.
239
三０七三頁三行行三十[二]度據集解引錢大昕說補。
The tens digit is restored so Mercury’s evening apparition covers thirty-two degrees of ecliptic travel.
240
三０七三頁五行 (如) [加]星合日度余據集解引錢大昕說改。
Page ref. Gloss. The verb 'add' is restored so the reader folds the star’s conjunction longitude remainder into the stepped position.
241
三０七三頁五行 (術) [行]分母乘之據集解引李銳說改。
Page ref. (method) Li Rui’s reading 'motion denominator' replaces a corrupt compound so daily motion fractions multiply against the correct divisor.
242
三０七三頁六行分 (日) 如[日]度法而一據集解引李銳說改。
Lemma. Gloss. Li Rui reads the phrase as 'compare to the day-degree method as unity,' so leftover parts fold against the standard degree divisor.
243
三０七三頁六行不盡如 (法) 半[法]以上據集解引盧文弨說改。
Lemma. Gloss. Lu Wenchao emends 'half the divisor' so fractional remainders that reach halfway or more carry to the next whole unit.
244
三０七三頁一０行 (日) [月]名據集解引李銳說改。 按：下表排列依李銳漢四分術改定。
Page ref. Column gloss. The column header is emended to read ‘month name’ for the almanac table. Editor states the following grid follows Li Rui’s Quarter-day reconstruction.
245
三０七四頁三行斗二十六四分[一]退二據集解引李銳說補。
Li Rui adds the quarter-degree fraction and retreat notation for Dipper.
246
三０七四頁三行女十二進 (二) [一]據集解引李銳說改。
Lemma: Girl 12° advance. Gloss digit two. Advance correction is +1 per Li.
247
三０七四頁三行虛十進 (三) [二]據集解引李銳說改。
Emptiness 10° advance. Gloss three. Advance +2.
248
三０七四頁四行危十 (六) [七]據集解引李銳說改。
Rooftop width lemma. Gloss six. Rooftop 16°→17° correction.
249
三０七四頁四行室十六進 (二) [三]據集解引李銳說改。
Hall 16° advance. Gloss two. Advance +3.
250
三０七四頁四行壁 (十) [九]進 (三) [一]汲本、殿本「進三」作「進二」。 集解引李銳說，謂「壁十」當作「壁九」，「進二」作「進一」。 今據改。 按：集解引李銳說，謂案此赤道度即太初星距見於三統術者是也。 自漢以後相沿承用，至唐大衍術始改畢、觜、參、鬼四宿，後漢施行四分，未嘗改測，則二宿度數不得與三統術異。 今本作「危十六」「壁十」者，與下文黃道度相涉而誤也。
Wall lodge. Gloss ten. Wall 9° advance. Gloss three. [one] Ji edition and palace edition read advance three as advance two. Li Rui’s preferred Wall 9° +1. Adopted Li’s numbers. Li identifies the table with Santong lodge widths. Historical continuity argument for fixed widths. The present text’s ‘Rooftop sixteen’ and ‘Wall ten’ cross-contaminate the Yellow-Path column and are wrong.
251
三０七四頁六行婁十二 (進) [退]一汲本、殿本「進一」作「進二」。 集解引李銳說，謂當作「退一」。 今據改。
Lou 12°. Gloss. "[retreat] one Ji edition and palace edition read advance one as advance two." "Collected Commentary cites Li Rui, saying it should read retreat one." Fixed.
252
三０七四頁六行胃十四 (進二) [退一]據集解引李銳說改。
Stomach 14°. Gloss. Advance/retreat pair corrected.
253
三０七四頁六行昴十一 (進) [退]二據集解引李銳說改。
Pleiades 11°. Gloss. Retreat −2.
254
三０七四頁七行畢十六 (進) [退]三汲本、殿本「進三」作「進二」。 集解引李銳說，謂當作「退三」。 今據改。
Net 16°. Gloss. "[retreat] three Ji edition and palace edition read advance three as advance two." "Collected Commentary cites Li Rui, saying it should read retreat three." Fixed.
255
三０七四頁一０行翼十八進 (一) [二]據集解引李銳說改。
Wing 18° advance. Gloss one. Advance +2.
256
三０七四頁一三行尾十八 (進) [退]三據集解引李銳說改。
Tail 18°. Gloss. Retreat −3.
257
三０七五頁二行斗二十四 (進一) [四分一]據集解引李銳說改。
Dipper 24°. Gloss. Quarter-degree fraction.
258
三０七六頁一行夜漏減 (三) [之二]百而一據集解引李銳說改。
Lemma night drip. Gloss. [of it] two hundred and one: emended per Li Rui in the Collected Commentary.
259
三０七六頁二行如法為少[二為半三為太]據集解引李銳說補。
Bracketed strength labels supplied.
260
三０七六頁五行赤道橫帶渾天之腹去極九十一度十[六]分之五御覽無「渾」字。 又「分」上原無「六」字； 占經、御覽作「十九分」，亦非是。 今依算理補。
Page 3076, line 5: the equator as a belt across the belly of the armillary sphere, ninety-one degrees and five sixteenths from the pole: the Taiping Yulan omits the graph ‘armillary.’ Also above ‘fen’ the original lacked ‘six’; the Astrological Classic and Yulan read ‘nineteen fen,’ which is also wrong. Now supplied according to calculation.
261
三０七六頁七行就夏歷景去極之法以為率也按：「夏歷景」開元占經作「夏至歷景」，影印宋本御覽引作「夏歷晷景」，鮑刻本作「夏至晷景」。
Page 3076, line 7: taking the Xia-calendar shadow method for polar distance as the rate: Note: ‘Xia-calendar shadow’ appears in the Kaiyuan Astrological Classic as ‘summer-solstice shadow’; the Song Yulan manuscript reads ‘Xia-calendar gnomon shadow’; Bao’s edition reads ‘summer-solstice gnomon shadow.’
262
三０七六頁九行取北極及衡各 (誠) [針]□之為軸據嚴可均輯全後漢文改。
Instrument lemma. (needle) [needle] pin as the axis: emended per Yan Kejun's reconstruction of the complete Later Han text.
263
三０七六頁一０行以為[百]八十二度八分之五據開元占經補。
Page 3076, line 10: making [one hundred] eighty-two degrees and five eighths: supplied per the Kaiyuan Astrological Classic.
264
三０七六頁一一行視篾之半際 (夕) 多[少]黃赤道幾也集解引盧文弨說，謂「夕」字衍。 今按：「夕」乃「少」字之形鬥，又顛倒其文耳。 下云「其所多少」，可證也。 開元占經引作「視篾半之際多少黃赤道幾何也」。
Lemma sighting. (evening) how much [less] ecliptic and equator differ: Collected Commentary cites Lu Wenchao, saying the graph ‘evening’ is superfluous. Now examined: ‘evening’ is a miswritten form of ‘less,’ with graphs reversed. The line below says ‘how much more or less,’ which confirms it. The Astrological Classic reads: ‘sight the middle of the strip’s half—how much do Yellow Path and equator differ?’
265
三０七六頁一二行從 (此) [北]極數之據汲本、殿本改。
Page 3076, line 12: from (this) The collators restore the graph 'north' so the instruction counts arc from the celestial pole rather than a corrupt homophone.
266
三０七六頁一二行則 (無) [去]極之度也據開元占經引改。
Lemma. Gloss. The Kaiyuan quotation supplies 'distance from the pole,' clarifying that the entry lists ecliptic polar distance in whole degrees.
267
三０七六頁一五行故使中道三日之中 (若) [差]少半也據開元占經改。
Lemma. (as if) [differ] by a little less than half: emended per the Astrological Classic.
268
三０七七頁三行而度猶[雲]退者集解引盧文弨說，謂「猶」下當有「雲」字。 今據補。
Page 3077, line 3: yet the degrees still [as-if] retreat: Collected Commentary cites Lu Wenchao, saying below ‘still’ there should be ‘as-if.’ Supplemented.
269
三０七七頁四行而以赤道 (重廣) [量度]黃道據開元占經引改。
Lemma. (double breadth) [to] measure the Yellow Path: emended per the Astrological Classic quotation.
270
三０七七頁五行以赤道為 (強) [距]耳據開元占經引改。
Lemma. (strong) [distance] ear: emended per the Astrological Classic quotation.
271
三０七七頁五行故於黃道亦[有]進退也據開元占經補。
The Astrological Classic adds the phrase 'there is advance and retreat' to stress that solar longitude along the Yellow Path speeds and slows relative to the equator.
272
三０七七頁九行斗二十一度八分退二原作斗二十度百一十分八分退二斗，徑據集解引錢大昕說改正。 按：錢雲因下有「百一十五」之文而重出耳。 此以三十二為度法，分滿法即進為度，無有過三十一分者。
Major corruption in solstice line fixed. Qian’s explanation of duplication. Calendrical note on 32-part system.
273
三０七七頁一０行女二度七分進一「進」下原脫「一」字，王先謙謂李本作「進一」，今徑補。
Wang Xianqian and Li's edition agree that the advance notation needs an explicit 'one,' so the fractional step for Girl lodge is corrected to advance one.
274
三０七七頁一一行百一十原作「百一十一」，鬥。 王先謙謂李本作「百一十」，逕據改。
Page 3077, line 11: one hundred ten—originally ‘one hundred eleven,’ miswritten. Wang Xianqian says Li’s edition reads one hundred ten—emended directly.
275
三０七七頁一二行危十度原作「危七度」，鬥，徑據集解引錢大昕說改正。
Page 3077, line 12: Rooftop ten degrees—originally 'Rooftop seven degrees,' miswritten, corrected per Qian Daxin in the Collected Commentary.
276
三０七七頁一二行百六少強「少強」原作「少弱」，鬥。 王先謙謂李本作「少強」，逕據改。
Page 3077, line 12: two hundred six lesser-strong—‘lesser-strong’ originally ‘lesser-weak,’ miswritten. Wang Xianqian says Li’s edition reads lesser-strong—emended directly.
277
三０七七頁一二行畢五少弱退三「少弱」原作「少強」，鬥，徑據汲本改正。
Page 3077, line 12: Net five lesser-weak retreat three—‘lesser-weak’ originally ‘lesser-strong,’ miswritten, corrected per Ji edition directly.
278
三０七七頁一三行室八度二十八分進三「進三」原作「退三」，鬥。 王先謙謂李本「退」作「進」，逕據改。
Page 3077, line 13: Hall eight degrees twenty-eight fen advance three—‘advance three’ originally ‘retreat three,’ miswritten. Wang Xianqian says Li’s edition reads ‘advance’ for ‘retreat’—emended directly.
279
三０七七頁一三行箕大弱退三「箕」下原有大字「六」，鬥。 王先謙謂李本無「六」字，逕據刪。
Page 3077, line 13: Winnowing Basket lesser-weak retreat three—below ‘Winnowing Basket’ the original had a large graph ‘six,’ miswritten. "Wang Xianqian states that the Li edition lacks the character "the cited text" and deletes it outright on that authority."
280
三０七八頁一行八十九強「強」原作「少強」，鬥。 王先謙謂李本無「少」字，徑據刪。
Collation: "the manuscript wrongly read the cited text where the table should show the cited text alone for the eighty-ninth degree mark." Wang Xianqian: "the Li text has no the cited text; the redundant character is removed."
281
三０七八頁一行斗十一弱退二「弱」原作「強」，鬥。 王先謙謂李本作「弱」，徑據改。
Collation fixes Dipper 11: "the weak grade was misprinted as the cited text." "Wang Xianqian states that the Li edition reads "the cited text" and emends to it outright on that authority."
282
三０七八頁二行胃一度十七分退一「退一」原作「退二」，鬥。 王先謙謂李本作「退一」，逕據改。
Stomach’s motion is corrected from retreat two to retreat one. "Wang Xianqian states that the Li edition reads "the cited text" and emends to it outright on that authority."
283
三０七八頁二行星四大進一「進」下原脫「一」字，王先謙謂李本「進」下有「一」字，逕據補。
The lacuna after the cited text is filled with the cited text to match the Li edition’s planet entry.
284
三０七八頁三行張十七進一「進一」原斗「進二」，逕據汲本改正。 按：王先謙謂李本作「大進一」。
Extended Net’s advance is set to the cited text, not the cited text, following the Ji imprint. Comment: "Wang Xianqian states that the Li edition reads "the cited text"."
285
三０七八頁四行畢六度「六」原作「八」，鬥，徑據汲本改正。
Net’s width is emended from eight to six degrees per the Ji edition.
286
三０七八頁四行女十少進一「進」原作「弱」，鬥。 王先謙謂李本「弱」作「進」，逕據改。
Page 3078, line 4: Girl ten the cited text advance one: ""the cited text" was originally written "the cited text", edition conflict." The Li edition’s the cited text replaces the erroneous the cited text.
287
三０七八頁五行角大弱「大」原作「六」，大字，鬥。 王先謙謂李本「六」作「大」，小字，逕據改。
Collation: "the large-script the cited text grade was misprinted as the numeral the cited text." "Wang Xianqian states that in the Li edition "the cited text" is written "the cited text", small character, and emends to it outright on that authority."
288
三０七八頁七行室十二少弱進三「進三」原作「退三」，鬥。 王先謙謂李本作「進三」，逕據改。
Rooftop’s motion is advance three, not retreat three. "Wang Xianqian states that the Li edition reads "the cited text" and emends to it outright on that authority."
289
三０七八頁九行星四度二分進一「二分進一」原作「三分進二」，鬥。 王先謙謂李本作「二分進一」，逕據改。
Page 3078, line 9: planet four degrees two fen advance one: ""the cited text" was originally written "the cited text", edition conflict." Li Rui’s the cited text reading is accepted.
290
三０七八頁一０行胃九大弱退一「退一」原作「退二」，鬥。 王先謙謂李本作「退一」，逕據改。
Stomach nine’s retreat is one, not two. "Wang Xianqian states that the Li edition reads "the cited text" and emends to it outright on that authority."
291
三０七八頁一一行翼九度十六分進二「進二」原作「退二」，鬥。 李本作「進一」，亦誤。 依算理應為「進二」，今徑改。
Wings’ motion is advance two, not retreat two. Even the Li text’s the cited text is rejected as incorrect here. "According to computational principle it ought to be "the cited text", and it is emended directly to that."
292
三０七八頁一一行斗十少退二「退」下原脫「二」字，王先謙謂李本作「退二」，徑據補。
Page 3078, line 11: Dipper ten the cited text retreat two: "below "the cited text" the text originally omitted the character "the cited text"; Wang Xianqian states that the Li edition reads "the cited text", and it is supplied outright on that authority."
293
三０七八頁一二行軫六度二十三分進一「進一」原作「退一」，鬥。 王先謙謂李本作「進一」，逕據改。
Page 3078, line 12: Chariot End six degrees twenty-three fen advance one: ""the cited text" was originally written "the cited text", edition conflict." "Wang Xianqian states that the Li edition reads "the cited text" and emends to it outright on that authority."
294
三０七八頁一二行斗二十一強退二「退」下原脫一字。 汲本、殿本作「退一」，鬥。 王先謙謂李本作「退二」，逕據補。
Page 3078, line 12: Dipper twenty-one the cited text retreat two: "below "the cited text" one character was originally missing." "The Ji edition and the palace edition read "the cited text", edition conflict." "Wang Xianqian states that the Li edition reads "the cited text" and supplies it outright on that authority."
295
三０七八頁一四行亢八度五分退一「退一」原作「退三」，鬥。 王先謙謂李本作「退一」，逕據改。
Neck’s retreat is one fen, not three. The Li edition’s the cited text is adopted.
296
三０七八頁一四行九十六大強「大強」原作「少強」，鬥。 王先謙謂李本作「大強」，逕據改。
Page 3078, line 14: ninety-six great the cited text: ""the cited text" was originally written "the cited text", edition conflict." the cited text is the Li edition’s grade reading.
297
三０七九頁一行氐十四度十二分退二「十二分」原作「十三分」，鬥。 錢大昕謂「三」當作「二」，王先謙謂李本作「十二分」，逕據改。
Root’s fraction is twelve fen, not thirteen. "Qian Daxin states that "the cited text" ought to be "the cited text"; Wang Xianqian states that the Li edition reads "the cited text" and emends to it outright on that authority."
298
三０七九頁一行虛六大進二「進二」原作「進一」，鬥。 王先謙謂李本作「進二」，逕據改。
Page 3079, line 1: Emptiness six great advance two: ""the cited text" was originally written "the cited text", edition conflict." "Wang Xianqian states that the Li edition reads "the cited text" and emends to it outright on that authority."
299
三０七九頁二行尾四度「尾」原作「房」，鬥。 王先謙謂李本作「尾」，逕據改。
Page 3079, line 2: Tail four degrees: ""the cited text" was originally written "the cited text", edition conflict." the cited text is the Li edition’s lodge label.
300
三０七九頁二行丈「丈」下原有「四寸二分」四字。 集解引李銳說，謂案祖沖之術二至晷景與此同。 其至前後各氣晷景，以此至前後晷景兩兩相加，折半得之。 如此術大雪景丈二尺五寸六分，小寒景二尺三寸，相加半之，得沖之術大雪、小寒景一丈二尺四寸三分是也。 覆檢此文，惟立冬一氣不合。 案祖沖之稱四分志立冬中景長一丈，立春中景九尺六寸，相加半之，得九尺八寸，與沖之術立春、立冬景正合。 然則此文立冬晷景丈四寸二分，誤衍「四寸二分」四字耳。 今徑據刪。
Page 3079, line 2: "after the character "the cited text" the text originally carried the four characters "the cited text"." Li Rui shows Zu Chongzhi’s solstice shadows agree with this passage. Li explains how to derive mid-season shadows by averaging paired solstice-adjacent readings. The worked example matches Zu’s figures for Major Snow and Minor Cold when averaged correctly. Only Lìdōng’s shadow line is inconsistent with the rest. Zu’s own mid-term values for winter and spring solstitial pairs average to his Lìchūn/Lìdōng shadows. "Therefore in this text the Lìdōng gnomon shadow of one zhang four cun two fen merely has the four characters "the cited text" erroneously appended." It is now deleted outright on that authority.
301
三０七九頁二行張十五大強進一「進一」原作「進二」，汲本無「進一」二字。 王先謙謂李本多「進一」二字，殿本同，逕據改。
Collation balances Extended Net’s the cited text against Ji (which omits it) and other witnesses. "Wang Xianqian states that the Li edition has the extra two characters "the cited text", the palace edition agrees, and it is emended to that reading outright on that authority."
302
三０七九頁三行室三原作「室二」，鬥。 王先謙謂李本「室二」作「室三」，逕據改。
Page 3079, line 3: "Rooftop three was originally written "the cited text", edition conflict." Li’s the cited text is adopted for Rooftop.
303
三０七九頁四行斗六度一分退二「退二」原作「退三」，鬥。 王先謙謂李本作「退二」，逕據改。
Page 3079, line 4: Dipper six degrees one fen retreat two: ""the cited text" was originally written "the cited text", edition conflict." "Wang Xianqian states that the Li edition reads "the cited text" and emends to it outright on that authority."
304
三０七九頁四行軫十五弱進一「弱」原作「少強」，鬥。 李本作「少弱」，亦鬥。 依算理應作「弱」，逕改。
Page 3079, line 4: Chariot End fifteen the cited text advance one: ""the cited text" was originally written "the cited text", edition conflict." Li’s the cited text is likewise rejected. "According to computational principle it ought to read "the cited text", and it is emended directly to that."
305
三０七九頁一五行振寒 (溫) [洞]洩據汲本、殿本改。
Lemma the cited text in the seasonal list (collation block). Gloss marking the cited text as related to the cited text in the apparatus. The compound is read the cited text per Ji and Dian witnesses.
306
三０八０頁二行[國]有大喪據汲本、殿本補。
The phrase the cited text is restored from Ji and Dian.
307
三０八０頁一四行五疸「疸」原斗「疽」，逕據殿本、集解本改正。
Page 3080, line 14: five the cited text: ""the cited text" was originally confused with "the cited text", corrected per the palace edition and the Collected Commentary edition."
308
三０八０頁一六行每次三十 (二) 度三十 (三) [二]分之十四據集解引錢大昕說刪改。
Lemma: thirty each interval. Gloss digit two in the fractional apparatus. Continuation: thirty degrees. Gloss digit three. Qian’s emendation yields two fourteenths for the fraction.
309
三０八０頁一六行自危十度至壁 (八) [九]度據集解引錢大昕說改。 下「自壁八度至胃一度」同。
Span lemma: Emptiness 10° through Wall. Gloss digit eight. Wall’s endpoint is nine degrees per Qian. "Below, 「from Wall eight degrees to Stomach one degree」 is the same."
310
三０八０頁一七行立春驚蟄居之按：殿本「驚蟄」作「雨水」，下「雨水」作「驚蟄」。 集解引錢大昕說，謂此以驚蟄為正月中氣，雨水為二月節，依古法也。 四分術以雨水為正月中氣。
Page 3080, line 17: Lìchūn and Jīngzhé occupy it-comment: "in the palace edition "the cited text" reads "the cited text", and below "the cited text" reads "the cited text"." Qian explains the old ordering: Jīngzhé as mid-month qi, Yǔshuǐ as the term. The Quarter-day calendar instead places Yǔshuǐ in the mid-qi slot.
311
三０八一頁一行清明穀雨居之集解引盧文弨說，謂清明穀雨當互易。 今按：證以月令問答，惟驚蟄、雨水用三統，余皆用四分，易之非是。
Lu argues Qīngmíng and Gǔyǔ should swap positions. The editor rejects Lu’s swap: only the rain terms use Triple Concordance; the rest match Quarter-day.
312
三０八一頁一０行強正弱 (直) [負]也集解引李銳說，謂「直」當作「負」，負猶背也。 今據改。
Lemma on strong versus weak gradings. Gloss character the cited text in the note. Li Rui reads the cited text for the cited text, explaining the cited text as ‘opposite / backs away’. It is now emended on that authority.
313
三０八二頁七行追太初前 (世) [卅]一元據集解引盧文弨說改。 按：前志謂太初元年距上元十四萬三千一百二十七歲，正為太初前卅一元，「卅」與「世」形近而鬥。
Lemma opening the retrograde count from Grand Inception. Gloss: the character the cited text in the manuscript line. Lu emends the cited text to the cited text for ‘thirty’ cycles. Comment: "the preceding Treatise states that Grand Inception year 1 stands 143,127 years from the high origin-exactly thirty-one cycles before Grand Inception, and "the cited text" was confused with "the cited text" because the graphs resemble each other."
314
三０八二頁九行追漢 (三) [四]十五年庚辰之歲據集解引錢大昕說改。
Lemma for the Han-era retrojection. Gloss digit three. Qian corrects the count to forty-five years to the gēngchén year.
315
三０八三頁二行趨走目下遂竟端右出相好藩按：集解引惠棟說，謂邕集「目」作「陛」，「竟」作「由」，「好」作「外」。
Hui Dong quotes variant diction from Cai Yong’s anthology for this memorial phrase.
316
三０八三頁三行一旦 (披) [被]章據汲本、殿本改。
Page 3083, line 3: once. Gloss character the cited text. the cited text ‘receive the rescript’ replaces the cited text per standard editions.
317
三０八三頁四行非臣無狀所敢[復]望據汲本、殿本補。
Page 3083, line 4: it is not what your servant’s unworthiness would dare [again] to hope for—supplied per the Ji edition and the palace edition.
318
三０八三頁五行[減罪]一等「一等」上疑有脫文，今據嚴可均輯全後漢文補「減罪」二字。
The editor inserts the cited text before the cited text following Yan’s reconstruction.
319
三０八三頁六行[邕為]郡縣促遣集解引盧文弨說，謂脫「邕為」二字。 今據補。 按：惠棟補注謂「郡縣」上邕集有「邕為」二字。
Page 3083, line 6: [Yong was] urged on by commandery and county officials-the Collected Commentary cites Lu Wenchao’s argument that the two characters "the cited text" are missing. They are now supplied on that authority. Hui Dong’s gloss confirms Yong’s collection reads the cited text.
320
三０八三頁六行□於吏手按：集解引惠棟說，謂「□」邕集作「迫」。
Hui Dong fills the damaged character with the cited text per Cai Yong’s parallel text.
321
三０八三頁七行但[懷]愚心據集解引盧文弨說補。
Page 3083, line 7: only [cherish] a foolish heart—supplied per Lu Wenchao as cited in the Collected Commentary.
322
三０八三頁八行略以所有舊事[與臣]據集解引盧文弨說補。
Page 3083, line 8: briefly using what old matters I have [with your servant]—supplied per Lu Wenchao as cited in the Collected Commentary.
323
三０八三頁一０行 (所使元順) [其]難者皆以付臣集解引惠棟說，謂邕集無「所使元順」四字，有「其」字。 今據改。
Collation header only: apparatus continues on page 3083, line 10. Apparatus gloss marking the four-character phrase the cited text for emendation. Hui Dong: Yong’s anthology reads the cited text without the cited text. The text is adjusted to match Cai Yong’s anthology.
324
三０八三頁一一行請太 (師) [史]舊注據集解引盧文弨說改。
Lemma opening the line ‘request the Grand…’ in the memorial. (Preceptor). [Astrologer] old commentary: emended per Lu Wenchao as cited in the Collected Commentary.
325
三０八三頁一二行尋繹適有頭角集解引盧文弨說，謂「尋繹」下脫「度數」二字。 按：如盧說增「度數」二字，則當於「尋繹度數」絕句。
Lu notes a lacuna after the cited text: "supply the cited text." Comment: if one adds the two characters "the cited text" per Lu’s view, the sentence should break after "the cited text".
326
三０八三頁一二行 (遂) [逐]放邊野集解引惠棟說，謂邕集「遂」作「逐」。 今據改。
Collation header for line 12 on page 3083. (Then). the cited text ‘drive into exile’ replaces the cited text per Yong’s text. It is now emended on that authority.
327
三０八三頁一三行 (故) [胡]廣所校據汲本、殿本改。
Collation header for line 13 on page 3083. (Therefore). [Hu] Guang’s collation: emended per the Ji edition and the palace edition.
328
三０八三頁一三行不能違望按：集解引盧文弨說，謂「違望」一作「自達」。
Page 3083, line 13: unable to go against what is hoped-comment: the Collected Commentary cites Lu Wenchao’s argument that "the cited text" has one variant reading "the cited text".
329
三０八三頁一五行不 (言四) [意西]夷相與合謀據集解引盧文弨說改。
Lemma fragment before the emended phrase. (speak of the four). [the western] barbarians plotting together—emended per Lu Wenchao as cited in the Collected Commentary.
330
三０八三頁一六行謹先顛踣按：集解引惠棟說，謂「謹」邕集作「恐」。
Hui Dong: Yong’s text has the cited text rather than the cited text.
331
三０八三頁一六行臣欲 (制) 刪定者一據集解引盧文弨說刪。
Lemma opening ‘your servant wishes…’. (compile). ‘one who deletes and fixes’—deleted per Lu Wenchao as cited in the Collected Commentary.
332
三０八三頁一七行臣欲著者 (三) [五]集解引惠棟說，謂「三」邕集作「五」。 盧文弨亦謂「三」當作「五」。 今據改。
Lemma: the works Yong proposes to compose. Gloss digit three. Hui Dong: five works in Yong’s anthology, not three. "Lu Wenchao also states that "the cited text" ought to be "the cited text"." The numeral is changed from three to five.
333
顓頊曰重黎。
Under Zhuanxu the calendrical officials were Chong and Li.
334
唐﹑虞﹑夏﹑商曰羲和。
Later ages styled the calendar office Xihe.
335
月令章句曰：「帝舜叶時月正日，湯、武革命，治歷明時。 言承平者叶之，承亂者革之。」
The gloss ties Shun’s harmonization to the Shang founders’ calendar reform. Peace calls for adjustment; upheaval calls for a new system."
336
即是起捨合朔。
The commentator explains aligning intercalation with lunation (the cited text).
337
月令章句曰：「紀，還復故歷。」
The Monthly Ordinances commentary says: ‘A ji cycle means returning to the former calendar.’
338
月令章句曰：「七十六歲為蔀首。」
The Monthly Ordinances commentary says: ‘Seventy-six years form the head of a bu.’
339
月令章句曰：「十九歲七閏月為一章。」
The Metonic-style rule: "nineteen years with seven leap months per zhang."
340
月令章句：「孟春以立春為節，驚蟄為中。 中必在其月，節不必在其月。 據孟春之驚蟄在十六日以後，立春在正月； 驚蟄在十五日以前，立春在往年十二月。」
First spring month: "Lìchūn as term, Jīngzhé as mid-month qi. Mid qi stays tied to its civil month; terms may spill across months. If Jīngzhé is late in the month, Lìchūn stays in month one. If Jīngzhé is early, Lìchūn shifts back into the prior year’s twelfth month.
341
四分退二
Quarter-day: retreat two.
342
張衡渾儀曰：「赤道橫帶渾天之腹，去極九十一度十[六]分之五。 黃道斜帶其腹，出赤道表裡各二十四度。 故夏至去極六十七度而強，冬至去極百一十五度亦強也。 然則黃道斜截赤道者，則春分、秋分之去極也。 今此春分去極九十少，秋分去極九十一少者，就夏歷景去極之法以為率也。 上頭橫行第一行者，黃道進退之數也。 本當以銅儀日月度之，則可知也。 以儀一歲乃竟，而中閒又有陰雨，難卒成也。 是以作小渾，盡赤道黃道，乃各調賦三百六十五度四分之一，從冬至所在始起，令之相當值也。 取北極及衡各 (誠) [針]□之為軸，取薄竹篾，穿其兩端，令兩穿中閒與渾半等，以貫之，令察之與渾相切摩也。 乃從減半起，以為[百]八十二度八分之五，盡衡減之半焉。 又中分其篾，拗去其半，令其半之際正直，與兩端減半相直，令篾半之際從冬至起，一度一移之，視篾之半際 (夕) 多[少]黃赤道幾也。 其所多少，則進退之數也。 從 (此) [北]極數之，則 (無) [去]極之度也。 各分赤道黃道為二十四氣，一氣相去十五度十六分之七，每一氣者，黃道進退一度焉。 所以然者，黃道直時，去南北極近，其處地小，而橫行與赤道且等，故以篾度之，於赤道多也。 設一氣令十六日者，皆常率四日差少半也。 令一氣十五日不能半耳，故使中道三日之中 (若) [差]少半也。 三氣一節，故四十六日而差今三度也。 至於差三之時，而五日同率者一，其實節之閒不能四十六日也。 今殘日居其策，故五日同率也。 其率雖同，先之皆強，後之皆弱，不可勝計。 取至於三而復有進退者，黃道稍斜，於橫行不得度故也。 春分、秋分所以退者，黃道始起更斜矣，於橫行不得度故也。 亦每一氣一度焉，三氣一節，亦差三度也。 至三氣之後，稍遠而直，故橫行得度而稍進也。 立春、立秋橫行稍退矣，而度猶雲進者，以其所退減其所進，猶有盈餘，未盡故也。 立夏、立冬橫行稍進矣，而度猶[雲]退者，以其所進，增其所退，猶有不足，未畢故也。 以此論之，日行非有進退，而以赤道 (重廣) [量度]黃道使之然也。 本二十八宿相去度數，以赤道為 (強) [距]耳，故於黃道亦[有]進退也。 冬至在斗二十一度少半，最遠時也，而此歷斗二十度，俱百一十五，強矣，冬至宜與之同率焉。 夏至在井二十一度半強，最近時也，而此歷井二十三度，俱六十七度，強矣，夏至宜與之同率焉。」
Zhang Heng’s Armillary Sphere says: ‘The red band spans the belly of the whole heaven, ninety-one degrees and five sixteenths from the pole. The ecliptic tilts twenty-four degrees from the equator on each side. Solstitial polar distances follow from the twenty-four-degree obliquity. Equinox polar distance lies where ecliptic and equator intersect. The tabulated ninety / ninety-one figures use the gnomon method as proxy. That row lists ecliptic motion against the equator. Ideal measurement uses a metal armillary and solar/lunar observations. A year of weather interrupts continuous observation. A desktop sphere marks both circles at 365.25° from the winter solstice point. Assembly step: mount pole and transverse ring. (needle). A bamboo strip on a needle axle traces the sphere’s great circle. Half the year’s arc is set to 182° 5/8 along the crossbar scale. The bisected strip pivots from solstice one degree per step. (weak). how much the yellow and red paths differ.’ That difference is the tabulated advance or retreat. Lemma: ‘from…’ continuing the polar-distance calculation. (this). Count from the north celestial pole along the strip. Gloss: supplied reading for polar distance. The result is angular distance from the pole. Twenty-four solar terms, each fifteen and 7/16 degrees, one degree ecliptic motion per qi. Near solstices the ecliptic’s slant makes equatorial spacing appear larger. A sixteen-day qi would imply a uniform daily shortfall. Actual qi length is under fifteen and a half days. (as if). The three-day mid-interval is short by a fraction. Each three-qi node accumulates about three degrees of discrepancy. Five-day runs equalize when the forty-six-day rule breaks down. Remainder days explain the five-day equal-rate segments. Early segments read ‘strong,’ late ones ‘weak,’ in endless variation. Obliquity prevents whole-degree steps on the equator. At equinox the ecliptic angle steepens, so motion ‘retreats’ in the table. The same three-degree node rule applies after equinox. Past the oblique zone, motion steadies and advances. Net motion still advances though the oblique component retreats. Net motion still retreats though the oblique component advances. Apparent solar motion is a projection effect of the equatorial grid. (double breadth). Measuring the ecliptic against the equator produces the apparent variation. Lodge spans are reckoned along the equator as (strong). [interval]—therefore on the yellow path there is also advance and retreat.’ Solstice longitude and polar distance should match the tabulated Dipper entry. Summer values likewise should align Well’s longitude with polar distance.
343
月令章句曰：「冬至之為極有三意焉：晝漏極短，去極極遠，晷景極長。 極者，至而還之辭也。」
Dong Zhongshu’s gloss: solstice extremes of day length, polar distance, and shadow. Ji means the turning point after maximum."
344
月令章句曰：「夏至之為極有三意焉：晝漏極長，去極極近，晷景極短。」
Summer solstice extremes: "longest day, nearest pole, shortest shadow."
345
蔡邕戍邊上章曰：「朔方髡鉗徒臣邕稽首再拜上書皇帝陛下：臣邕被受陛下尤異大恩，初由宰府備數典城，以叔父故□尉質時為尚書，召拜郎中，受詔詣東觀著作，遂與腢儒並拜議郎。 沐浴恩澤，承荅聖問，前後六年。 質奉機密，趨走目下，遂竟端右，出相好藩，還尹輦轂，旬日之中，登躡上列。 父子一門兼受恩寵，不能輸寫心力，以效絲發之功，一旦 (披) [被]章，陷沒辜戮。 陛下天地之德，不忍刀鋸截臣首領，得就平罪，父子家屬徙充邊方，完全軀命，喘息相隨。 非臣無狀所敢[復]望，非臣罪惡所當復蒙，非臣辭筆所能復陳。 臣初決罪雒陽詔獄，生出牢戶，顧念元初中故尚書郎張俊，坐漏洩事，當伏重刑，已出谷門，復聽讀鞠，詔書馳救，[減罪]一等，輸作左校。 俊上書謝恩，遂以轉徙。 [邕為]郡縣促遣，□於吏手，不得頃息，含辭抱悲，無由上達。 既到徙所，乘塞守烽，職在候望，憂怖焦灼，無心復能操筆成草，致章闕庭。 誠知聖朝不責臣謝，但[懷]愚心，有所不竟。 臣自在布衣，常以為漢書十志，下盡王莽，而世祖以來，唯有紀傳，無續志者。 臣所師事故太傅胡廣，知臣頗識其門戶，略以所有舊事[與臣]，雖未備悉，粗見首尾，積累思惟，二十餘年。 不在其位，非外吏庶人所得擅述。 天誘其衷，得備著作郎，建言十志皆當撰錄，遂與議郎張華等分受之， (所使元順) [其]難者皆以付臣。 先治律歷，以籌筭為本，天文為驗，請太 (師) [史]舊注，考校連年，往往頗有差舛，當有增損，乃可施行，為無窮法。 道至深微，不敢獨議。 郎中劉洪，密於用筭，故臣表上洪，與共參思圖牒。 尋繹適有頭角，會臣被罪， (遂) [逐]放邊野。 臣竊自痛，一為不善，使史籍所闕， (故) [胡]廣所校，二十年之思，中道廢絕，不得究竟。 慺慺之情，猶以結心，不能違望。 臣初欲須刑竟，乃因縣道，具以狀聞。 今年七月九日，匈奴始攻郡鹽池縣，其時鮮卑連犯雲中、五原，一月之中，烽火不絕。 不 (言四) [意西]夷相與合謀，所圖廣遠，恐遂為變，不知所濟。 郡縣鹹懼，不守朝旦。 臣所在孤危，懸命鋒鏑，湮滅土灰，呼吸無期。 誠恐所懷隨軀腐朽，抱恨黃泉，遂不設施，謹先顛踣。 科條諸志，臣欲 (制) 刪定者一，所當接續者四，前志所無，臣欲著者 (三) [五]，及經典腢書所宜捃摭，本奏詔書所當依據，分別首目，並書章左。 臣初被考，妻子迸竄，亡失文書，無所案請。 加以惶怖愁恐，思念荒散，十分不得識一，所識者又恐謬誤。 觸冒死罪，披 (散) [瀝]愚情，願下東觀，推求諸奏，參以璽書，以補綴遺闕，昭明國體。 章聞之後，雖肝腦流離，白骨剖破，無所復恨。 惟陛下省察。 謹因臨戎長霍圉封上。 臣頓首死罪稽首再拜以聞。」 其所論志，志家未以成書，如有異同，今隨事注之於本志也。
Cai Yong, shorn and collared exile on the northern frontier, bowed his head twice and submitted a memorial to His Majesty the Emperor: ‘Your servant Yong received Your Majesty’s extraordinary grace: at first through the minister’s office I was assigned to magistracies; because my uncle Zhi then served as Commandant of the Guards while he was Minister, I was summoned and appointed Gentleman of the Palace; I received an edict to go to the Eastern Pavilion to compile texts, and so together with erudite scholars I was appointed Counselor. He served six years answering imperial queries. His uncle’s rapid rise from court to frontier and back mirrors Yong’s own promotion. They could not repay the grace before disaster struck. (receive). [I received] the rescript and was plunged into guilt and execution.’ The emperor commuted execution to family exile on the frontier. Yong disclaims any right to expect further mercy. He recalls Zhang Jun’s reprieve from death to hard labor in Yuanchu. Zhang Jun thanked the emperor and was then reassigned. Urged onward by local officials and trapped in clerks’ hands, he could not rest or get his plea to the court. At his frontier post he guarded beacons and could no longer draft memorials for the capital. He knows the court does not demand thanks, yet he still grieves the work left undone. Since his commoner days he had noted the Later Han lacked treatises to match Ban Gu’s ten monographs. Hu Guang had shared materials with him for over twenty years, enough to see the shape of a continuation. Without rank, a private scholar may not author such state papers. Once appointed at the Eastern Pavilion, he proposed the ten treatises and split the labor with Zhang Hua and colleagues. Apparatus gloss for the four-character phrase later emended per Cai Yong’s anthology. The hardest sections fell to him alone. He began with law and the calendar, grounding the work in computation and checking it against the sky. (Preceptor). The Grand Astrologer’s old commentary needed years of collation before it could serve as a durable norm. The technical material was too subtle to settle privately. He brought in Liu Hong, expert at rod calculus, to work through the tables with him. The project had barely taken shape when he was condemned. (Then). Banishment to the border cut the work short. He blames his crime for leaving the dynastic record incomplete. (Therefore). Hu Guang’s twenty-year editorial effort was broken off mid-course. He still cannot abandon the hope of finishing the treatises. He had planned to report through frontier channels once his sentence ended. In the seventh month, Xiongnu and Xianbei raids kept the frontier beacons burning without pause. Negation at line start. Spurious characters the cited text in the received text, struck per Lu Wenchao. He fears the western tribes are conspiring on a large scale and may soon rise—outcome unknown. Local defenses might not last the night. His own life on the frontier hangs by a thread amid the fighting. Lest his plans die with him, he risks this memorial though it may mean his fall. He outlines the treatise plan: first, (to compile). One treatise to redact, four to continue, and new sections for gaps in the old Han shu. Gloss: manuscript three (emended to five). Five new monographs plus sources from classics and edicts, with headings listed on the left. His arrest scattered his family and destroyed his papers. Terror has erased nine-tenths of what he remembered; the rest may still be wrong. He risks death to open his breast and begin the plea that follows. (pour out). He asks the court to lodge the plea at the Eastern Pavilion and cross-check memorials and sealed edicts to repair the record. Once the emperor has seen it, he accepts any execution without regret. He begs the emperor’s judgment. The memorial is sealed and sent through the frontier captain Huoyu. Closing formula: obeisance and submission." Editorial closing: Cai Yong’s treatise plan was never finished as a book; divergent details are annotated here in the present monograph.
← Previous Chapter
Back to Chapters
Next Chapter →