Lightning

《河圖》曰：“激陽為電。”《易稽覽圖》曰：“陰陽和合，其電耀耀也，其光長。”《元命包》曰：“陰陽激為電。”《河圖》曰：“大電繞樞星，炤郊野，感符寶而生黃帝。”《天官書》曰：“電者，陰陽之動也。”《莊子》曰：“陰氣伏於黃泉，陽氣上通於天，陰陽交爭，故為電。”《釋名》曰：“電，殄也，乍見則殄威也。”京房曰：“電生火，有所擊，極陰生陽之象。”董仲舒曰：“太平世，電不眗目，霣示光耀而已。”

The Yellow River Chart says: "When yang energy is stimulated, it becomes lightning." The Yi Ji Lian Tu says: "When yin energy and yang energy harmonize and interact, the lightning flashes brilliantly with long light trails." The Yuan Ming Bao says: "When yin energy and yang energy interact, it becomes lightning." The Yellow River Chart also says: "Great lightning surrounds the pivot star of the Big Dipper, illuminating the outskirts, and responded to Fu Bao to give birth to the Yellow Emperor." The Records of the Grand Historian, Tian Gao Book says: "Lightning is the movement of yin and yang energies." The Zhuangzi says: "Yin energy lies hidden beneath the Yellow Spring, while yang energy ascends to the heavens; when yin and yang contend, it becomes lightning." The Ximing says: "Lightning is tian, which flashes briefly and its power vanishes instantly." Jing Fang says: "Lightning can generate fire; any striking is a symbol of yin reaching its extreme and yang emerging." Dong Zhongshu says: "In times of peace, lightning does not pierce the eyes, but only shows a faint gleam."

京房曰：“凡霆者，金餘氣也，金者，內鏡而外冥，人君即位，乃先敬讓，而內尚賊心，陰行眾物，舉事發微，自以為明，虛言妄據，深依以考，凡事亦自以為能矣。故無雲而霆，霆或為火者，此人君以暴罰也；霆或聚火者，光起而從，此人君將聚之象也；霆或中天而見，此人君自以為明也；霆或將如交蛇，光明之而一正者，此人君行明行直；霆或正赤，下至地而復上者，此人君聽讒言也；

Jing Fang said: All thunder is the leftover energy of metal. The nature of metal is bright inside like a mirror but dark outside. When a ruler first ascends the throne, he first shows himself respectful and humble, but his inner heart still harbors harmful intentions. He secretly employs various methods, and his every action starts from the subtlest details. He believes himself to be observant, speaks empty words and takes things for granted, and deeply relies on others to examine them, believing himself capable of handling everything. Therefore, there is thunder without clouds. Thunder sometimes transforms into fire, which signifies the ruler's use of cruel punishments. Thunder sometimes gathers into fire, with light coming first and then following immediately, which is a symbol of the ruler's intention to accumulate wealth. Thunder sometimes appears in the middle of the sky, which indicates the ruler's belief in his own discernment. Thunder sometimes appears like entwined snakes, bright and upright, which signifies the ruler's actions being clear and upright. Thunder sometimes appears purely red, descending to the ground and then returning upward, which signifies the ruler's belief in slanderous words.

霆或東西南北，皆有霆者，此人君行役，不避四時也；霆或正直而長，光明者，此人君行微，人不知曲直；霆或正黃澤者，此人君行明，雖得事實，取其中也；或霧而霆者，此人君默行，不得事實，而亂不息，下將不制；霆瞬瞬暉暉者，此人君質，不知自明也；霆或奕奕明之而復息者，此人君譏問內直言之事；霆或無雲而以昏數見者，此人君誠以微言而以害人也，一云：默言；霆或將如蝶蛇，明久而微正者，此人君行不明，莫有非也；天陰不雨，但為霆者，此人君陰行，欲以來事實也。”

Thunder sometimes appears in all four directions east, west, south and north, indicating that the sovereign conscripts laborers without avoiding the farming seasons; sometimes it is straight and long, shining brightly, indicating that the sovereign acts secretly, and others cannot discern right from wrong; sometimes it appears yellow and moist, indicating that the sovereign acts clearly, even though he knows the truth, he still follows the middle path; sometimes there is mist and thunder, indicating that the sovereign acts secretly without knowing the truth, thus disorder continues without restraint; thunder that flickers and dims is indicative of the sovereign being simple-minded and not seeking clear understanding; thunder that shines brightly and then fades is indicative of the sovereign secretly inquiring about matters of direct speech and remonstrance; thunder that appears repeatedly in darkness without clouds is indicative of the sovereign using subtle words to harm others, or acting silently without making a sound; thunder that resembles a butterfly or snake, shining for a long time and slightly straight, indicates that the sovereign acts unclearly, yet no one dares to point out his faults; when the sky is cloudy but no rain falls, only thunder occurs, indicating that the sovereign acts secretly, intending to obtain the truth.

上元甲子，距今麟德元年甲子，歲積二十六萬九千八百八十，算外。上元甲子，距今大唐開元二年甲寅，歲積二十六萬九千九百三十，算外。章歲，三萬九千五百七十一。總法，一千三百四十。章月，四十八萬九千四百二十八。半總，六百七十。章閏，一萬四千五百七十六。辰率，三百三十五。閏分，一千二百一十四。（奇八）奇率，一十二。歷周，四十四萬三千七十七。旬周，六十。月法，二十九日，餘七百一十一。弦法，七日，餘五百一十二。（奇九）氣法，一十五日，餘二百九十二。（奇十）曆法，二十七日，餘七百四十三。（奇一）

Starting from the beginning of the first year, the Jiazi, and calculating up to the year Jiazi of the Lín dé era, the total number of years accumulated is twenty-six thousand nine hundred and eighty years, and the term "calculated outside" means the last year is not counted. From the beginning of the first year, the Jiazi, to the year Jiayin of the second year of the Kaiyuan era of the Tang dynasty, the total number of years accumulated is twenty-six thousand nine hundred and ninety-three years, with the last year also not counted. The following are the constant numbers used in this calendar calculation: the chapter year is thirty-nine thousand five hundred and seventy-one, the total method is one thousand three hundred and forty (the common denominator for all fractional calculations), the chapter month is forty-eight thousand nine hundred and forty-eight, half of the total method is six hundred and seventy (half of the total method), the chapter leap year is eleven thousand four hundred and seventy-six, the rate of the day is three hundred and thirty-five, the leap year fraction is one thousand two hundred and fourteen (note in the original text: the odd number is eight), the odd rate is twelve, the cycle of the calendar is forty-four thousand three hundred and seven (the number of days required for the moon to complete its slow and fast motion cycle), the cycle of ten days is sixty (the number of days in a cycle of heavenly stems and earthly branches), the month method is twenty-nine days and a remainder of seven hundred and eleven (the length of a synodic month), the crescent method is seven days and a remainder of five hundred and twelve (the length from new moon to first quarter, note in the original text: the odd number is nine), the solar term method is fifteen days and a remainder of two hundred and ninety-two (the length of a solar term, note in the original text: the odd number is ten), and the calendar method is twenty-seven days and a remainder of seven hundred and forty-three (note in the original text: the odd number is one).

推朔端術：置上元甲子距今所求年，算以章月乘為歲分，章歲除為積月，餘為閏餘。以閏餘減歲分為朔分，總除為積日，餘為小餘。（以天正小餘滅朔分，餘為總實。）求次朔，以月法加；求閏餘，以閏餘加；求弦，以弦法各加其朔；得上弦，以次轉加望，下弦，及來月朔。

The method to determine the starting point of the new moon: list the accumulated years from the beginning of the cycle, Jiazi, to the desired year, multiply by the chapter month to get the "years' division"; then divide by the chapter years to get the "accumulated months", with the remainder being the "leap remainder". Subtract the leap remainder from the years' division to get the "new moon division"; divide the new moon division by the total method to get the "accumulated days", with the remainder being the "small remainder". (Original note: subtract the small remainder of the first month from the new moon division to get the "total actual".) To find the next new moon, add one month method; to find the leap remainder, add one leap remainder; to find the crescent day, add one crescent method to the new moon of each month, first obtaining the upper crescent, then successively adding to get the full moon, lower crescent, and the new moon of the next month.

推入曆術，以歷周去總實，餘以變奇率乘之，滿變小餘。加其餘，總法除為日，不盡為餘命日，算外。即所求年天正恆朔入曆日及餘。

The method of calculating the entry into the calendar: subtract the calendar cycle from the total reality, then multiply the remainder by the changing odd rate. When the product is full of one change, it becomes the small remainder, which is added to the remainder. Then divide by the total method to get the 'day', and the remainder is the 'leftover'. The day numbers are named in sequence, with the last one not counted. This gives the required number of days and the remainder for the beginning of the month of the first month of the year in the heavenly calendar.

求次朔及弦望準前術，（滿歷日及餘奇則去之）求夜半入曆，各以其恆小餘減。

To calculate the next new moon and the entries of the first quarter, full moon, and last quarter, follow the same method as before. (Original note: When the number of days and the remainder and fraction complete a full cycle, subtract them.) To calculate the entry of the new moon at midnight, subtract each of their mean new moon fractions.

推中氣術，各以閏餘加其恆朔小餘，即其氣節，以氣法加。

The method of calculating the middle qi: add the leap year remainder to the small remainder of the month's mean new moon, and you will find the position of the qi for that month; to calculate the next qi, add another qi value.

推加時術，各以六乘其小餘，辰率除，殘以五乘四百二，除為辰殘。又以六乘，如前除為刻分。命時起子半算，即加時所在辰及刻分。

To determine the added time: multiply each small remainder by six, then divide by the rate of the hour. The remainder is multiplied by five and divided by four hundred and two to get the remainder of the hour. Then multiply by six again and divide as before to get the minutes. The hours start from the middle of the Zi hour, and the last digit is not counted, thus obtaining the hour and minutes of the added time.

求定氣術，冬夏二至，即以恆氣為定，自外各以氣下消息數，息減、消，加其恆氣小，（滿若不足，進退其日。）各得所求。（加者為盈氣，減者為縮氣。定其盈縮所在，故曰定氣。）

The method of determining the fixed solar terms: On the days of winter solstice and summer solstice, the mean solar term is directly taken as the fixed solar term. For other solar terms, the number listed under each term is used to adjust the mean solar term. If the term is marked as 'xi' (cessation), subtract the number; if it is marked as 'xiao' (increase), add the number. The addition or subtraction is applied to the small remainder of the mean solar term. (Original note: If the small remainder is full or insufficient, advance or retreat one day accordingly.) In this way, each solar term is determined. (Original note: The added part is called the surplus solar term, and the subtracted part is called the deficient solar term. Since the purpose is to determine the location of surplus and deficiency, these are called fixed solar terms.)

推日中影及消息盈縮定數術，各置其所入氣後氣數，互相減，餘以乘所入，如算十五，以各減其數為定。（前少加之，前多減之，其求入曆，定遲速及星者仿此，入氣十五，算者如十六。）

The method for calculating the length of the shadow at noon and determining the fixed numbers of expansion and contraction: list the numerical values of the current qi and the next qi, subtract the previous number from the following one, multiply the resulting difference by the number of days already entered, divide by fifteen, and use this result to subtract from the original qi number, giving the fixed number. (Original note: if the previous number is smaller, add it; if the previous number is larger, subtract it. The methods for calculating the entry into the calendar, determining the rate of change, and calculating stars are all done in the same way. If the entry into the qi exceeds fifteen, it is calculated as sixteen.)

推朔弦望盈縮經辰所入術，各以其日所入盈縮，定積。盈加縮減其恆經辰所入，餘即各所求。

The method to calculate the entry of the new moon, crescent, and full moon along with their variations in the mean longitude: for each day, accumulate the fixed numbers of excess or deficiency. For excess, add it; for deficiency, subtract it. Add or subtract these from the fixed mean longitude entry. The remainder is the required number for each case.

推朔弦望定遲速術，置所入日遲速數，分後日數，互相減，餘以入曆日，餘乘以總法，除得以加減遲速數為定。

The method for determining the fixed speed by considering the new moon, first quarter, and full moon: list the speed number for the day entered, subtract the numbers of the previous and next days respectively, multiply the remainder by the day's remainder of the calendar, divide by the total method, and add or subtract the result from the speed number to obtain the fixed number.

推定朔弦望術，各以其日遲速盈縮定數，歷速減遲加，氣盈如縮，減恆小餘，滿若不足，進退其日，命以甲子，算外，即各所求。

To determine the fixed new moon (定朔), fixed quarter (定弦), and fixed full moon (定望), use the daily rate of slow and fast motion and the excess and deficiency of the orbit. In terms of the calendar, subtract the fast and add the slow. In terms of the solar term, treat the excess as if it were deficiency, and use it to add or subtract from the mean new moon's small remainder. If the small remainder is full or insufficient, adjust the day forward or backward by one day. Then use the Jiazi system to name the day, ignoring the last digit, to obtain the required fixed new moon, fixed quarter, and fixed full moon.

推定月大小術，其定朔日名，十干與來月同者，大；不同者，小；月無中氣者，為閏。（若正月交，有時正見者，消息前後各一兩月，以定月以之大小，今虧在晦二，其弦望亦隨事消息，月朔大小若過類三者，觀定小餘，近夜半者，量定其弦望，以五乘定小餘，秒法除為刻，即各其辰所入刻數，皆減其晨刻，不盡為辰後刻，不滿前刻者，從前日注於嗣後。）

The method to determine the size of a month is as follows: look at the day name of the fixed new moon. If the heavenly stem of the fixed new moon day matches the heavenly stem of the fixed new moon of the next month, then this month is a large month; if they do not match, it is a small month. If a month does not contain a solar term, it is designated as a leap month. (Original book note: If the new moon occurs in January, sometimes the moon can be seen clearly, so adjustments are made within one or two months before and after to determine the size of each month. Now, if the moon is missing on the 2nd day of the last month, the crescent and full moon are adjusted accordingly. If the size of the new moon exceeds three categories, the small remainder is considered. If it is close to midnight, the time of the crescent and full moon is determined by multiplying the small remainder by five and dividing it by seconds to get the number of hours. This gives the number of hours for each time period. Subtract the morning hours from it; if it does not subtract completely, it is considered as the time after the hour. If it is less than the previous hour, it is recorded from the previous day to the next day.)

推土王術，置四季之節，加大餘十二，小餘二百三十五，即各其氣后土王日。

The method to determine the day when the Earthly Sovereign (土王) governs: list the solar terms at the end of each season, then add a large remainder of twelve and a small remainder of two hundred thirty-five, thereby obtaining the day when the Earthly Sovereign governs after each solar term.

推沒日術，氣小餘一千四百七巳上，以九十乘之；又以十五乘小分，從以減沒分十二萬二千三百五十七。

The method of calculating the day of disappearance: if the excess of the solar term is more than 1407, multiply it by 90; then multiply the small fraction by 15, combine the two results, and subtract them from the total disappearance fraction of 122,357. The rest of this method follows, and is continued below.

餘，以沒法一千七百五十七，除為日，不盡為餘，以日數加氣大餘，即各氣內沒日。

Using the remainder obtained from the previous calculation, divide it by 1757 as specified in the "Lindeng Calendar": the quotient represents the number of full days, and the remainder is kept as the fractional part. Then add this number of full days to the "Da Yu" (the total number of full days accumulated from the beginning of the calendar to that solar term). The result is the day on which the "Mei Ri" (non-existent day) falls within that solar term. The term "Mei" refers to a fictitious day name used in the calendar to adjust for fractional days: when the accumulated fractional days reach a certain amount, a "Mei Ri" is marked, but it does not correspond to an actual day when the sun rises and sets. This passage only covers the latter part of the method for calculating the "Mei Ri." The earlier steps, which involve setting a number and multiplying by another to find the remainder, were explained in the previous section. This passage continues from there.

推每定氣初日夜半日所在定度術，各以其定小餘，減其氣日度分，命以宿次，即其夜半日度及分。求次日，日加一度，又各以其日躔差率，進加退減度分數。

The method to determine the actual degree of the Sun on the Yellow Path at the midpoint of the night before a fixed solar term: take the "fixed remainder" of that term, which is the corrected portion less than a full day, and subtract it from the degree and fraction of the Sun's position on that day. The remainder is then successively reduced according to the order of the twenty-eight constellations until it falls within a specific constellation; this gives the degree and fraction of the Sun's position in that constellation at the midpoint of the night. To calculate the degree for the next day, add one degree to the previous day's result. Additionally, each day's "solar anomaly rate" (the correction for the Sun's actual motion compared to its average motion) must be applied. If the correction is to be added, include it; if it is to be subtracted, deduct it.

推朔望加時所在度術，各以定小餘，加其日夜半日度分，即加時所在。推朔望加時月所在度術，朔加時日所在，為合朔。日月同度，加其望日，加時度一百八十二分百三十四四十二。

The method of calculating the degree of the Star of brightness and prestige at the moment of conjunction or opposition is: take the 'fixed small remainder' of the day and add it to the degree and minute of the Star of brightness and prestige at midnight of that day. The result is the 'added time' (i.e., the exact moment of conjunction or opposition). The degree of the Moon at conjunction or opposition is calculated as follows: the degree of the Star of brightness and prestige at the moment of conjunction is the position of conjunction, because at conjunction the Sun and Moon are at the same degree. If the position of the Moon on the day of opposition is required, add 182 degrees, 134 minutes, and 42 seconds to this, which is half a full circle, because at opposition the Sun and Moon are exactly opposite each other.

推朔定日夜半月所在度術，各以其定小餘，乘所入曆日離程總法而一，為夜半後分，以六十七除，為度。餘，為度分。以減其日加時所在夜及分，命以宿度。求次日各以離程加。（若至其朔，即別起其端，端不得團前而加之。）

The method to determine the degree of the moon at the midnight of the fixed new moon day: take the "fixed small remainder" of that day, multiply it by the "departure distance" of the day when the moon enters the slow and fast calendar, then divide by the "total method" denominator, the quotient is the fraction of the moon's movement after midnight. Divide this fraction by 67 to get the degrees, with any remainder remaining as the fraction under the degree. Use this to subtract from the degree and minute of the moon's position at the added time of that day. The remainder is then successively subtracted according to the order of the twenty-eight constellations to find the moon's position in the constellation at midnight. To calculate for the next day, add the "departure distance" of the next day respectively. (Note from the original text: if the calculation reaches the next new moon day, start anew, and the new set of numbers cannot be continued from the previous set.)

推定氣日晝夜漏及日出沒時，倍其氣晨前刻及分，分漏法從刻為日不見漏，以減百刻為日見漏，加日見漏五刻，為晝漏刻，以晝漏刻減百刻，餘為夜時刻。四刻十二分加晨前漏刻，命起於刻算外，即日出晨刻，以日見漏加日出辰刻，以次如前，即日沒所在晨刻。（以二十五日除夜漏，得每更一算之數，以二刻三十六分加日沒晨刻，即甲夜，即卯晨。又以更籌數加之，得甲夜一籌數，以次累滿，晨去之，即五更籌所當晨刻及分也，以配二十一前夜漏漏法。）

The method for calculating the length of day and night and the times of sunrise and sunset for each solar term is as follows: first, add the number of divisions and fractions of the "morning before" time to double it. The fractions are converted into divisions using the denominator "leifafa" (leak rate), resulting in the "sun not visible" time, which is the duration the sun is below the horizon. Subtract this from 100 divisions to get the "sun visible" time, which is the duration the sun is above the horizon. Then add five divisions to the sun visible time to get the daytime divisions. Subtract the daytime divisions from 100 to get the nighttime divisions. Take four divisions and twelve fractions and add them to the morning before time, starting from the divisions but not counting the current division, to get the time of sunrise. Then add the sun visible time to the sunrise time, following the same method step by step, to get the time of sunset. (Note from the original text: divide the nighttime divisions by twenty-five to get the number of divisions for each watch. Add two divisions and thirty-six fractions to the sunset time to get the start of the first watch, which is the time before the卯 (rabbit) hour. Then add the divisions for each watch in sequence to get the time for each watch. Continue this until a full day and night cycle is completed, then subtract accordingly. Thus, the divisions and fractions for each of the five watches can be determined, which can be matched with the previously listed nighttime divisions and leak rate. The phrase "twenty-one before" is likely a missing or incorrect character.)

推昏明去中星度術，每日求其晝夜漏刻數，以乘歲實二百，乘總法而除之，得昏去其中星度。以減周天度，餘為晨去中星度。以昏明去中星度，加其晨所在，即各其日中星度數。（其梗概粗舉者，加其夜半日度，亦各其中中星宿度。）因求次日者，各置其日刻差七十三乘之，二百八十八而一，為度差，滿十為度。冬至後加，夏至後減，隨日加減，各得每日去中度。晨昏所距，日在黃道，中星準度以赤道。

The method for calculating the degree of the celestial pole from sunset to dawn is: first determine the number of hours for daylight and night each day, multiply them by the "annual measure" (total number of days in a year), then divide by 200 times the "total method" as the divisor. The result is the degree of the celestial pole at sunset. Subtract this number from the full circle degree to get the degree of the celestial pole at dawn. Add the degree of the celestial pole at sunset and dawn respectively to the degree of the sun at dawn to obtain the celestial pole degree for that day. (Note from the original text: if only a rough calculation is needed, adding it to the degree of the sun at midnight will also yield the celestial pole degree at that time.) To predict the next day, take the daily difference of 73, multiply it, and divide by 288. The result is called the "degree difference." When it reaches ten, it is converted into one degree. After the winter solstice, add it; after the summer solstice, subtract it. Continue adding or subtracting daily to get the degree of the celestial pole for each day. Note: the degree of the celestial pole at sunrise and sunset is measured according to the sun's position on the Yellow Path, while the degree of the celestial pole itself is measured according to the equator. They are based on different coordinate systems.

歷日、離程、速數；歷日、離程、遲數。一日九百八十五，增一百四十四，速初；一十五日百一十，增一百一十八，遲二十九。二日九百七十四，增一百一十七，速一百三十四；十六日八百一十九，增一百一十五，遲一百五十七。三日九百六十二，增九十九，速二百五十一；十七日八百三十三，增九十五，遲二百七十二。四日九百四十八，增七十八，速三百五十；十八日八百四十六，增七十四，遲三百六十七。

The following is a table of the moon's slow and fast motion over a period, divided into six columns. The first three columns represent the first half of the moon's fast motion, and the last three columns represent the second half of the moon's slow motion. The column names are in order: the day in the slow and fast calendar (indicating the day number in the slow and fast calendar), the distance (the expected distance of the moon on that day in fractions), the fast number (the accumulated difference when the moon is faster than the mean motion), and the day in the slow and fast calendar, the distance, and the slow number (the accumulated difference when the moon is slower than the mean motion). On the first day, the distance is 985 fractions, which is 144 more than the mean, and the fast number is recorded as "initial fast" indicating the starting point. On the fifteenth day, the distance is 810 fractions (the original text says "one hundred and ten", and it is suspected that the character "eight" is missing), increasing by 118, and the slow number is 29. On the second day, the distance is 974, increasing by 117, and the fast number is 134. On the sixteenth day, the distance is 819, increasing by 115, and the slow number is 157. On the third day, the distance is 962, increasing by 99, and the fast number is 251. On the seventeenth day, the distance is 833, increasing by 95, and the slow number is 272. On the fourth day, the distance is 948, increasing by 78, and the fast number is 350. On the eighteenth day, the distance is 846, increasing by 74, and the slow number is 367.

五日九百三十三，增五十六，速四百二十八；十九日八百六十一，增五十二，遲四百四十一。六日九百一十八，增四十三，速四百八十四；二十日八百七十七，增二十八，遲四百九十三。七日九百二，增九，速五百一十七；二十一日八百九十三，增四，遲五百二十一。八日八百八十六，減一十四，速五百二十六；二十二日九百九，減二十，遲五百二十五。九日八百七十，減三十八，速五百十二；二十三日九百二十五，減四十四，遲五百五。

On the fifth day, the distance covered was 933 units, with an increase of 56, and the speed total reached 428; on the nineteenth day, the distance covered was 861 units, with an increase of 52, and the slow total reached 441. On the sixth day, the distance covered was 918 units, with an increase of 43, and the speed total was 484; on the twentieth day, the distance covered was 877 units, with an increase of 28, and the slow total was 493. On the seventh day, the distance covered was 902 units, with an increase of 9, and the speed total was 517; on the twenty-first day, the distance covered was 893 units, with an increase of 4, and the slow total was 521. On the eighth day, the distance covered was 886 units, changed to a decrease of 14, and the speed total was 526; on the twenty-second day, the distance covered was 909 units, with a decrease of 20, and the slow total was 525. On the ninth day, the distance covered was 870 units, with a decrease of 38, and the speed total was 512; on the twenty-third day, the distance covered was 925 units, with a decrease of 44, and the slow total was 505. It can be seen that between the seventh and eighth days, the increase turned into a decrease, and both the speed total and the slow total reached their peaks and then declined, which is exactly the turning point where the moon's movement shifted from the fastest to a steady pace.

十日八百五十四，減六十二，速四百七十四；二十四日九百四十一，減六十八，遲四百六十一。十一日八百三十九，減八十五，速四百一十二；二十五日九百五十，減八十九，遲三百九十三。十二日八百二十六，減百四，速三百二十七；二十六日九百六十八，減百八，遲三百四。十三日八百一十五，減百二十一，速二百二十三；二十七日九百七十九，減百二十五，遲百九十六。十四日八百八，初減一百二，末減二十九，速一百二；二十八日九百八十五，減百三十四，初減七十一，遲七十一。

On the tenth day, the distance traveled was 854 parts, reduced by 62, with a speed number of 474; on the twenty-fourth day, the distance traveled was 941, reduced by 68, with a slow number of 461. On the eleventh day, the distance traveled was 839, reduced by 85, with a speed number of 412; on the twenty-fifth day, the distance traveled was 950, reduced by 89, with a slow number of 393. On the twelfth day, the distance traveled was 826, reduced by 104 (the original text says 'one hundred four'), with a speed number of 327; on the twenty-sixth day, the distance traveled was 968, reduced by 108, with a slow number of 304. On the thirteenth day, the distance traveled was 815, reduced by 121, with a speed number of 223; on the twenty-seventh day, the distance traveled was 979, reduced by 125, with a slow number of 196. On the fourteenth day, the distance traveled was 808, which was calculated in two segments for that day, the first segment reduced by 102 and the second segment reduced by 29, with a speed number of 102; on the twenty-eighth day, the distance traveled was 985, reduced by 134, with the first segment reduced by 71 and a slow number of 71. The fourteenth and twenty-eighth days are the beginning and end of the transition between the slow and rapid motion in the calendar, so the numbers must be taken in two parts, 'initial' and 'final', and cannot be considered as a whole for the entire day.

二十日氣中影消息數，盈縮數，晨前刻，（及分）昏去中度，定日度，及寄冬十一月，一丈二尺七寸五分，息初先盈，末三十刻八十二度二分，斗十二度。

This line includes the column titles of the twenty-four solar terms table along with the first set of data, but the original text is damaged and disordered. The columns are roughly in the following order: name of the solar term, length of the noon shadow, the number of days for the shadow to change (the amount by which the shadow lengthens or shortens with each solar term), the excess or deficiency of the shadow compared to the average, the number of hours and fractions before sunrise, the degree of the sun's departure from the meridian at sunset, and the fixed degree of the sun for that solar term. The surviving line corresponds to the solar term of Winter Solstice in November: the noon shadow is one zhang two chi seven cun and five fen, and the shadow begins to shorten from this point, indicating the start of the "cessation" phase. The initial excess or deficiency is recorded as excess, the time before sunrise is thirty hours, the degree of the sun's departure from the meridian at sunset is eighty-two degrees and two fen, and the fixed degree of the sun is twelve degrees in the Dipper mansion. The first two characters "twenty" are likely missing "four solar terms," and "及寄" should be "Winter Solstice" or similar. The data for the remaining twenty-three solar terms is entirely missing and cannot be corrected.

赤道度：斗二十六，牛八，女十四，虛十，危十七，室十六，壁九，北方九十八度。斗分三百二十八。奎十六，婁十三，胃十四，昴十一，畢十六，觜二，參九，西方八十度。井三十三，鬼四，柳十四，星七，張十八，翼十八，軫十八，南方一百一十二度。角十二，亢九，氐十五，房五，心五，尾十八，箕十一，東方七十五度。

The following lists the equatorial distances of the Twenty-Eight Mansions, i.e., the number of degrees each mansion occupies along the equator. Northern Black Tortoise seven mansions: Dou 26 degrees, Niu 8 degrees, Nu 14 degrees, Xu 10 degrees, Wei 17 degrees, Shi 16 degrees, Bi 9 degrees, totaling 98 degrees in the north; in addition to Dou, there is a remainder of 328 degrees, which is the leftover portion of the full circle assigned to the Dou mansion. Western White Tiger seven mansions: Gui 16 degrees, Lou 13 degrees, Wei 14 degrees, Mao 11 degrees, Bi 16 degrees, Zi 2 degrees, Shen 9 degrees, totaling 80 degrees in the west. Southern Vermilion Bird seven mansions: Jing 33 degrees, Gui 4 degrees, Liu 14 degrees, Xing 7 degrees, Zhang 18 degrees, Yi 18 degrees, Zhen 18 degrees, totaling 112 degrees in the south. Eastern Azure Dragon seven mansions: Jiao 12 degrees, Kang 9 degrees, Di 15 degrees, Fang 5 degrees, Xin 5 degrees, Wei 18 degrees, Ji 11 degrees, totaling 75 degrees in the east. Adding all four directions together gives 365 degrees, and adding the remainder completes the full circle.

黃道度：斗二十四及分，牛七，女十一，虛十，危十六，室十八，壁十，北方九十六度。奎十七，婁十三，胃十五，昴十一，畢十六，觜三，參十，西方八十三度。井三十三，鬼四，柳十四，星七，張十七，翼十九，軫十八，南方一百九度。角十二，亢十，氐十六，房五，心五，尾十八，箕十，東方七十七度。

The following lists the degrees of the Twenty-Eight Mansions along the Yellow Path. Since the Yellow Path intersects the Celestial Equator at an angle, the same mansion occupies different degrees on the two paths, so another table is needed. The seven mansions in the north: Dou 24 degrees with some additional fractions, Niu 7 degrees, Nu 11 degrees, Xu 10 degrees, Wei 16 degrees, Shi 18 degrees, Bi 10 degrees, totaling 96 degrees in the north. The seven mansions in the west: Kui 17 degrees, Lou 13 degrees, Wei 15 degrees, Mao 11 degrees, Bi 16 degrees, Zi 3 degrees, Can 10 degrees, totaling 83 degrees in the west. The seven mansions in the south: Jing 33 degrees, Gui 4 degrees, Liu 14 degrees, Xing 7 degrees, Zhang 17 degrees, Yi 19 degrees, Chen 18 degrees, totaling 109 degrees in the south. The seven mansions in the east: Jiao 12 degrees, Kang 10 degrees, Di 16 degrees, Fang 5 degrees, Xin 5 degrees, Wei 18 degrees, Ji 10 degrees, totaling 77 degrees in the east. The previous text explains that for calculating the sun's position, the degrees on the Yellow Path are used, while for calculating the position of the stars, the degrees on the Celestial Equator are used. These numbers are taken separately from these two tables.

終率：一千九十三萬九千三百一十三。奇率：三百。約終：三萬六千四百六十四。奇：一百一十三。交中：一萬八千二百三十二。奇：五十六半。虧朔：三千一百六。奇：一百八十六。望實：一萬九個七百八十五。奇：一百五十。後準：一千五百五十三。奇：九十三半。前準：一萬六千六百七十八。奇：二百六十三。

The following are several constants used in the calculation of solar and lunar eclipses. "Zonglv" (终率) is 10939313, representing the total fractional value of the synodic month. "Qilv" (奇率) is 300, the denominator of a more detailed fraction beneath the remainder. "Yuezhong" (约终) is 36464, with a remainder of 113, representing the simplified version of the Zonglv. "Jiaozhong" (交中) is 18232, with a remainder of 56 and a half, exactly half of Yuezhong, indicating the fractional value of the time it takes for the Sun and Moon to move from one node to the next. "KuiShuo" (亏朔) is 3106, with a remainder of 186, representing the difference between a synodic month and Jiaozhong, used to calculate the entry into the node for the next new moon. "WangShi" (望实) is 19785 (originally written as "wan ji ge qi bai ba wu" which is likely a miswriting of "wan qi ge qi bai ba wu", meaning ten thousand seven hundred eighty-five), with a remainder of 150, representing the fractional addition used when calculating from new moon to full moon. "Houzhun" (后准) is 1553, with a remainder of 93 and a half, and "Qianzhun" (前准) is 16678, with a remainder of 263, serving as two boundary values to determine whether an eclipse can occur before or after the node.

推泛交分術，置總實，以終率去之，不足去者，奇率乘之，滿，終率又去之，奇率約之，為天正恆朔夜半入交分。不盡為奇。（以天正恆朔小餘，朔泛交分，交加分，即天正恆。）

The method for calculating the "general intersection fraction" (the uncorrected intersection fraction) is as follows: first, determine the "total real" which is the total fraction accumulated from the epoch to the desired year. Subtract the "final rate" of 10939313 repeatedly until the remainder is less than one final rate. If the remainder is not enough to subtract another final rate, multiply it by the "odd rate" of 300. Then subtract the final rate from this product. Finally, simplify using the odd rate to obtain the intersection fraction at the midnight of the first lunar month of the year. Any remainder left after division is called the "remainder" or the smaller fraction. (Original note: Add the small remainder of the first lunar month, the general intersection fraction of the new moon, and the corrected intersection fraction to get the intersection fraction of the first lunar month.) "Heavenly beginning" refers to the eleventh month, which is the month of the winter solstice. All calculations in calendrical computations start from this month.

求望泛交分，以望實加之，求以朔者，以虧朔加之，滿，約終去之。推朔望定交分術，以六十一乘定遲速數七十，七十七除之，所得為限數及入氣盈縮，加減泛交分，為定交分。

To calculate the general intersection fraction for the full moon, add 19785 to the general intersection fraction of the new moon. To calculate the next new moon, add 3106 to the previous new moon's general intersection fraction. If the total exceeds one "approximate cycle" (36464), subtract it and keep only the remainder less than one cycle. The method to calculate the fixed intersection fraction for new moons and full moons is as follows: multiply 61 by the fixed slow-fast number of the day, then divide by 70 and 77 (the original text says "multiply by sixty-one, divide by seventy and seventy-seven," which is likely a miswriting and should be interpreted as multiplying by sixty-one and dividing by seventy-seven). The result is called the "limit number." Add or subtract this limit number along with the ingress excess or deficiency number of the corresponding solar term to the general intersection fraction. Add if needed, subtract if needed, and the result is the "fixed intersection fraction," which represents the actual fraction of the distance between the sun and moon from the lunar node after correction.

推入蝕限術，入交定分，如交中已下為月在外道，交中已上，以交中減之，餘為月在內道。內外分如後準巳上，前準已上者，為入蝕限，望則月蝕朔入限。月在裡者，蝕入限如後準已下，為交後分，前準已上，反減交中，餘為交前分。置前後分，以一百一十二約之，為去交時數。

The method to determine whether an eclipse is within the limit: look at the fixed value of the entry into the intersection. If it is below 18232, the moon is outside the Yellow Path, called "outer path"; if it is above the intersection, subtract one intersection, and the remainder indicates the moon is inside the Yellow Path, called "inner path." Whether it is inner or outer path, if the value is above 1553 and above 16678, it is considered within the eclipse limit. If the full moon is within the limit, there will be a lunar eclipse; if the new moon is within the limit, there will be a solar eclipse. If the moon is on the inner path, and the value within the eclipse limit is below the posterior standard, it is called "post-intersection value," meaning it has just passed the node. If the value is above the anterior standard, reverse the calculation by subtracting the intersection from it, and the remainder is called "pre-intersection value," meaning it is approaching the node. Divide the obtained pre-intersection or post-intersection value by 112 to get the "time to the node," which indicates how many hours are left until the node.

推日月蝕所在辰術，如求律氣，應加時法，得加時所去辰。月在沖辰蝕。昔非正見者，於日出及日沒前十二刻半內其初末，以候之。

The method for determining the time of a solar or lunar eclipse: use the same method as for calculating the time of a solar term and its added hours. Convert the remaining minutes into hours to determine the exact hour when the eclipse occurs. For a lunar eclipse, observe the 'opposite' hour of the sun's position, as during a lunar eclipse the sun and moon are opposite each other. If the eclipse does not occur at the meridian and is not easily visible, one should wait during the period of twelve and a half hours after sunrise and before sunset to observe the beginning of the moon's darkening and its restoration to full.

求月在表朔蝕，準夏至初去交前後二百四十八分為初準已下，加則午正前後七刻內者蝕。又夏至前後，每一日損初準二分，畢於前後九十四日，各為每日變準。若變已下加時如前者蝕，又以末準六十，減初準及變準，餘以十八約之，為刻準，以算。

To determine whether an eclipse occurs when the moon is on the outer path (outside the Yellow Path) during the new moon, take the summer solstice as the reference point. The initial standard is set at 248 parts before and after the node. If the distance to the node is less than this initial standard, and the conjunction of the new moon occurs within seven hours before or after noon, an eclipse will occur. From the summer solstice, each day subtract two parts from the initial standard until reaching 94 days before and after, and each day's result is called the "changed standard." If the distance to the node is less than the changed standard for that day, and the conjunction time meets the previous condition, an eclipse will occur. Take the final standard of 60, subtract it from the initial and changed standards, and divide the remainder by 18 to get the "time standard," which is used for further calculations.

午正前後七刻數，百時準加時去時準內去交，如末準已下，前蝕。又置末準，每一刻加十八，為差準。加時刻去午正前後，如差準刻已下去交分，如差準已下者，甚蝕。秋分至春分，去交如末準已下，加時南方三辰者，二蝕。

Taking the seven hours around noon as a base, it is matched with the standard hour measure. If the time of the new moon conjunction falls within the standard hour measure, and the value of the intersection difference is less than 60 in the final standard, then an eclipse occurs before noon. Then take the final standard, and for each additional hour, add 18; the result is called the 'difference standard'. If the time of the new moon conjunction is within the difference standard before or after noon, and the intersection difference is also less than the difference standard, then the eclipse is very deep. During the six months from the autumn equinox to the spring equinox, if the intersection difference is less than the final standard and the conjunction time falls within the three hours of the southern direction, which are Si, Wu, and Wei, then a solar eclipse can be seen to a degree of about two parts. This passage has many errors and omissions in the original text, with terms like 'hundred hour standard', 'before eclipse', and 'two eclipses' not being very clear. It can only be roughly explained based on the general algorithm.

求月在裡朔不蝕，準夏至日，去交一千三百七十三初為準，初準已上，加時午正前後十八刻，內或不蝕。又夏至後，每一日益初準一分半，畢前後九十四日，各為每日變準。以初準減變準，餘，十而一為刻準，以減午正前後十八刻，餘為時準。去交在變準已上，加時準，內或不蝕。

To determine the conditions when the moon is within the Yellow Path (inner path) and a solar eclipse does not occur on the new moon day: take the summer solstice as the reference point. A fixed value of 1373 parts is defined as "initial standard." If the lunar distance from the node is above this initial standard, and the conjunction of the new moon occurs within eighteen moments before or after noon, then a solar eclipse may not occur. After the summer solstice, each day adds 1.5 parts to the initial standard, continuing until ninety-four days before and after, with each day's result being the "variable standard." Subtract the variable standard from the initial standard, divide the remainder by ten to get the "moment standard." Then subtract this moment standard from the eighteen moments before and after noon, and the remainder is called the "time standard." If the lunar distance from the node is above the variable standard for that day, and the conjunction time falls within the time standard, then a solar eclipse may not occur. This rule is opposite to the previous one: one determines that an eclipse is certain, while this one determines that an eclipse may not occur.

推月蝕分術，置去交前後定分，冬交前皆去二百二十四；春交後去一百，交前去二百；夏不問前後去五十；秋交後去二百，交前去一百，不足去，蝕既。有蝕者，以減後準一百四而一，餘，半已下為半弱，半已上為半強，命以十五滿限，得月蝕之大分。

The method to calculate the extent of a lunar eclipse is as follows: first, determine the fixed fraction before or after the intersection. Then, subtract a fixed number according to the season. In winter, regardless of whether it is before or after the intersection, subtract 224. In spring, subtract 100 if it is after the intersection and 200 if it is before. In summer, subtract 50 regardless of whether it is before or after. In autumn, subtract 200 if it is after the intersection and 100 if it is before. If the fixed fraction is insufficient for subtraction, it indicates that the moon is exactly at the node, which is called 'complete eclipse' or 'total eclipse'. If there is a remainder after subtraction, subtract it from 1553, the 'post-standard'. The remainder is divided by 104. If the remainder is less than half of 104, it is recorded as 'half weak'; if it exceeds half of 104, it is recorded as 'half strong'. The diameter of the moon is fixed at fifteen parts. The resulting number is placed within fifteen parts to determine the 'great fraction', which indicates how much of the moon's diameter is eclipsed.

月在內道蝕，（初起東南，蝕甚西南。）月在外道蝕。（初起東北，食甚西北。十三分巳上正東起，皆據正南而言之。）

The direction of the beginning of a lunar eclipse and its maximum obscuration depends on whether the moon is within or outside the Yellow Path. If the moon occurs within the Yellow Path and an eclipse takes place, the obscuration begins at the southeast corner of the moon and reaches its maximum at the southwest. If the moon occurs outside the Yellow Path and an eclipse takes place, the obscuration begins at the northeast corner and reaches its maximum at the northwest; if the eclipse exceeds thirteen parts, it begins at due east. All these directions are based on the observer facing due south. The reason for distinguishing between inside and outside the Yellow Path is that whether the moon passes through the Earth's shadow from the south or north side determines which side of the moon's edge is first touched by the shadow.

推日食分術，月在內道，冬至畢雨水，秋分畢大雪，以五百五十八為差。（日入春分巳後，日損六分，畢芒種。）日夏至初日無差。（自後日益六分，畢白露。）

The method for calculating the extent of a solar eclipse is as follows: when the moon is within the Yellow Path, during the periods from winter solstice to rain water and from autumn equinox to great snow, the fixed number to be subtracted, called "difference," is taken as 558. (Original note: After the spring equinox, 6 parts are subtracted each day from this difference until the beginning of summer.) On the day of summer solstice, the difference is exactly reduced to zero. (Original note: After summer solstice, 6 parts are added each day until the beginning of autumn.) This shows that the difference continuously increases and decreases with the solar terms. In the winter half-year it is largest, and at summer solstice it is smallest, reflecting the variation in solar eclipse conditions throughout the year due to the relationship between the ecliptic and the celestial equator.

置去交前後定分，皆以差減，不足減，反減差為不蝕餘。小滿畢小暑，加時午正前後七刻外，去不蝕餘，一時三刻內即加一時。大寒畢立春，交前五時外。大暑畢立秋立後五時外，皆去不蝕餘，一時五時內，即加一時。月在外道，冬至初日無差，自後日益六分，為畢雨水。自入春畢白露，皆以五百二十二為差。（自入秋分巳後，日損六分，畢大雪，所損為差。）

When setting up the fixed portions before or after the intersection, use the previously calculated difference to subtract from it; if the fixed portion is insufficient for subtraction, reverse the process and subtract the difference from the fixed portion, the remainder is called "non-eclipse remainder," indicating how much more is needed to reach the eclipse. From Minor Full Moon to Minor Heat, if the conjunction occurs within seven hours before or after noon, subtract the non-eclipse remainder; if the conjunction occurs within one and a half hours, add one and a half hours. From Greater Cold to Start of Spring, take those five hours before the intersection; from Greater Heat to Start of Autumn, take those five hours after the Start of Autumn: both of these periods also subtract the non-eclipse remainder. If the conjunction occurs within one to five hours, add one hour. When the moon is outside the Yellow Path, on the day of Winter Solstice the difference is zero, and it increases by six parts each day until Rain Water; from Start of Spring to White Dew, the difference is always fifty-two parts. (Original note: After Autumn Equinox, the difference decreases by six parts each day until Greater Snow, and the remaining amount after subtraction is the daily difference.)

以差加去交定分，為蝕分，以減後準，餘，為不食分。各置差十五約之，以減一百四，餘，為定法，以除不餘分，得為分餘，半法已上為半強，已下為半弱，以減十五，餘。為蝕之大分。

Add the previously calculated difference to the value of the intersection fraction, and the result is called "蚀分" (the amount of the moon's obscuration). Then subtract this "蚀分" from "后准" (1553), and the remainder is called "不食分" (the part of the moon not obscured), that is, the part of the moon not covered. Take that difference again, divide it by 15, subtract the result from 104, and the remainder is called "定法" (the fixed divisor), which represents the value of each fraction. Divide the "不食分" by the "定法" to get the fraction of the moon not obscured. If the remainder exceeds half of the "定法", it is called "半强" (more than half), and if it is less than half, it is called "半弱" (less than half). Finally, subtract this fraction of the moon not obscured from 15, and the remainder is the "大分" (the major fraction) of this solar eclipse, indicating how many of the fifteen parts of the sun were obscured.

月在內道日蝕。（初起西南，蝕甚東北。）月在外道日蝕。（初起西南，食甚東南。十三分已上正西起，亦據正南方而言之也。）

The direction from which the eclipse begins and reaches its maximum is determined by whether the moon is within or outside the Yellow Path. If the moon is inside the Yellow Path and an eclipse occurs, the loss begins at the southwest corner of the sun and reaches its maximum in the northeast. If the moon is outside the Yellow Path and an eclipse occurs, the loss begins at the southwest corner and reaches its maximum in the southeast; if the eclipse exceeds thirteen parts, it begins to lose from the due west. These directions are all based on the observer facing due south. An eclipse is when the moon covers the sun, and since the moon moves eastward and catches up with the sun, the loss always starts from the west, which is opposite to the moon eclipse, which begins to lose from the east.

推日月蝕虧初及復末時刻術，置朔望蝕大分為率，五已下增二，已上增三，十已上增四，十三分已上增五，為蝕用刻。以四乘，十除，得減蝕甚刻為虧初。又六乘，十而一，以加蝕甚刻，為復末辰刻。（其日蝕甚初末更籌，因其日月所入辰刻及分，依前定氣所遇夜刻更籌術，求其初末及其時更籌。）

The method for calculating the initial eclipse and the time of re-emergence of a solar or lunar eclipse begins by taking the "great fraction" of the new moon as the base. If the eclipse fraction is less than five parts, add 2; if more than five parts, add 3; if more than ten parts, add 4; if more than thirteen parts, add 5. The result is called "the time of eclipse," which represents the total number of moments the eclipse lasts. Multiply this by 4 and divide by 10, then subtract this from the time of greatest eclipse to find the time of initial eclipse. Multiply the same number by 6 and divide by 10, then add this to the time of greatest eclipse to find the time of re-emergence. The reason for using four parts before and six parts after, rather than dividing evenly, is that the re-emergence occurs more slowly after the eclipse has fully developed. (Original note: On that day, the times of greatest eclipse, initial eclipse, and re-emergence are determined by the specific watch and division of the night, according to the previously calculated time divisions for the night, and thus the initial and final times can be determined as to which watch and division they fall into.)