Speaking of it, the ancient sages put a lot of effort into establishing the calendar! They carefully observed celestial phenomena, such as the movement of celestial bodies, the trajectories of the sun, moon, and stars, as well as their changing speeds, the length of shadows, the direction of the Dipper, the position of the Azure Dragon constellation, etc. Through these intricate numerical relationships, they formulated the calendar.
The celestial bodies complete one rotation in one day and night. The stars rotate from east to west along with the celestial sphere, while the sun moves in the opposite direction, from west to east. The sun's path in the sky forms degrees, which then become dates in the calendar. The sun passes through a total of twenty-eight constellations (four sevens twenty-eight), using the terms "Jia" and "Yi" to mark the days, totaling sixty days. The sun and moon chase each other, with the sun moving slowly and the moon moving quickly. When they reach the same position, it is called the new moon (the lunar New Moon). When the sun moves slowly and the moon moves quickly, it's a waxing crescent when they are close together and a waning crescent when they are farther apart. When the sun and moon are in opposite positions, it is called the full moon (the fifteenth day of the lunar month). When the moon catches up with the sun and its light is obscured, it is called the dark moon (the thirtieth day of the lunar month). The phases of the new moon, full moon, and dark moon cycle continuously throughout the month, with the direction of the Dipper changing accordingly, forming one month. The movement of the sun and moon determines the changing seasons: when the sun moves north, it signals winter; when it moves west, it signals spring; when it moves south, it signals summer; and when it moves east, it signals autumn. When the sun reaches the farthest point to the south, with the longest shadow, it is the winter solstice; when the sun reaches the closest point to the North Pole, casting the shortest shadow, it is the summer solstice. Between the winter solstice and summer solstice, when the sun is in the middle, the shadow length is just right; this marks the spring and autumn equinoxes.
The sun orbits the sky, going through a cold season and a hot season, completing the four seasons in a year, and all things change accordingly. The positions of the stars in the circumpolar region change, and the positions of the Qinglong constellation also change, which constitutes a year. The beginning of the year is the winter solstice, and the start of a month is the new moon (the first day). When the winter solstice and the new moon fall on the same day, this phenomenon is referred to as "Zhang"; when the winter solstice and the new moon both fall at the beginning of the month, it is called "Bu"; when one "Bu" ends, it marks sixty years, known as "Ji"; the start of a new year is termed "Yuan." Therefore, we use "day" to calculate specific dates, "month" to arrange leap months, "hour" to divide time periods, "year" to calculate years, "Zhang" to mark special days, "Bu" to divide longer periods, "Ji" to record longer periods of time, and "Yuan" to indicate the beginning of a cycle. This way, despite the countless changes in time and the emergence of various situations, they can all be arranged and corrected according to these rules.
Speaking of celestial phenomena, the internal workings of these phenomena are intricate, and their external manifestations are also varied. For example, by following the sun's movements with instruments like the Xuanji (a celestial sphere) and Yuheng (a balance), one can observe the patterns of sunrise and sunset, and thus deduce the changes in the light path. Additionally, using a water clock and a floating arrow for timing, by watching the water clock drip and the floating arrow move, one can deduce the movements of the central star, and thus determine the length of day and night.
The sun has its orbit, while the moon has nine phases. These nine phases overlap with each other, which explains the moon's phases from full to crescent. The new moon and full moon positions are adjacent to their intersection points, which is why the moon appears to change. Just as the moon waxes and wanes, stars also exhibit conjunctions and retrograde motion. These phenomena ultimately point to a rule, which is the basis for our calendar calculations. The movements of Venus and Mercury are influenced by the sun, their speeds sometimes fast and sometimes slow. When they are fast, they move ahead of the sun, and when they are slow, they lag behind, resulting in retrograde motion. Retrograde motion is opposite to the direction of the sun's movement, and after moving in the opposite direction to the sun, their speed increases, causing them to move ahead of the sun again. This interplay of speed and direction forms the phenomenon of sunrise and sunset that we observe. The sun, moon, and five planets all have their own rules and cycles of movement, from which the rules of the seven celestial bodies can be deduced. The hidden rules of movement and the degrees of their movement can be calculated. By organizing and classifying these irregular phenomena, reconciling their variations, we can calculate their cycles of movement. Through continuous exploration and research, delving deeper, no hidden phenomenon is beyond explanation through precise rules. This is how yin and yang can be distinguished, the seasons have solar terms, the heavens and earth have their regularity, and the sun and moon have their rules. As for those responsible for establishing the calendar, they inherit the legacy of the ancient calendar makers, carrying the mission of the holy emperor, as monumental as the heavens. They grasp the movements of the sun, moon, and stars, guiding people's livelihoods, establishing leap months and solar terms, marking the glorious achievements of the Xihe era.
During the reigns of the Shang Dynasty's Tang and King Wu of Zhou, a new calendar system was created based on the movements of Venus and Mars, which made the calendar more accurate and aligned with celestial events, benefiting the populace and marking the pinnacle of calendar advancement. However, as royal power declined, incompetent monarchs disrupted court affairs, and ignorant historians neglected their duties, the calendar also declined. During the Xia Dynasty, Xi He indulged in pleasure, neglecting the calendar, leading to a disruption of timekeeping, ultimately being punished by Emperor Yin. King Zhou of Shang was even more depraved and disregarded the calendar, ultimately being destroyed by King Wu.
Those who stay true to the right path and actively promote it will flourish in their pursuits; while those who stray from this path and eventually decline will meet with swift and total ruin. Establishing a calendar and maintaining the order of heaven and earth is a monumental task for emperors and leaders, so sages hold it in high regard, and gentlemen study it earnestly. "It is as majestic as the guiding principles of heaven and earth and the grand endeavors of rulers."
Throughout history, sages have possessed six extraordinary abilities: the ability to grasp the essence of things according to their nature, to perfect their form by mastering various laws, to understand their symbolic meanings by examining their categories, to handle matters according to the timing, to speculate on the origins of things based on the past, and to predict the future development of things based on the present. These significant endeavors rely on these abilities, and thus fortunes and misfortunes stem from them. Therefore, gentlemen consult thoroughly before embarking on significant endeavors, and once they accept a mission, they will not go against it.
Speaking of this, nothing is more crucial than the "Monthly Ordinances" for those who follow celestial timing, implement education based on the time, issue edicts from the grand hall, and seek the well-being of the people. Emperors use this to ensure thorough preparations, allowing all affairs to be carried out successfully. Outside of this context, gentlemen are less informed about the intricate taboos and constraints.
Among the 28 constellations of the Dipper, it is the farthest constellation from the North Star. When the sun reaches this point, it marks the winter solstice, signaling the beginning of growth for all living things. Therefore, the first note in the musical scale is "Huang Zhong," the calendar begins with the winter solstice, the first month is Zi, and the time is midnight. When Emperor Gaozu Liu Bang of the Han Dynasty was appointed, he was 45 years old; the sun was in the constellation of Kang, and the moon was in the constellation of Xu. On the night of the Jiazi day in November of that year, the new moon occurred on the winter solstice, marking the start of calculating lunar phases, leap months, and establishing a new calendar known as the "Han Calendar." Additionally, it calculated the two cycles (yuan) and the patterns of lunar eclipses and the movements of the five visible planets, all starting from this point.
It is said that in ancient times, people studied time by first setting up an instrument to observe the length of the sun's shadow. A longer shadow indicates that the sun is farther away from us, marking the starting point of celestial motion. The sun starts from this point and completes one orbit in a year, but the shadow does not return to its original length right away. It takes 1461 days for the shadow to return to the starting point, representing one complete cycle of the sun's movement. Dividing this cycle by the number of days gives a year of 365 and one-quarter days. The sun moves one degree per day, which also serves as a measure of celestial motion.
Observing the sun and moon starting from the same point and running simultaneously, the sun completes 19 revolutions while the moon completes 254 revolutions before they return to the starting point at the same time; this marks one complete lunar cycle. By dividing the sun's revolution by the moon's revolution, the number of celestial movements in a year can be calculated. Subtracting the sun's revolution, the remaining value indicates how many more times the moon has revolved compared to the sun, which corresponds to the number of months in a year. Dividing this number by the number of days in a year gives the number of days in a month. When the excess days from the moon's revolutions accumulate sufficiently, they form a month, adding an extra month to the year and making it a leap year. The movement of the moon determines the changing of the four seasons, so twelve solar terms are established to determine the position of each month. If only a new moon occurs without a corresponding solar term, this is designated as a leap month. The beginning of a solar term is called a solar month. The twelve solar terms and twelve lunar months together form the twenty-four solar terms. Dividing the number of days in a year by the twenty-four solar terms gives the duration of each solar term. These excess days accumulate to form a period called "mei," and by calculating "mei" and the remaining days of the solar terms, the total "mei" in a year can be determined. The year ends at the winter solstice, and when the remaining days of the winter solstice accumulate to a certain extent, they form a day, completing a cycle every four years. The waxing and waning of the moon leads to leap months, and after seven leap months, a cycle of 19 years is completed, called a "zhang" (章). After four "zhang," a cycle of 76 years is completed, called a "bu." Multiplying the total days in a year by 76 yields the total days in a "bu."
The traditional 60-year cycle known as "jiazi" is used for dating, with 20 "bu" constituting a "ji." The "ji" year concludes before the onset of the Blue Dragon year, and after three cycles of "ji," the year of the Blue Dragon begins again, known as a "yuan."
Yuan method: 4560.
Ji method: 1520.
Ji month: 18800.
Bu method: 76.
Bu month: 940.
Zhang method: 19.
Zhang month: 235.
Zhou Tian: 1461.
Day method: 4.
Bu day: 27759.
Mei number: 21.
Tong method: 487.
Mei method: 7, due to leap zhang.
Day remainder: 168.
Zhong method: 32.
Da Zhou: 34335.
Yue Zhou: 1016.
Studying lunar eclipses involves recording their occurrences. Approximately every 23 lunar eclipses, another lunar eclipse occurs, with about 135 months in between. By dividing these numbers, it can be calculated that a lunar eclipse occurs approximately every five months and 23 days. Dividing this number by the number of months in a year, it can be calculated that there are approximately two lunar eclipses in a year, more precisely 513.055 times. The remaining calculations are then compared to the cycle of "bu," resulting in the numbers four and twenty-seven, which when multiplied together equal two thousand and fifty-two, and twenty of these cycles make up a "yuanhui."
A "yuanhui" consists of forty-one thousand and forty.
Let's first look at these numbers. The cycle lasts for 2520 years, the duration is 513 years, there have been a total of 181 solar eclipses and 135 lunar eclipses, with an interval of 23 days between each solar eclipse.
The method of calculation is as follows: first, subtract the value of the upper element from the value of the element method (specific numerical values not mentioned here), then divide the remaining numbers by the record method (also unspecified units). The resulting value, starting from the celestial record, indicates the era in which it falls. If it does not divide evenly, the remaining numbers represent the number of years. Then divide by the "bu" method (also unspecified units), and the resulting value, starting from the 60-year cycle, is the designation of the year within that era, and inside is the position of that year's "tai sui."
Next is the method of calculating in which year of the "bu" cycle lunar eclipses occur: first, subtract the upper element from the value of the element method (also unspecified units), then divide the remaining numbers by the "bu" cycle, multiply the result by 27, then divide by 60 and take the remainder, and then divide the remainder by 20. Beginning with the celestial record, the result indicates the era in which it is located. If it does not divide evenly, the remaining numbers are counted from the 60-year cycle, and the result is the "bu" cycle in which it is located. If the "bu" cycle does not divide evenly at the beginning, the remaining numbers represent the number of years in the "bu" cycle. Finally, based on the calculation results, the position of that year's "tai sui" can be determined.
The following table lists the year names for different epochs (Heavenly Calendar, Earthly Calendar, Human Calendar) across different cycles of the Stems and Branches. The table records the year names for each of the three epochs from Jiazi One to Yiyou Twenty, covering a total of twenty cycles. This table allows us to identify specific years and heavenly stems based on our calculations.
There is another method to calculate the precise date of the new moon. "One method, multiply the years by Da Zhou, subtract the product of the Zhou days and the remainder of the leap year; if the remaining number equals the number of months in a lunar year, then it is the exact date of the new moon." This is a simple method, using Da Zhou (a unit of time) multiplied by the number of years, subtracting the product of the days in a year and the remainder of the leap year, and if the remaining number is exactly the number of months in a lunar year, then it is the exact date of the new moon.
Next, let's talk about how to calculate the twenty-four solar terms. "The method of calculating the twenty-four solar terms states: subtract one from the number of years, multiply it by Da Yu, if the result equals the total number of solar terms, it is called Da Yu; if not, it is called Xiao Yu. If Da Yu exceeds sixty, it is divided by sixty, the remainder is named after the month, and after the calculation, it corresponds to the winter solstice of the previous year." To calculate the twenty-four solar terms, subtract one from the number of years, multiply it by Da Yu, if the result equals the total number of solar terms, it is called Da Yu; if not, it is called Xiao Yu. Divide Da Yu by sixty, name the remainder after the month. After the calculation, it corresponds to the winter solstice of the previous year.
To calculate the next solar term. "To find the next term, add fifteen to Da Yu, add seven to Xiao Yu, divide it as before; this corresponds to the day of Xiao Han." To calculate the next solar term (Xiao Han), add 15 to Da Yu, add 7 to Xiao Yu, and then calculate it using the same method.
How to calculate the leap month. "To determine the location of the leap month, subtract the chapter method from the leap year remainder, multiply the remainder by twelve; if the result equals the number of leap months, or more than four, it is the leap month. Starting from November of the previous year, calculate until the result indicates the month of the leap month. It can also be adjusted based on the solar terms."
Method for calculating the lunar phases (on the 8th and 23rd day of each lunar month). "Calculate the waxing, full, and waning phases of the moon; add 7 to the remainder of the size of the new moon, and add 359.75 to the small remainder. If the small remainder completes a full month, add the large remainder. Following this method, you can calculate the waxing, full, and waning phases, as well as the next month's new moon. If the small remainder of the lunar phases is less than 260, use 'bai ke' (unit of time measurement) to calculate; if not enough, take the closest time to midnight of the solar terms.
Lastly, the method of calculating solar eclipses. "Using the 'mei mie' method, subtract one from the year of the new moon, multiply it by the number of solar eclipses; if the full day method is reached, it is termed 'ji mei'; if not fully obtained, it is 'mei yu'. Multiply the 'ji mei' by the common method; if the full 'mei' method is reached, it is called 'da yu'; if not fully obtained, it is 'xiao yu'. Divide the 'da yu' by sixty, name the remainder after the year of the new moon, and determine the date of the eclipse that occurred before the winter solstice of the previous year. To find the next eclipse, add 69 to 'da yu' and 4 to 'xiao yu', and continue this pattern. If there are no decimals, it is a solar eclipse; if there are decimals, it is a lunar eclipse.
There is also a simpler method for calculating solar eclipses. "Using the 'yi shu' method, multiply 15 by the small remainder from the winter solstice, subtract the common method; if the result equals the full 'mei' method, then the eclipse occurs after Tianzheng."
In conclusion, this passage describes the ancient astronomical calendar calculation methods as complex and ingenious, showcasing the ingenuity of ancient laborers.
First, we need to calculate the position of the new moon (lunar first day) by multiplying the number of days from the new moon by the number of days in a lunar cycle, then dividing by the number of days in the lunar cycle; the remainder represents the position of the new moon in degrees. Then add 21 degrees and 235 minutes, and divide by the degree of a constellation. If the remainder is less than the degree of a constellation, then it indicates the degree of the new moon within that constellation. To calculate the next new moon, add 29 degrees and 499 minutes to this degree; if the minutes exceed a cycle, add one degree, then divide by 235 minutes.
Another method is to multiply the remaining days by 360 (the number of degrees in a full circle), then subtract the remainder of a lunar cycle; if it exceeds a lunar cycle, add one degree, then add 21 degrees, 4 minutes, and 1 second. This will allow you to determine the position of the sun and moon conjunction at the vernal equinox.
Next, let's calculate the position of the sun at midnight. Multiply the number of days by the number of days in a lunar cycle, then divide by the number of days in the lunar cycle; the remainder is the degree of the sun. Then add 21 degrees and 19 minutes, and divide by the degree of a constellation; you will find out which constellation the sun occupies at midnight. To calculate the position of the sun the next day, add one degree; to calculate the position for the next month, add 30 degrees for a full month and 29 degrees for a short month, then divide by 19 minutes.
Another method is to subtract the degree and minutes of the new moon from the remainder of the new moon; this will give you the position of the sun at noon. Divide the minutes by 235, then multiply by 19.
Then calculate the position of the moon. Multiply the number of days by the number of days in a lunar cycle, then divide by the number of days in the lunar cycle; the remainder is the degree of the moon. Then add 21 degrees and 19 minutes, and divide by the degree of a constellation; you will find out which constellation the moon occupies at midnight. To calculate the position of the moon the next day, add 13 degrees and 28 minutes; to calculate the position for the next month, add 35 degrees and 61 minutes for a full month, and 22 degrees and 33 minutes for a short month. If the minutes exceed a cycle, add one degree, then divide by 19 minutes. At the end of winter, the moon is positioned near the Zhang and Xin constellations, indicating that daylight is nearly over.
Another method is to divide the number of days in a lunar cycle by the remainder of the new moon, subtract the degree at noon from the result, and then subtract the minutes; this will give you the position of the moon at midnight.
Finally, let's calculate the time of sunrise and sunset. To calculate the time of sunrise, multiply the number of nighttime hours of the solar terms in the month by the number of days in the cycle, then divide by 200 to get the fraction of time from midnight to sunrise. Add this fraction to the midnight position of the sun to get the position of the sun at sunrise.
To calculate the time of sunset, subtract the number of days in the cycle from the fraction of time from midnight to sunrise; the remaining fraction is the time from midnight to sunset. Add this fraction to the midnight position of the sun to get the position of the sun at sunset.
First, let's talk about how to calculate the position of the moon. According to the book, to determine the degree at which the moon is located, multiply the number of nighttime hours on the night of the solar terms by the number of days the moon orbits the Earth in a lunar month, then divide by 200 to get a value. When this value completes a "蔀" (a cycle), add one degree to the midnight position, which is the position of the moon.
To calculate the position of the moon at dusk, subtract the number of days the moon orbits the Earth in a lunar month from the value calculated earlier. If the remaining value completes a "蔀," add one degree to the midnight position, which is the position of the moon at dusk.
Now, let's figure out where the waxing moon, full moon, and waning moon show up in the constellations. Add 7 degrees, 59 minutes, and 4 seconds to the degree of the new moon, then divide by the degree of the constellation to get the constellation and degree of the waxing moon. The same method applies to calculating the full moon and waning moon, rounding down for decimal values and rounding up by one degree for whole numbers.
The method for calculating where the waxing moon, full moon, and waning moon appear in which constellation is similar. Add 98 degrees, 65 minutes, and a half to the degree of the new moon, then divide by the degree of the constellation to get the constellation and degree of the waxing moon. The same method applies to calculating the full moon and waning moon, rounding up by one degree for a "蔀."
Now let's talk about how to calculate a lunar eclipse. According to the book, first subtract one from the number of years since the last "shu," then multiply by the number of lunar eclipses. If it completes a "year" (one year), it's counted as a "cumulative eclipse"; if not, it's recorded as "eclipse remaining." Then multiply by the number of months to get the "cumulative eclipse"; if it completes a "month law" (the lunar eclipse cycle), it's counted as a "cumulative month"; if not, it's recorded as "month remaining fraction." Then divide the "cumulative month" by the "chapter month" (one cycle); the remainder gives you the "entry chapter month number." Subtract the "entry chapter leap" (leap month) first, then divide by 12; if it's less than 12, treat it as November. After calculating, if there is still a remainder, it is the month of the lunar eclipse before November of the previous year.
To find the "entry chapter leap," multiply the "entry chapter month" by the "chapter leap"; if it completes a "chapter month," you get the "entry chapter leap number." If the remainder falls between 224 and 231, then the lunar eclipse occurs in the leap month. Whether the leap month is early or late depends on the new moon day. To calculate the next lunar eclipse, add 5 minutes and 20 seconds; complete a cycle to advance a month, and so on until all minutes are used up.
Finally, let's talk about how to calculate the new moon day of a lunar eclipse. First, multiply the "cumulative month" by 29 to get the "cumulative day." Then multiply by 499; if it completes a "shu," add 1, then add this number to the "cumulative day," and divide by 60; the remainder indicates the date based on the "shu." After calculating, if there is still a remainder, it is the new moon day of the lunar eclipse before the Tianzheng of the previous year.
To calculate the exact date of a lunar eclipse, add a big remainder of 14 and a small remainder of 719.5. When the small remainder completes a "shu," it converts into a big remainder, and the algorithm for the big remainder is the same as before. To calculate the new moon day and date of the next lunar eclipse, add a big remainder of 27 and a small remainder of 615. If the "month remaining fraction" is less than 20, add another 20 days, with a small remainder of 499. The leftover remainder should be calculated using the drip method; if the night drip isn't finished, use the calculated date.
First, let's talk about how to calculate the Lantern Festival. Subtract the year of the Lantern Festival from your current age to get a number; then multiply by 112, subtract the total number of full moons, and the remaining is the number of post-Tianzheng eclipses.
Next, let's calculate the extra time. Multiply 12 by the decimal we calculated earlier, subtract half first to determine a time. Then divide the remaining result by the earlier decimal, and this will give us the additional time.
Next is the calculation of water leaks. Multiply 100 by the decimal; if the result is enough for a full leak, count it as one leak; if it’s not enough for a full leak but sufficient for a tenth, count it as one minute. Subtract half of the nighttime water leak count from the calculated daytime leak count, and the remainder is the daytime water leak count. Then subtract the daytime water leak count from the total daytime water leak count, and the remaining is the nighttime water leak count. If the nighttime water leak count is less than half of the total nighttime water leak count, then subtract it, and the remaining is what was not leaked last night, which relates to the lunar calendar phases (Lunar New Year's Day 7 or 8, 22 or 23).
The calculation method for the five stars is as follows: the movements of the five stars must be documented in the calendar and related to the degrees of the annual cycle (360 degrees) to form a ratio. Multiply a coefficient by the annual cycle (360 degrees) to derive the monthly calculation method; then multiply the chapter month (a coefficient) by the daily rate (a coefficient). If the result matches the monthly calculation method, you obtain the accumulated monthly total. Multiply the monthly day count by the accumulated month to get the remaining balance; then multiply by a number to get the remaining day balance. Multiply the daily rate (a coefficient) by the annual cycle to get the daily degree method. Subtract the daily rate from the annual cycle, multiply the remaining by the week; if the result matches the daily degree method, you get the accumulated degree balance. After reducing the daily rate, you get the number 299011621582300, the cycle of the five stars' movement corresponds to the time unit, aligning with the defined time unit.
Next are the specific parameters:
The weekly rate is 4327.
The daily rate is 4725.
The combined monthly total is 13. The monthly balance is 41660.
The monthly method is 82213.
The big balance is 23.
The small balance is 847.
The virtual division is 93.
The entry month day is 15.
The daily balance is 14641.
The daily degree method is 17380.
The accumulated degree is 33.
The degree balance is 1314.
Fire:
The weekly rate is 879.
The daily rate is 1876.
The combined monthly total is 26.
The monthly balance is 6634.
The monthly method is 16710.
The big balance is 47.
The small balance is 754.
The virtual division is 186.
The entry month day is 12.
The daily balance is 1872.
The daily degree method is 3516.
The accumulated degree is 49.
The degree balance is 114.
Earth:
The weekly rate is 9996.
The daily rate is 9415.
The combined monthly total is 12.
The monthly balance is 138637.
The monthly method is 172824.
This text describes a complex calendar calculation method involving many astronomical parameters and intricate calculation steps. A deeper understanding of astronomy and calendar knowledge is required to grasp the specific meanings of these parameters and formulas.
The Major Remainder is 54, the Minor Remainder is 348, the Virtual Fraction is 592, the Entry Month Day is 24, the Day Remainder is 2163, the Daily Synodic Rate is 36384, the Product Degree is 12, and the Degree Remainder is 29451. The synodic period of Venus is 5830, and the daily synodic rate is 4661. The combined synodic period of Venus is 9, the lunar remainder is 98450, the lunar calculation method is 117770, the Major Remainder is 25, the Minor Remainder is 731, the Virtual Fraction is 290, the Entry Month Day is 26, the Day Remainder is 281, the Daily Synodic Rate is 33320, the Product Degree is 292, and the Degree Remainder is 281. The synodic period of Mercury is 11980, and the daily synodic rate is 1889, the combined synodic period of the moon is 1, the lunar remainder is 217663, the lunar calculation method is 226252, the Major Remainder is 29, the Minor Remainder is 499, the Virtual Fraction is 441, the Entry Month Day is 28, the Day Remainder is 44850, the Daily Synodic Rate is 47632, the Product Degree is 57, and the Degree Remainder is 44850.
Next is the method of calculating the positions of the five classical planets: since the start of the calendar, multiply the year to be calculated by the synodic period; if it can be divided by the daily synodic rate, you will get a "combined synodic period," indicating a conjunction event; if it cannot be divided, the remainder is the "synodic remainder." Divide the synodic remainder by the synodic period; if it can be divided, it means that the celestial body will be in conjunction in that year; if it cannot be divided, you need to calculate the year backwards until you find the year of conjunction. For example, if the first conjunction occurs before a certain year, the second conjunction will be in the year one year earlier. The synodic period of Venus and Mercury, odd numbers are morning stars, and even numbers are evening stars. If it is less than the synodic period, subtract it in reverse, and the remainder is the degree.
To calculate the conjunction of celestial bodies and the moon, first multiply the combined synodic period by the synodic period to get a small product; then multiply the lunar remainder by the synodic period. If it can be divided by the lunar calculation method, subtract the multiple of the division from this small product, and the remaining portion is the synodic month, while the undivided portion is the lunar remainder. Subtract the recorded month from the synodic month, and the remainder is the entry month. Then multiply the intercalary month by the entry month; if it can be divided by the chapter month, it means there is a leap month; the undivided portion is the leap remainder. Subtract the leap remainder from the entry month, then divide by 12; the remainder is the entry year month number, starting from Tianzhen November; this month marks the conjunction of the celestial body and the moon. If the leap remainder occurs between 224 and 231, the conjunction of the celestial body and the moon is in the leap month. Whether the leap month occurs early or late depends on the date of the new moon.
Finally, the method for calculating the new moon day is: multiply the daily value by the monthly value; if it can be evenly divided by the monthly value, the result is the accumulated days count; the remainder that cannot be evenly divided is the small difference. When the accumulated days total sixty, subtract sixty; the remainder is the large difference, named by the sexagenary cycle. This marks the new moon day when the celestial bodies align with the moon.
First, calculate the days of the month: multiply the daily value by the monthly difference, then multiply the monthly calculation method by the new moon small difference, add the results together, and then divide by 4465. The quotient is the monthly days, and the remainder is the daily difference. Use the sexagenary cycle to record the monthly days; this completes the calculation, and the stars and the sun are aligned.
Next, calculate the alignment degree: multiply the weekly cycle by the degree minutes, and divide by the daily degree method. The quotient is the accumulated degree, and the remainder is the degree difference. Use the dipper (214th) to record the degree; this completes the calculation, and the stars align at that degree.
There is another method: first calculate the number of years to add or subtract, subtract it from the previous element; if it can be evenly divided by 80, divide it. The remaining number is then multiplied by the number of years and divided by the daily method; the quotient is the large difference, and the remainder is the small difference. Use the sexagenary cycle to record the large difference; this indicates the stars' alignment with the winter solstice. Then multiply the weekly rate by the small difference, add the degree difference, and divide by the daily degree method. The quotient is the number of days until the next alignment with the sun, starting from the winter solstice.
To calculate the next alignment with the moon, add the accumulated months to the annual months, add the monthly difference to the annual months; if it can be evenly divided by the monthly calculation method, divide it, subtract twelve when the annual months reach twelve, and remember if there is a leap month. Record the remaining number using the previously mentioned method; this completes the calculation, and the next alignment with the moon is obtained. Venus and Mercury add to the evening and become the morning; add the morning and become the evening (this does not need explanation, transcribed directly from the original text).
To calculate the new moon day, add the large and small differences to the number calculated now; if the monthly difference is enough for a month, then add 29 to the large difference and 499 to the small difference. If the small difference can be evenly divided by the monthly value, divide it, then add the large difference and record the large difference according to the method just mentioned.
To calculate the monthly days, add the daily difference to the number calculated now; if it can be evenly divided by the daily degree method, divide it, and record the remaining number according to the method just mentioned. If the previous alignment with the moon new moon small difference can be evenly divided by the virtual division, add one more day; if the number of days reaches a month, subtract 29 first. If the new moon small difference in the back is not less than 499, subtract one more day and record the remaining number according to the method just mentioned.
Finally, calculate the alignment degree: add the accumulated degree and the degree difference to the number calculated now; if it can be evenly divided by the daily degree method, divide it, and record the remaining number according to the method just mentioned. The division using the dipper method is equivalent to the weekly rate.
In the morning, Jupiter begins to set. After about 16 days and 7,320.5 minutes (which is approximately 16 days and 5 hours), it travels 23,811 minutes and hides a little over 13 degrees behind the sun before reappearing in the east. While moving, it travels 58 and 11/58 degrees per day, and covers 11 degrees every 58 days. Sometimes its movement may slow down slightly, moving 9 minutes per day and covering 9 degrees every 58 days. Sometimes it can stop and remain stationary for 25 days. Then it reverses direction, moving backward by 7 and 1/84 degrees per day, retreating 12 degrees after 84 days. It then stops for another 25 days before resuming its forward movement, covering 9 degrees in 58 days, followed by 11 degrees in another 58 days, finally reappearing a little over 13 degrees in front of the sun before disappearing in the west in the evening. Excluding the time it moves backward, Jupiter completes one cycle in 366 days, during which it travels 28 degrees. It then disappears again for about 16 days and 7,320.5 minutes, moving 23,811 minutes before finally aligning with the sun. Jupiter completes a full orbit in 398 days and 14,641 minutes, covering 33 degrees and 1,314 minutes, averaging 398/4,725 degrees per day.
In the morning, Mars begins to set. After about 71 days and 2,694 minutes, it travels 55 degrees and 2,254.5 minutes, hiding a little over 16 degrees behind the sun before reappearing in the east. While moving, it travels 23 and 14/184 degrees per day, and covers 112 degrees every 184 days. Sometimes its movement may slow down slightly, moving 12 minutes per day and covering 48 degrees every 92 days. Sometimes it can stop and remain stationary for 11 days. Then it reverses direction, moving backward by 62 and 17/62 degrees per day, retreating 17 degrees after 62 days. It then stops for another 11 days before resuming its forward movement, covering 48 degrees in 92 days, followed by 112 degrees in another 184 days, finally reappearing a little over 16 degrees in front of the sun before disappearing in the west in the evening. Excluding the time it moves backward, Mars completes one cycle in 636 days, during which it travels 333 degrees. It then disappears again for about 71 days and 2,694 minutes, moving 55 degrees and 2,254.5 minutes before finally aligning with the sun. A complete orbit of Mars takes 779 days and 1,872 minutes, covering 414 degrees and 993 minutes, averaging 997/1,876 degrees per day.
In the morning, Saturn transits behind the Sun, a process that takes 19 days and 181.5 minutes, covering a distance of 34725.5 degrees while being about 15 degrees away from the Sun. Then, Saturn rises in the east. Saturn moves forward, covering 43.33 degrees each day, with a total of 6 degrees over 86 days. After that, Saturn remains stationary for 33 days. Then Saturn retrogrades, moving 17.12 degrees each day and regressing by 6 degrees each day. It stops moving again for 33 days. Saturn then resumes its forward motion, covering 6 degrees over 86 days and being about 15 degrees away from the Sun, appearing in the west in the evening. The total duration from Saturn's retrograde movement to its reappearance is 340 days, during which it covers 6 degrees. Saturn once again transits behind the Sun, lasting for 19 days and 181.5 minutes, covering a distance of 34725.5 degrees, and aligns with the Sun. One cycle is completed, totaling 378 days and 2163 minutes, with the planet covering 12 degrees and 29451 minutes in its orbit. On average, it covers 319/9415 degrees each day.
Venus transits behind the Sun in the morning, a process that lasts for 5 days, regressing by 4 degrees and being about 9 degrees away from the Sun, then appearing in the east. Venus retrogrades, moving 0.6 degrees each day and regressing by 6 degrees after 10 days. Venus stops moving for 8 days. Then Venus moves forward, covering 46.33 degrees each day for a total of 33 degrees over 46 days. The speed increases to 1.91 degrees per day, totaling 106 degrees over 91 days. The speed further increases, with Venus covering 1.22 degrees each day and 113 degrees over 91 days, being about 9 degrees away from the Sun and appearing in the east in the morning. The total duration from the start of Venus's retrograde to its reappearance is 246 days, during which it covers 246 degrees. Venus transits behind the Sun, lasting for 41 days and 281 minutes, covering 50 degrees and 281 minutes, and aligns with the Sun. One alignment cycle lasts for 292 days and 281 minutes, while the planet's orbital angle remains unchanged.
In the evening, Venus transits behind the Sun, a process that lasts for 41 days and 281 minutes, covering 50 degrees and 281 minutes, being about 9 degrees away from the Sun, and appearing in the west. Venus moves forward at a fast speed, covering 1.91 degrees each day and 113 degrees over 91 days. The speed slightly decreases, with Venus covering 1.15 degrees each day and 106 degrees over 91 days. The speed slows down further, with Venus covering 0.46 degrees each day and 33 degrees over 46 days. Venus stops moving for 8 days. Then Venus retrogrades, moving 0.6 degrees each day and regressing by 6 degrees after 10 days, being about 9 degrees away from the Sun, and appearing in the west in the evening. The total duration from the start of Venus's retrograde to its reappearance is 246 days, during which it covers 246 degrees. Venus transits behind the Sun, lasting for 5 days, regressing by 4 degrees, and then aligns again. The two alignment cycles conclude, totaling 584 days and 562 minutes, while the planet's orbital angle remains unchanged. On average, it covers 1 degree each day.
Mercury, it conceals itself in the morning, retrograding seven degrees after nine days, sixteen degrees behind the Sun, and then it appears in the east. If it retrogrades, it retreats one degree after a day. If it stops moving, it's two days. If it progresses, it moves eight-ninths of a degree each day, covering eight degrees after nine days. If it moves fast, it moves one and a quarter degrees each day, covering twenty-five degrees after twenty days, sixteen degrees behind the Sun, and then appears in the east in the morning. From the beginning of its retrograde motion to its reappearance, a total of thirty-two days, covering thirty-two degrees in forty-four thousand eight hundred and fifty minutes, remaining hidden for sixteen days, then meets the Sun. Each conjunction lasts a total of fifty-seven days and forty-four thousand eight hundred and fifty minutes; this is how Mercury moves.
In the evening, Mercury conceals itself for sixteen days and forty-four thousand eight hundred and fifty minutes, covering thirty-two degrees in forty-four thousand eight hundred and fifty minutes, sixteen degrees ahead of the Sun, and then appears in the west. If it progresses and moves fast, it covers one and a quarter degrees each day, reaching twenty-five degrees after twenty days. If it moves slow, it covers eight-ninths of a degree each day, reaching eight degrees after nine days. If it stops moving, it's two days. If it retrogrades, it retreats one degree after a day, sixteen degrees ahead of the Sun, and then appears in the west in the evening. From the beginning of its retrograde motion to its reappearance, a total of thirty-two days, covering thirty-two degrees, remaining hidden for nine days, retreating seven degrees, and then reuniting. Two conjunctions count as one cycle, totaling one hundred and fifteen days and forty-one thousand nine hundred and seventy-eight minutes; this is how Mercury moves. On average, it moves one degree each day.
The method of calculating the position of Mercury is to calculate the degrees and minutes of Mercury's daily movement in steps, add the remaining degrees after the conjunction of the celestial body with the sun, and determine the position of the celestial body using the previously mentioned method. Multiply this degree by the denominator value, calculate the minutes based on the degrees of daily movement; if it does not divide evenly, apply the half method to derive a result, then add this result to the daily movement degrees, and when the denominator is complete, you obtain one degree. The denominators for retrograde and direct motion differ; multiply the current denominator by the original fraction to get the original denominator, which equals one. For stationary positions, carry over the degrees from the previous day, subtract for retrograde, and do not record degrees for hidden positions. Calculate longitude using the movement's denominator; one quarter corresponds to one degree. The fractions may increase or decrease, balancing each other out. Use the equator to determine degrees: add for advancement and subtract for retreat.
Months:
November (Tianzhen), December, January, February, March, April, May, June
Winter Solstice, Major Cold, Rain Water, Vernal Equinox, Grain Rain, Minor Fullness, Summer Solstice, Major Heat
July, August, September, October
End of Heat, Autumn Equinox, Frost Descent, Minor Snow
Dipper 26° (retreat by one quarter), Ox 8, Girl 12 (advance by one), Void 10 (advance by two)
Danger 17 (advance by two), Room 16 (advance by three), Wall 9 (advance by one)
Northern 98 degrees and one quarter
Kui 16, Lou 12 (retreat by one), Stomach 14 (retreat by one), Pleiades 11 (retreat by two)
Tail 16 (retreat by three), Beak 2 (retreat by three), Can 9 (retreat by four)
In the Western region, there are a total of 80 degrees; Well 33 degrees (retreat by three degrees), Ghost 4 degrees, Willow 15 degrees, Star 7 degrees (advance by one degree). Zhang 18 degrees (advance by one degree), Wing 18 degrees (advance by two degrees), Axle 17 degrees (advance by one degree).
Next, we have the Southern region, which has a total of 112 degrees. Angle 12 degrees, Kang 9 degrees (retreat by one degree), Di 15 degrees (retreat by two degrees), Room 5 degrees (retreat by three degrees), Heart 5 degrees (retreat by three degrees), Tail 18 degrees (retreat by three degrees), Ji 11 degrees (retreat by three degrees).
In the east, the total is 75 degrees. The equatorial circle measures 365 degrees and a quarter. The Dipper is 24 degrees (a quarter degree), the Ox is 7 degrees, the Girl is 11 degrees, the Empty is 10 degrees, the Dangerous is 16 degrees, the Room is 18 degrees, and the Wall is 10 degrees.
Finally, in the north, the total comes to 96 degrees and a quarter. The Lyre is 17 degrees, the Lou is 12 degrees, the Weaving Basket is 15 degrees, the Pleiades is 12 degrees, the Big Dipper is 16 degrees, the Beak is 3 degrees, and the Canopy is 8 degrees.
Next, let's discuss the west, which measures 83 degrees this time. The Well is 30 degrees, the Ghost is 4 degrees, the Willow is 14 degrees, the Star is 7 degrees, the Zhang is 17 degrees, the Wing is 19 degrees, and the Zhen is 18 degrees.
In the south, the total is now 190 degrees. The Horn is 13 degrees, the Neck is 10 degrees, the Di is 16 degrees, the House is 5 degrees, the Heart is 5 degrees, the Tail is 18 degrees, and the Winnowing Basket is 10 degrees.
In the east, it is 77 degrees this time. The ecliptic also has a total of 365 degrees and a quarter.
Next, we will explain how to calculate the lengths of daylight and twilight: to calculate the length of daylight, the distance from the ecliptic to the celestial pole must be determined, as well as the instruments and observation tables. To calculate the time of the sundial, multiply the difference in distance from the ecliptic to the celestial pole and the difference in solar terms. For instance, if this distance difference leads to a quarter-hour discrepancy on the sundial, it must be adjusted accordingly. To calculate twilight duration, multiply the celestial degrees by the daytime sundial time, subtract the nighttime sundial time, then divide by 200 to get a fixed degree. Subtract this fixed degree from the celestial degrees, and the remaining time represents the duration of daylight; adding the fixed degree and one degree determines the twilight duration. If the remaining parts are not divisible, multiply by three and classify as "strong." If the remaining part exceeds half, it is calculated as "strong." Three "strong" equals one "weak," and four "weak" equals one degree. "Strong" times two equals "weak." Additionally, the leftover solar degrees are used to calculate "weak strong," and then added accordingly.
