This text describes the calculation methods of ancient calendars, which are quite complex. Let's break it down sentence by sentence and try to explain it in simple terms.
First paragraph: The length of each day varies; some days are longer, some days are shorter, just like the four seasons in a year, with gains and losses. The calculation method is as follows: first, add up the time of each day. If it exceeds a day's standard time, subtract 1338 (this number is a constant in ancient calendars); if it does not exceed, multiply the total time by 1338, then add 837, and divide by 899. Next, adjust according to the daily variations in time and continue to calculate the next day. This part is about calculating the calendar, involving many constants; the specific meanings need to be deeply studied in ancient calendars to understand.
Second paragraph: According to the pattern of daily time changes, calculate the rate of daily time changes. If the calculated time is insufficient, add it back instead of subtracting. This paragraph briefly describes the method of calculating the rate of daily time changes, as well as how to handle cases where the calculation results are insufficient.
Third paragraph: Calculate the time of sunrise and sunset each day. The method is to multiply the monthly movement angle by the time of sunrise and sunset each day (using a water clock), then divide by 200 to get the time of sunrise. The time of sunset is obtained by subtracting this value from the monthly movement angle. The calculation method is similar to the previous one, accumulating time and then making adjustments. This explains how to calculate the time of sunrise and sunset each day, also involving some proportion coefficients and constants.
Fourth paragraph: There are four tables used in a month (possibly referring to four different calculation methods or tables) and three calculation steps. These steps are interwoven, ultimately calculating the number of days in each month. This section describes the lunar calendar calculation method, involving concepts such as "four tables" and "three ways," which need to be understood in conjunction with knowledge of ancient calendars. The method of calculating days is based on the movement pattern of the moon and the cycle of the synodic month. "Shuohefen" (new moon calculations) and "Tuifen" (waxing moon calculations) refer to some intermediate results in the calculation.
From the fifth section to the end: This section lists some important parameters in the lunar-solar calendar, including "decay," "profit and loss rate," and so on, for each day. It provides specific numerical examples showing how to calculate "decay" and "profit and loss rate" based on different days. Special cases in the calculations are explained in parentheses, such as "exceeding extreme loss," which refers to the need for corresponding adjustments when the moon reaches a certain position in its orbit. Finally, some important constants are listed, such as "calendar week," "differential rate," "new moon combined fraction," "differential," and their calculation methods. These figures are crucial components of ancient calendrical calculations. These numbers and formulas demonstrate the complexity of ancient calendrical calculations, requiring in-depth study to fully grasp their implications.
In summary, this text describes a very complex method of ancient calendrical calculations, involving a large number of constants and intricate calculation steps. To fully grasp its implications, one needs to delve into ancient astronomical and calendrical knowledge. This text reads more like a technical document rather than an easily understandable explanation.
Let’s first discuss how to calculate how many days have passed from the Lantern Festival to a certain day. I start by multiplying the new moon day (the first day of each month) by the number of days in each month, then add the decimal part (differential) to the integer part (combined fraction). If the combined fraction exceeds one week (360 degrees), I subtract one week, and the remainder is the number of days in the solar calendar; if the combined fraction is less than one week, the remainder is the number of days in the lunar calendar. This is done for each month, and the final calculated number of days, if there is still a decimal part, represents the remaining fractional days.
Adding two days, the remaining days are 2580 days, and the decimal part is 914. According to the method, if the calculated number of days exceeds thirteen, I subtract thirteen days, and the remainder is calculated based on the fractional days. In this way, the lunar and solar calendars convert into one another; whichever calendar comes first, the remaining days are placed first; whichever calendar comes later, the remaining days are placed later, just like the moon running halfway through its orbit.
Next, we need to take into account the differences in the magnitude of gains and losses for each month. Using a coefficient (count) multiplied by the decimal part (small fraction) to obtain the differential, I then add the gains and losses to the remaining days in the lunar and solar calendars. If there are too few or too many remaining days, I adjust the days. Then, I multiply the remaining days by the profit and loss rate; if the result equals the number of days in a month, I use the total number of gains and losses to figure out the extra hours.
Multiply the difference rate by the remaining brief period of time on the new moon day, calculate a unit using differential methods, then subtract the remaining days. If it's not enough, add the number of days in a month and subtract again, then subtract one day. Combine the decimal parts, use the total to approximate the differential value to obtain the small fraction, thus determining which calendar the new moon midnight belongs to.
To calculate the next day, add one day; the remaining days are 31 days, and the small fraction is also 31 days. If the small fraction exceeds the total, subtract the number of days in a month and add one day. If the calendar ends and the remaining days exceed the number of days in the decimal part, subtract to get the first day of the month. If the remaining days do not exceed the number of days in the decimal part, retain them, then add 272, and the small fraction is 31 days, which gives the date of the next calendar.
Multiply the total by the profit and loss of the midnight of the late and early calendars and the remaining days. If the remaining days are more than half a week, consider it as a small fraction. Use the gains to adjust the remaining days of the yin and yang calendars; if the remaining days are insufficient or excessive, make the necessary adjustments. Then multiply the remaining days by the gain-loss rate; if the result equals the number of days in a month, use the total of gains and losses to determine the additional hours at midnight.
Multiply the gain-loss rate by the time during the night of the recent solar term, with 1/200 for brightness, and subtract the gain-loss rate for darkness, using the midnight gain-loss number to determine the times of brightness and darkness.
Combine the additional hours and the brightness-darkness constants, divide by twelve, and one-third of the remaining part is considered small; if it's not enough for one, it's strong; two smalls are weak. The result obtained is the angle between the moon and the ecliptic. For the solar calendar, subtract the extremum from the ecliptic calendar where the additional day is located; for the lunar calendar, add the extremum to the ecliptic calendar where the additional day is located. This way, you can calculate the angle between the moon and the extremum. Strong is positive, weak is negative; add same names and subtract different names. When subtracting, same names cancel out, different names add up; if there are no correspondences, they cancel each other out, with two strong adding one small subtracting one weak.
From the year of the Metal Ox in the Yuan Dynasty to the year of the Metal Dog in the Jian'an period, a total of 7,378 years have elapsed.
Metal Ox, Earth Tiger, Fire Rabbit, Wood Dragon, Fire Snake, Wood Horse, Water Goat.
Ren Shen, Xin You, Geng Xu, Ji Hai, Wu Zi, Ding Chou, Bing Yin, these are all the following years. Next are the Five Elements: Wood corresponds to Jupiter, Fire corresponds to Mars, Earth corresponds to Earth, Metal corresponds to Venus, and Water corresponds to the Star of Chen. The speed of each star's movement is related to the degrees in the sky, represented using the circumference and the daily rate. Multiplying the annual cycle by the circumference gives the monthly method, multiplying the monthly value by the daily rate gives the monthly remainder, and dividing the monthly remainder by the monthly method gives the month number. Multiplying the month number by the universal method gives the daily degree method. The fraction of the Dipper is calculated by multiplying the Dipper fraction by the circumference. (The daily degree method is calculated by multiplying the calendar method by the circumference, so fractions are used here for multiplication.)
Next is the calculation method for the five stars' major and minor remainders. Multiply the universal method by the month number, then divide by the daily method to get the quotient as the major remainder and the remainder as the minor remainder, finally subtracting the major remainder from sixty. The calculation method for the five stars' entry into the month and daily remainder is as follows: multiply the universal method by the monthly remainder, then multiply the combined monthly method by the minor remainder, add these two results together, simplify, and finally divide by the daily degree method to get the result. The calculation method for the degrees and degree remainder of the five stars is: first subtract the excess, the remainder is the degree remainder fraction, then multiply by the weekly sky to get the degree remainder fraction, and then simplify using the daily degree method; the quotient is the degree, and the remainder is the degree remainder. If it exceeds the weekly sky, subtract the weekly sky and Dipper fraction.
Next are some specific values:
- Calendar month: 7285
- Chapter leap: 7
- Chapter month: 235
- Year: 12
- Universal method: 43026
- Daily method: 1457
- Count: 47
- Weekly sky: 215130
- Dipper fraction: 145
Below are the specific data for Jupiter: the circumference is 6722, the daily rate is 7341, the combined month number is 13, the monthly remainder is 64810, the combined monthly method is 127718, the daily degree method is 3959258, the major remainder is 23, the minor remainder is 1370, the entry into the month day is 15, the daily remainder is 3484646, the minor fraction is 150, the Dipper fraction is 974690, the degree is 33, and the degree remainder is 2509956.
Then there is the data for Mars: the circumference is 3447, the daily rate is 7271, the combined month number is 26, the monthly remainder is 25627, the combined monthly method is 64733, the daily degree method is 206723, the major remainder is 47, the minor remainder is 1157, the entry into the month day is 12, the daily remainder is 973113, the minor fraction is 300, the Dipper fraction is 494115, and the degree is 48. These numbers represent the principles of celestial movements, which are used to forecast celestial phenomena.
This pile of numbers is overwhelming! Simply put, it's for counting days. First, let's look at "degree remainder," which is 1,991,760. Then, for Saturn, the annual lunar period is 3,529, and the daily cycle is 3,653. There are 12 months in a year, resulting in an excess of 53,843 days. Calculation method: a total of 6,751 months, 278,581 days. The new moon day has a large surplus of 54 and a small surplus of 534, with an excess of 166,272 days for the 24th of each month. The new moon day is divided by 923, and the Dipper is divided by 511,750, with a degree of 12 and a degree remainder of 1,733,148.
Next is Venus, with an annual lunar period of 9,022 and a daily cycle of 7,213. A year is defined as 9 months, resulting in an excess of 152,293 days. Calculation method: a total of 171,418 months, 5,313,958 days. The new moon day has a large surplus of 25 and a small surplus of 1,129, with an excess of 56,954 days for the 27th of each month. The new moon day is divided by 328, and the Dipper is divided by 1,308,190, with a degree of 292 and a degree remainder of 56,954.
Finally, for Mercury, the annual lunar period is 11,561 and the daily cycle is 1,834. A year consists of one month, resulting in an excess of 211,331 days. Calculation method: a total of 219,659 months, 6,809,429 days. The new moon day has a large surplus of 29 and a small surplus of 773, with an excess of 6,419,967 days for the 28th of each month. The new moon day is divided by 684, and the Dipper is divided by 1,676,345, with a degree of 57 and a degree remainder of 6,419,967.
After performing these calculations, we also need to factor in the "Upper Yuan," which refers to a specific starting point in time, by multiplying the lunar period by the year to get a "product sum." If it doesn't divide evenly, the remainder is the "remaining surplus." Dividing the lunar period by the remaining surplus allows us to determine which year's celestial phenomena it corresponds to. If it doesn't divide evenly, we look at the remainder to find out which year it is. The remaining surplus minus the lunar period gives us the degree minutes. For Venus and Mercury's product sum, an odd number indicates morning, while an even number indicates evening. This is practically a codebook for astronomical calculations!
First, let's calculate the conjunction dates of celestial bodies. Multiply the months and the surplus of months separately and combine them. If it meets the criteria of the "combined month method" (referring to a fixed value, which is not specified in the text), count by months; if not, record it as month surplus. Then, subtract the accumulated months from the recorded month value, and the remainder is the "entry month." Next, multiply by the "zhang run" (possibly referring to coefficients related to leap months). If it reaches "zhang month" (possibly referring to a cycle), we get a leap month, which is used to subtract from the "entry month." The remaining value is then subtracted from a year, resulting in the conjunction result outside of the celestial calculation (possibly referring to a certain astronomical calendar). If it is at the boundary of a leap month, we use the new moon day (the first day of the lunar calendar) to make adjustments.
Next, multiply the lunar remainder using a standard calculation method, then use the lunar conjunction method to multiply by the small remainder of the new moon, add these two results together, and simplify using the cycle count. If you meet the solar degree method, you will obtain the date of the star conjunction; if not, the remainder is the solar remainder, noted outside the new moon calculation. Then multiply the degrees by the week (360 degrees); if it meets the solar degree method, it equals one degree; if not, it is recorded as a remainder, using the five positions before the ox for recording. The above is the method for determining the conjunction of the stars.
To calculate the conjunctions for the following years, add the months and lunar remainders separately; if it is met by the combined lunar method, count as one month; if not, calculate for that year, if it is met, subtract it, and account for any leap months; the remaining portion is carried over to the following year; if it is met again, carry it over to the next two years. For Venus and Mercury, if seen in the morning, adding a day changes it to evening sighting; adding a day to evening sighting changes it back to morning sighting.
Next, add the magnitude of the new moon's remainder and the magnitude of the combined lunar remainder; if it is a big month, add 29, if it is a small month, add 773; when the small month satisfies the solar degree method, begin counting from the big month, using the same method as before.
Add the day of the month and the solar remainder together; if it meets the solar degree method, it equals one day; if the previous small remainder of the new moon satisfies the virtual division, subtract one day; if the later small remainder exceeds 773, subtract 29 days; if not, subtract 30 days, and the remainder indicates the day of the month for the next conjunction.
Finally, add the degrees and the degree remainder together; if it meets the solar degree method, it equals one degree. The following are the specific data for Jupiter, Mars, Saturn, and Venus:
Jupiter: retrograde for 32 days, 3484646 minutes; direct for 366 days; retrograde 5 degrees, 2509956 minutes; direct 40 degrees (12 degrees retrograde, actual movement 28 degrees).
Mars: retrograde for 143 days, 973113 minutes; direct for 636 days; retrograde 110 degrees, 478998 minutes; direct 320 degrees (17 degrees retrograde, actual movement 303 degrees).
Saturn: retrograde for 33 days, 166272 minutes; direct for 345 days; retrograde 3 degrees, 1733148 minutes; direct 15 degrees (6 degrees retrograde, actual movement 9 degrees).
Venus: morning retrograde in the east for 82 days, 113908 minutes; direct in the west for 246 days (6 degrees retrograde, actual movement 246 degrees); morning retrograde 100 degrees, 113908 minutes; seen in the east (solar degree like the west, retrograde for 10 days, 8 degrees retrograde).
Mercury appears in the morning for thirty-three days. It has traveled a distance of six million one hundred twenty-five thousand five hundred fifty minutes. Then, it appears in the west and remains visible for thirty-two days. (Subtracting one degree of retrograde motion, it ultimately covers thirty-two degrees.) It retrograded sixty-five degrees, totaling six million one hundred twenty-five thousand five hundred fifty minutes. After that, it appears in the east. Mercury's speed in the east is the same as in the west, retrograding for eighteen days and retreating fourteen degrees.
Calculate the degrees Mercury travels each day along with the remaining degrees; if the remaining degrees meet the daily travel standard, a cycle is established. Using this calculation method, you can determine when Mercury appears and how far it travels. Multiply the denominator of the celestial body's motion by the degrees Mercury appears; if the remaining degrees meet the daily travel standard, a cycle is established. If the denominator cannot be divided evenly and exceeds half, it is also considered a full cycle. Then add the degrees traveled each day; if the degrees reach the value of the denominator, increase by one degree. The denominators for retrograde and direct motion are different; multiply the current movement denominator by the remaining degrees. If the result equals the original denominator, that is the current movement degrees. "Leaving" refers to inheriting the value of the previous cycle, while for retrograde, you subtract. If retrograde has not completed the required degrees, use the "dou" (ancient timekeeping unit) to divide by the degrees, using the movement denominator as a proportion; the degrees will adjust based on one another. Any reference to "exact division" pertains to seeking precision in division; "remove and divide to take the remainder" refers to taking the remainder in division.
As for Jupiter, in the morning it aligns with the sun, then goes into retrograde motion, moving in the opposite direction for a total of sixteen days, traversing a distance of 1,742,323 minutes. The planet travels two degrees and 323,467 minutes, then appears in the east in the morning behind the sun. During its direct motion, the speed is fast, moving 11/58 degrees each day, covering 11 degrees in 58 days. Then, during its direct motion, the speed slows down, moving 9 minutes each day, covering 9 degrees in 58 days. It stops for 25 days, then resumes its motion. During retrograde motion, it moves 1/7 degrees each day, retreating 12 degrees in 84 days. It stops again for 25 days, then continues in direct motion, moving 9/58 degrees each day, covering 9 degrees in 58 days. During direct motion, the speed is fast, moving 1/11 degrees each day, covering 11 degrees in 58 days, appearing in the west in the evening in front of the sun. After sixteen days, this distance of 1,742,323 minutes is covered again, and the planet travels two degrees and 323,467 minutes, then aligns with the sun. The entire cycle takes 398 days, covering a distance of 3,484,646 minutes, with the planet traveling 43 degrees and 250,956 minutes.
In the morning, Mars and the sun appear together, then Mars appears to "hide" and begins its direct motion. It continues in direct motion for 71 days, covering a distance of 1,489,868 minutes, moving 55 degrees and 242,860.5 minutes along the ecliptic. Then, it becomes visible in the east each morning, positioned behind the sun. During direct motion, Mars moves 23/14 degrees each day, covering 112 degrees in 184 days. Next, the speed of direct motion slows down, moving 23/12 degrees each day, covering 48 degrees in 92 days. Then it stops for 11 days. After that, it begins its retrograde motion, moving 62/17 degrees each day, retreating 17 degrees after 62 days. It stops again for 11 days, then resumes direct motion, moving 12 minutes each day, covering 48 degrees in 92 days. During the subsequent direct motion, the speed increases, moving 14 minutes each day, covering 112 degrees in 184 days. At this point, it appears in front of the sun and is visible in the west during the evening. After 71 days, it covers a distance of 1,489,868 minutes, moving 55 degrees and 242,860.5 minutes along the ecliptic, and finally appears simultaneously with the sun. The entire cycle lasts 779 days and 973,113 minutes, during which it moves 414 degrees and 478,998 minutes along the ecliptic.
Saturn also rises in the morning at the same time as the sun, then conjuncts and begins to move in direct motion. It moves in direct motion for 16 days, covering a distance of 1,122,426.5 minutes, which is equivalent to moving 1 degree and 1,995,864.5 minutes along the ecliptic. It can then be seen in the eastern sky in the morning, positioned behind the sun. During its direct motion, Saturn moves 35/3 degrees each day, covering 7.5 degrees in 87.5 days. It then comes to a stop for 34 days. After that, it starts retrograde motion, moving 1/17 degree each day, retreating 6 degrees after 102 days. It stops again for 34 days, and then starts moving in direct motion again, covering 3 minutes each day and 7.5 degrees in 87 days. At this point, it is in front of the sun and can be seen in the western sky at night. Sixteen days later, it covers another distance of 1,122,426.5 minutes, moving 1 degree and 1,995,864.5 minutes along the ecliptic, and finally rises at the same time as the sun. Throughout this entire cycle, it covers a total of 378 days and 166,272 minutes, moving 12 degrees and 1,733,148 minutes along the ecliptic.
Venus, when it meets the sun in the morning, first conjuncts, indicating it is in retrograde, retreating 4 degrees in 5 days. It can then be seen in the eastern sky in the morning, positioned behind the sun. Continuing its retrograde motion, Venus moves 5/3 degrees each day, retreating 6 degrees in 10 days. It then pauses, remaining stationary for 8 days. Then it rotates, starting to move in direct motion at a slower speed, covering 46/33 degrees each day, moving 33 degrees in 46 days. Its speed increases to 1 degree and 91/15 minutes each day, covering 160 degrees in 91 days. Speeding up further, it moves 1 degree and 91/22 minutes each day, covering 113 degrees in 91 days, at which point it is behind the sun and rises in the eastern sky in the morning. Continuing in direct motion, it traverses 1/54 of a circle over 41 days, with the planet also moving 50 degrees and 1/54 of a circle, and finally meets the sun again. Each conjunction spans a total of 292 days and 1/54 of a circle, with the planet moving the same number of circles.
When Venus meets the Sun in the evening, it first "bows," this time moving forward. It completes one fifty-six-thousand nine-hundred fifty-fourth of a circle, traveling fifty degrees and appearing in front of the Sun in the west in the evening. It continues in the same direction, speeding up, moving two twenty-second degrees per day for ninety-one days, totaling one hundred thirteen degrees. Then the speed slows down, moving fifteen one-hundredths of a degree per day for ninety-one days, totaling one hundred sixty degrees, continuing in the same direction. The speed further slows down, moving three thirty-sixths of a degree per day for forty-six days. It then "pauses" for eight days. It then begins to move in the opposite direction, moving three-fifths of a degree per day for ten days, totaling six degrees, and appearing in front of the Sun in the west in the evening. It then speeds up, moving four degrees in total over five days, ultimately meeting the Sun again. Two meetings make up one cycle, totaling five hundred eighty-four days and one one-hundred-thousandth of a circle, with the planet completing the same number of orbits.
Mercury, when it meets the Sun in the morning, first "bows," indicating retrograde motion, retreating seven degrees for nine days, then appearing in the east in the morning behind the Sun. It continues moving backwards, speeding up, moving one degree per day. It then "pauses" for two days. It then begins to move forward, moving one and one-eighth degrees per day for nine days, totaling eight degrees. It then speeds up, moving one and a quarter degrees per day for twenty days, totaling twenty-five degrees, appearing in the east in the morning behind the Sun. It continues in the same direction, moving a total of thirty-two degrees over sixteen days, ultimately meeting the Sun again. One meeting totals fifty-seven days and one one-hundred-thousandth of a circle, with the planet completing the same number of orbits.
Speaking of Mercury, it sets alongside the sun and then disappears, moving in a direct path. Sixteen days later, it will be at a position of 32 degrees and 641,967 minutes along the ecliptic. At this point, it can be seen in the western sky in the evening, positioned ahead of the sun. When it moves forward, it travels quite fast, covering a quarter of a degree each day, and in twenty days, it can cover a total of twenty-five degrees. If it moves slowly, it only covers seven-eighths of a degree in a day, taking nine days to cover a total of eight degrees. If it remains stationary, it stays still for two consecutive days. When it moves in retrograde, its direction is reversed, retreating one degree each day, and at this point, it disappears in the western sky in the evening ahead of the sun. During retrograde motion, it moves slowly, taking nine days to retreat a total of seven degrees, and eventually aligns with the sun again. From its first conjunction with the sun to the next, Mercury takes a total of 115 days and 612,505 minutes to complete its orbit.
