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Written by: Realhistories

Category: Book of Jin （晉書）

Published: 02 January 2025

Created: 02 January 2025

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On the first day, we calculated that there are 2580 days remaining, plus 914 fractions. Using the calculation method, we subtract a full 13 days to determine the remaining days and fractions. The lunar and solar calendars are converted back and forth, with the entry date coming first, followed by the remaining days, which correspond to the midpoint of the month.

Next, we need to consider the rate of profit and loss in the calendar, multiply various profit and loss values by small fractions to obtain differentials, and then add or subtract the profit and loss to the remaining days in the lunar calendar. If there are insufficient or excessive remaining days, adjust the days. Then multiply the remaining days by the profit and loss rate; if the result equals the number of weekdays in a month, use the comprehensive profit and loss value to determine the overtime.

Multiply the difference rate by the remainder of the new moon day, calculate it according to the differential method, then subtract it from the remaining days in the lunar calendar. If it's insufficient, add the number of weekdays in a month, then subtract again, and finally subtract one day. Add the calculated differential to the day of the month and simplify it to obtain the small fractions, thereby determining the entry date at midnight on the new moon day.

Calculate the second day, add one day; the remaining days are 31, and the small fractions are also 31. If the small fractions exceed the total, subtract the number of weekdays in a month. Add another day; if the calendar calculation is completed, subtract the full moon day to get the starting date of the calendar. If it isn't a full moon day, retain it, then add 2720 days, with small fractions being 31, to get the next entry date.

Multiply the total days by the entry rate, the profit and loss at midnight, and the remaining days. If the remaining days exceed half of the number of weekdays, use it as small fractions. Add profits, subtract losses, and adjust the remaining days in the lunar calendar. If the remaining days are insufficient or excessive, adjust them based on the number of weekdays in the month. Multiply the adjusted remaining days by the profit and loss rate; if the result equals the number of weekdays in a month, use the comprehensive profit and loss value to determine the value at midnight.

Multiply the profit and loss rate by the midnight time of the nearest solar term, where 1/200 represents brightness; subtract this value from the profit and loss rate to get darkness, then use the value of the profit and loss at midnight to determine the values of darkness and brightness.

Combine the overtime number and twilight values, divide by 12 to obtain the degree; one-third of the remainder indicates a 'slight' value, while a remainder of less than one minute is considered 'strong'; two 'slight' values together indicate 'weakness'. The result is the degree at which the moon leaves the ecliptic. For the solar calendar, subtract from the ecliptic calendar that accounts for the added days, and for the lunar calendar, use subtraction to obtain the degree at which the moon leaves the ecliptic. Strong is positive, weak is negative; add or subtract the strong and weak values, add like terms, and subtract unlike terms. When subtracting, like terms cancel out, unlike terms are combined; if there is no corresponding term, swap them, adding two strong and one weak, and subtracting one weak.

From the year Ji-Chou of the Shangyuan era to the year Bing-Xu of the Jian'an era in Chinese history, a total of 7378 years have been accumulated. Ji-Chou, Wu-Yin, Ding-Mao, Bing-Chen, Yi-Si, Jia-Wu, Gui-Wei, Ren-Shen, Xin-You, Geng-Xu, Ji-Hai, Wu-Zi, Ding-Chou, Bing-Yin.

This passage outlines the methods of astronomical calculations used in ancient China. We will break it down sentence by sentence and explain what it means in modern terms. First, it defines the celestial bodies associated with the five elements: Jupiter is the Year Star, Mars is the Fiery Star, Saturn is the Earth Star, Venus is the White Star, and Mercury is the Morning Star. Then it mentions the daily movement degrees of these celestial bodies in the sky, and how to use these degrees to calculate the weekly rate, daily rate, to then determine the month and specific dates. This part is like saying: first, we document the daily movement distances of Jupiter, Mars, and other planets, then use this data to calculate their monthly and annual movement. "Chapter-year multiplied by weekly, for the monthly calculation method. Chapter-month multiplied by day, for the month division. Division according to the method, for the month number. Multiply the total number by the monthly calculation method; this is the daily calculation method." These sentences are calculation formulas, in modern terms: multiply the year by the weekly rate to obtain the monthly calculation method, then multiply the monthly calculation method by the number of days to get the month division, divide the month division by the monthly calculation method to get the month number, and finally multiply the total number of days by the monthly calculation method to get the daily calculation method.

Next, it explains how to calculate the big and small remainders of the five planets at new moon, as well as the dates they enter a certain constellation each month. "The big remainder and small remainder of the five planets at new moon. (Using the general method, multiply by the month number, and for the day method, divide by the day number to obtain the big remainder; the leftover is the small remainder. Subtract the big remainder from sixty.)" This means: calculate the remainder of the five planets at the beginning of each month, multiply the total number of days by the month number, then divide by the number of days; the integer part represents the big remainder, while the leftover is the small remainder, then subtract the big remainder from sixty. "The entry date and day remainder of the five planets. (Each uses the general method to multiply by the month remainder, and the combined month method to multiply by the new moon small remainder, then combine and round the numbers, and divide each by the day method.)" This section discusses calculating the date and remainder of the five planets entering a certain constellation using specific methods to determine the final date.

Next is the calculation of the degrees of the five planets and the remainder of those degrees. "The degrees and degree remainder of the five planets. (Subtract the excess to find the degree remainder, multiply by the week number, and round using the day degree method; the result is the degree, and the excess is the degree remainder, subtracting the week number and considering the斗分.)" This part explains how to calculate the degrees of planetary motion and the remainder exceeding the week number, taking into account the斗分 factor. Finally, it lists a series of numbers that are constants used in the calculations, such as the calendar month constant being 7285, the leap month being 7, the chapter month being 235, the year being 12, and so on. These numbers are like coefficients in a formula and are preset.

Next, it separately lists the parameters for Jupiter and Mars, including their orbital rates, daily rates, combined month numbers, month remainders, combined month methods, day degree methods, new moon big remainders, new moon small remainders, entry dates, day remainders, new moon virtual divisions,斗分, degrees, degree remainders, etc. These data result from calculations specific to each celestial body, used to more accurately predict the motion trajectories of the celestial bodies. This section illustrates how we apply the earlier calculation methods to Jupiter and Mars to derive their specific data.

In summary, this text describes an ancient astronomical calculation method that uses a series of complex formulas and constants to predict the motion trajectories of planets. While it may seem cumbersome today, this method was undoubtedly quite advanced in an era lacking precise instruments. It reflects the ancient astronomers' exploration and understanding of the laws of the universe.

The orbital period of Saturn is 3,529 days, with a daily motion of 3,653. In total, it takes approximately 12 months, with an excess of 53,843 days. The total for one month is 67,051 degrees, with a daily degree of 2,785,881. The remainder on the first day of the month is 54 and 534. In a month, the new moon occurs on the 24th day, leaving a remainder of 166,272 degrees. The remainder on the first day of the month is 923, with a division of 511,705 degrees. The degree is 12, with a remainder of 1,733,148.

The orbital period of Venus is 9,022 days, with a daily motion of 7,213. In total, it takes approximately 9 months, with an excess of 152,293 days. The total for one month is 171,418 degrees, with a daily degree of 531,958. The remainder on the first day of the month is 25 and 1,129. In a month, the new moon occurs on the 27th day, leaving a remainder of 56,954 degrees. The remainder on the first day of the month is 328, with a division of 130,819 degrees. The degree is 292, with a remainder of 56,954.

The orbital period of Mercury is 11,561 days, with a daily motion of 1,834. In total, it takes approximately 1 month, with an excess of 211,331 days. The total for one month is 219,659 degrees, with a daily degree of 680,942. The remainder on the first day of the month is 29 and 773. In a month, the new moon occurs on the 28th day, leaving a remainder of 641,967 degrees. The remainder on the first day of the month is 684 and 167,345 degrees. The degree is 57, with a remainder of 641,967.

Enter the date of the Shangyuan Festival for the year you want to calculate, multiply it by the weekly rate; if it can be exactly divided by the daily rate, this is referred to as the combined total, and the part that cannot be divided is called the remainder. Divide the remainder by the weekly rate; if it can be divided, you can determine which year's star combination it corresponds to. If it cannot be divided, keep calculating until it can be divided. Subtract the weekly rate from the remainder to find the degree. For the combination of Venus and Mercury, odd numbers indicate morning appearances, while even numbers indicate evening appearances.

First, let's calculate the lunar aspects. Multiply the number of months by the remainder of the month, and add up the results; if it is enough for a month, count it as a month; if not, record it as the remainder of the month. Then subtract the months that have passed from the total number of months; the remaining is the number of months for this month. Then multiply by the leap month factor; subtract one leap month if it is enough for a leap month, and put the remaining into the year, not included in astronomical calculations; this is the combined month. If it coincides with the leap month transition, adjust using the new moon.

Next, calculate the timing of the star's conjunction with the moon. Multiply the common method by the remainder of the month, then multiply the combined month method by the remainder of the new moon, add these two results, and then simplify the result. If the result is enough for a daily rate, it means the day of the star's conjunction with the moon has arrived; if not, the remaining is the remainder of the day, not included in the new moon calculation.

Then calculate the weekly day. Multiply the weekly day by the degree; count it as one degree if it meets the daily rate; if not, record it as a remainder, starting the count from the Ox Fifth. The above is the method for calculating the star's conjunction with the moon.

Next, let's calculate the other factors. Add up the number of months, and also add up the remainder of the month; count it as a month if it is enough for a combined month; if not, record it for this year; if there is a leap month this year, include it as well; the remaining will be left for the next year; if it is enough for a combined month, leave it for the next two years. Venus and Mercury, if they appear in the morning, will appear in the evening; if they appear in the evening, they will appear in the morning.

Add the remainders from the new moon and the combined month; if it exceeds a month, add a large remainder of twenty-nine and a small remainder of seven hundred seventy-three. If the small remainder is enough for a daily rate, subtract it from the large remainder; the method is the same as before.

Add the entry date and the remainder of the day. If it is enough for one day, record it as one day. If the small remainder from the preceding new moon is sufficient for its fractional division, subtract one day; if the small remainder from the latter remainder exceeds seven hundred seventy-three, subtract twenty-nine days; if not enough, subtract thirty days, and the remainder will be carried over to the next new moon's calculation, which is the entry date. Finally, add the degrees together, and also add the remainder of the degrees; if it is enough for one day, record it as one degree.

Jupiter: Hidden for thirty-two days, totaling three million four hundred eighty-four thousand six hundred forty-six minutes; appeared for three hundred sixty-six days; during hiding, it moved five degrees, two million nine thousand nine hundred fifty-six minutes; during appearance, it moved forty degrees. (Subtracting retrograde twelve degrees, the final movement totals twenty-eight degrees.)

Mars: Hidden for one hundred forty-three days, ninety-seven thousand three hundred thirteen minutes; appeared for six hundred thirty-six days; during hiding, it moved one hundred ten degrees, four hundred seventy-eight thousand nine hundred ninety-eight minutes; during appearance, it moved three hundred twenty degrees. (Subtracting retrograde seventeen degrees, the final movement totals three hundred three degrees.)

Saturn: Hidden for thirty-three days, sixteen thousand six hundred seventy-two minutes; appeared for three hundred forty-five days; during hiding, it moved three degrees, one hundred seventy-three thousand three hundred forty-eight minutes; during appearance, it moved fifteen degrees. (Subtracting retrograde six degrees, the final movement totals nine degrees.)

Venus: In the morning, it was hidden in the east for eighty-two days, eleven thousand three hundred ninety-eight minutes; appeared in the west for two hundred forty-six days. (Subtracting retrograde six degrees, the final movement totals two hundred forty-six degrees.) In the morning, during hiding, it moved one hundred degrees, eleven thousand three hundred ninety-eight minutes; appeared in the east. (The daytime and the west are the same. Hidden for ten days, retrograded eight degrees.)

Mercury: In the morning, it appeared for thirty-three days, covering a distance of six million twelve thousand five hundred five minutes. Then, it appeared in the west for thirty-two days. (Here, one degree of retrograde should be subtracted, resulting in a total movement of thirty-two degrees.) Next, it moved forward sixty-five degrees, still covering a distance of six million twelve thousand five hundred five minutes. Then, it appeared in the east. The degrees it moved in the east are the same as those in the west, it was hidden for eighteen days and retrograded fourteen degrees.

The method of calculating the movement of Mercury is as follows: add up the degrees Mercury moves each day and the remaining degrees of movement, then add up the remaining degrees when it aligns with the Sun. If the remaining degrees meet the daily movement standard, a complete cycle is achieved, as mentioned earlier, and the timing of Mercury's appearance and its degree of movement can be calculated. Multiply the denominator of Mercury's movement by its degrees of appearance. If the remaining degrees meet the daily movement standard, a complete cycle is achieved; if it doesn't divide evenly, count it as a complete cycle if it exceeds half. Then add up the degrees moved each day. If the degrees reach the denominator's standard, one degree is completed. The methods for calculating retrograde and direct motion differ. Multiply the current denominator of movement by the remaining degrees. If the result equals the original denominator, the current movement degrees are obtained. The remaining portion carries over the previous results. If in retrograde, subtract; if the remaining degrees are insufficient, use Dou Su (斗宿) to divide by the degrees, using the movement's denominator as a ratio. Degrees will increase or decrease, mutually constraining each other before and after. The phrase '如盈约满' signifies precise division, while '去及除之，取尽之除也' denotes exhaustive division.

As for Jupiter, it appears with the sun in the morning and then disappears. It moves forward, traveling 1,742,323 minutes in sixteen days, covering 2 degrees and traveling for 3,234,607 minutes. Then it appears in the east in the morning, behind the sun. It moves forward quickly, covering 11/58 of a degree each day, covering a total of 11 degrees in fifty-eight days. Then it continues to move forward, but at a slower speed, covering 9 minutes each day, covering a total of 9 degrees in fifty-eight days. After that, it stops for twenty-five days before starting to move again. It moves backward, covering 1/7 of a degree each day, retreating a total of 12 degrees in eighty-four days. It then stops again for twenty-five days before resuming its forward motion, covering 9/58 of a degree each day, covering a total of 9 degrees in fifty-eight days. It moves forward quickly, covering 11 minutes each day, covering a total of 11 degrees in fifty-eight days, appearing in front of the sun and hiding in the west in the evening. It travels 1,742,323 minutes in sixteen days, covering 2 degrees and traveling for 3,234,607 minutes, and then it meets with the sun again. A complete cycle lasts 398 days, during which it travels a total of 3,484,646 minutes, covering 43 degrees and traveling for 2,509,956 minutes.

In the morning, the sun and Mars appeared together, and Mars went into hiding. For 71 days, it traveled a total of 1,489,868 minutes, which corresponds to moving along the planet's orbit by 55 degrees and 242,860.5 minutes. After that, Mars became visible in the eastern sky, positioned behind the sun. While moving forward, Mars traversed 14/23 degrees daily, covering 112 degrees over the course of 184 days. Next, the forward speed slowed down, walking 12/23 degrees each day, covering 48 degrees in 92 days. Then it stopped, remaining stationary for eleven days. Then it moved in reverse, walking 17/62 degrees each day, moving back 17 degrees in 62 days. It stopped again, remaining stationary for eleven days. Then it resumed its forward motion, walking 1/12 degrees each day, covering 48 degrees in 92 days. Moving forward again, the speed increased, walking 1/14 degrees each day, covering 112 degrees in 184 days. At this stage, it passed in front of the sun, and could be spotted in the western sky at night. After 71 more days, covering a total of 1,489,868 minutes, which corresponds to moving along the planet's orbit by 55 degrees and 242,860.5 minutes, it appeared together with the sun again. This entire cycle lasted a total of 779 days and 973,113 minutes, traversing 414 degrees and 478,998 minutes along the planet's orbit.

Next, let's discuss Saturn. In the morning, the sun and Saturn appeared together, and Saturn began to hide. Then it started moving forward, walking for 16 days, covering a total of 1,122,426.5 minutes, which corresponds to moving along the planet's orbit by 1 degree and 1,995,864.5 minutes. After that, Saturn could be seen in the east in the morning, behind the sun. While moving forward, Saturn walked 3/35 degrees each day, covering 7.5 degrees in 87.5 days. Then it stopped, remaining stationary for 34 days. Then it moved in reverse, walking 1/17 degrees each day, moving back 6 degrees in 102 days. After an additional 34 days, it resumed its forward motion, walking 1/3 degrees each day, covering 7.5 degrees in 87 days. At this point, it moved in front of the sun, and could be spotted in the western sky at night. After 16 more days, covering a total of 1,122,426.5 minutes, which corresponds to moving along the planet's orbit by 1 degree and 1,995,864.5 minutes, it appeared together with the sun again. This entire cycle lasted a total of 378 days and 166,272 minutes, traversing 12 degrees and 1,733,148 minutes along the planet's orbit.

Venus, when it conjoins with the Sun in the morning, first conceals itself, then moves in retrograde, receding four degrees in five days. After that, it can be seen in the east in the morning; at this point, it is positioned behind the Sun. Continuing in retrograde, it moves three-fifths of a degree daily, retreating six degrees over ten days. Then, it remains stationary for eight days. Next, it turns to direct motion, moving slightly slower at three degrees and thirty-three minutes per day, completing thirty-three degrees in forty-six days while moving forward. The speed increases, moving one degree and ninety-one minutes each day, covering one hundred six degrees in ninety-one days. It accelerates further in direct motion, moving one degree and ninety-one minutes and twenty-two seconds each day, completing one hundred thirteen degrees in ninety-one days, at which point it is behind the Sun and appears in the east in the morning. It moves forward for forty-one days, covering one-fifty-six thousand nine hundred fifty-fourth of a circle, while the planet also moves fifty degrees one-fifty-six thousand nine hundred fifty-fourth of a circle, and then it conjoins with the Sun again. One conjunction cycle is two hundred ninety-two days and one-fifty-six thousand nine hundred fifty-fourth of a circle, with the planet following the same cycle.

In the evening, when Venus conjoins with the Sun, it first conceals itself, then moves forward, covering one-fifty-six thousand nine hundred fifty-fourth of a circle in forty-one days, while the planet moves fifty degrees one-fifty-six thousand nine hundred fifty-fourth of a circle, becoming visible in the west in the evening; at this point, it is positioned in front of the Sun. Continuing in direct motion, the speed increases, moving one degree and ninety-one minutes and twenty-two seconds each day, completing one hundred thirteen degrees in ninety-one days. Then, the speed decreases to one degree and fifteen minutes per day, covering one hundred six degrees in ninety-one days, and then it continues moving forward. The speed slows down, moving three degrees and thirty-three minutes per day, completing thirty-three degrees in forty-six days. It then pauses for eight days. Next, it turns to retrograde motion, moving three-fifths of a degree each day, retreating six degrees over ten days; at this point, it is positioned in front of the Sun and appears in the west in the evening. Continuing in retrograde, the speed accelerates, retreating four degrees in five days, and then it conjoins with the Sun again. Two conjunctions complete one cycle, totaling five hundred eighty-four days and one hundred thirteen thousand nine hundred and eight one-fifth of a circle, with the planet following the same cycle.

Mercury, when it conjoins with the Sun in the morning, first goes into hiding, then moves in retrograde, moving back seven degrees over nine days. After that, it becomes visible in the eastern sky in the morning, where it is positioned behind the Sun. It continues moving retrograde, accelerating to retreat one degree each day. Then it stops and remains motionless for two days. Next, it resumes direct motion, moving slightly slower, covering eight-ninths of a degree each day, covering eight degrees in nine days, moving forward. Its speed increases, moving one and a quarter degrees each day, covering twenty-five degrees in twenty days, at which point it is behind the Sun and appears in the eastern sky in the morning. After moving forward for sixteen days, it has completed one part in six million four hundred and nineteen thousand sixty-seventh of a circle, while the planet has moved thirty-two degrees and one part in six million four hundred and nineteen thousand sixty-seventh of a circle, and then it conjoins with the Sun again. The duration of a conjunction cycle is fifty-seven days and one part in six million four hundred and nineteen thousand sixty-seventh of a circle, and the planet has the same cycle.

When it comes to Mercury, when it conjoins with the Sun, it appears to be lying in wait, and then it moves along its orbit, traveling about thirty-two degrees and one part in six million four hundred and nineteen thousand sixty-sixth of a degree in approximately sixteen days. At this point, it can be seen in the western sky in the evening, located in front of the Sun. When it moves quickly, it can travel one and a quarter degrees in one day, covering twenty-five degrees in twenty days. When it moves slowly, it travels about eight-ninths of a degree in one day, covering eight degrees in nine days. If it stops, it remains motionless for two days. If it moves in retrograde, it will retreat one degree in one day, and at this time, it is still in front of the Sun, hiding in the western sky at dusk. When it moves in retrograde, it also moves slowly, retreating seven degrees in nine days, and ultimately it conjoins with the Sun again.

From one conjunction to the next, the entire cycle lasts one hundred fifteen days and one part in six million two thousand five hundred fifty-fifths of a day; this is how Mercury moves. This entire process is known in modern astronomy as Mercury's conjunction cycle.

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Written by: Realhistories

Category: Book of Jin （晉書）

Published: 02 January 2025

Created: 02 January 2025

Hits: 46

Let's first calculate the months from the Lantern Festival to the present. Multiply the combined and differential values of the new moon day (lunar calendar first day) by it separately. Subtract the differential from the combined when a full cycle is completed, and subtract 360 degrees when a full week (360 degrees) is completed. The remaining part that is less than a full week indicates the portion that enters the solar calendar; if it is full, subtract it, and the remaining is the part that enters the lunar calendar. For the remaining part, count one day for each complete lunar month of the week, and calculate these values in days according to the method; if the calculated month combined with the new moon day is less than one day, use the remaining days to represent it.

Add two days; the remaining days are 2580, and the differential is 914. Calculate these values in days according to the method, subtract 13 when it is full, and the remaining is the fractional day. This is how the lunar and solar calendars interrelate: the remaining parts before entering the calendar are in front, while the remaining parts after entering the calendar are behind. This indicates that the moon has reached the midpoint.

List the surplus and deficit sizes of the late or fast calendars separately. Multiply the reference number by the small part to get the differential, and adjust the remaining days of the lunar and solar calendars based on the surplus and deficit. If the surplus and deficit are insufficient, adjust the number of days to determine. Multiply the determined days by the profit and loss rate, counting one for every full month of the week, and use the total profit and loss to determine the additional time.

Multiply the difference rate by the small remainder of the new moon day, get one according to the method of differential, and use it to reduce the remaining days of entering the calendar. If it is insufficient, add one week of days, then subtract one day. Add the fractional day to its fraction, simplify the differential by the reference number to get the small part; this marks the moment when the new moon is recorded in the calendar at midnight.

To find the second day, add one day; the remaining days are 31, and the small part is 31. Subtract the small part from the remainder according to the reference number; subtract one week of days when the remainder reaches a full week, and add one more day. The calendar is calculated, and subtract the full fractional day from the remaining days; this is the beginning of entering the calendar. The remaining days that are not a full fractional day are retained; add the remaining 272, and the small part is 31; this is the time to enter the next calendar.

Multiply the common number by the surplus and remaining of the late and fast calendars at midnight; consider half a week as the small part when the remainder reaches a full week. Adjust the remaining days of the lunar and solar calendars with the surplus and deficit; if the surplus and deficit are insufficient, adjust the number of days using one week of days. Multiply the determined days by the profit and loss rate, counting one for every full month of the week, and use the total profit and loss to determine the midnight constant.

Multiply the profit and loss rate by the measured values from the recent solar terms at night; 1/200 is bright. Subtract it from the profit and loss rate to get dark; the profit and loss at midnight is the dark-bright constant.

The number of overtime hours or the fixed number of twilight (昏明定数) is divided by 12 to obtain the degree; one third of the remainder indicates weakness, while anything less than one indicates strength; two weak values together indicate weakness. The degree obtained is the degree to which the moon deviates from the ecliptic. The solar calendar uses the ecliptic corresponding to the day of the month to determine the extreme position, and the lunar calendar uses subtraction, which is the degree of the moon leaving the extreme. Strength is positive while weakness is negative; strong and weak values are added together, the same names are added, and different names are subtracted. When subtracting, subtract the same names, add the different names; if there is no corresponding match, they cancel each other out—two strong values add one weak and subtract one weak.

From the year Ji Chou in the First Yuan to the year Bing Xu in the eleventh year of Jian'an, a total of 7378 years have been accumulated.

Ji Chou, Wu Yin, Ding Mao, Bing Chen, Yi Si, Jia Wu, Gui Wei, Ren Shen, Xin You, Geng Xu, Ji Hai, Wu Zi, Ding Chou, Bing Yin—these are all years, and I will not explain further. Next are the Five Elements: Wood corresponds to Jupiter, Fire corresponds to Mars, Earth corresponds to Saturn, Metal corresponds to Venus, and Water corresponds to Mercury. Each star has its own degree of movement in the sky every day, and these degrees and cycles (weekly rate, daily rate) have fixed algorithms. Multiply the weekly rate by the year to obtain the monthly method; then multiply the monthly method by the number of days to get the month; divide the month by the monthly method to get the month. Then multiply the month by the common method to obtain the daily method. Finally, multiply the Dipper (斗分) by the weekly rate to calculate the Dipper value. It is noted that the daily method is obtained by multiplying the record method by the weekly rate, so here all calculations are done in minutes.

Next, we calculate the New Moon and Old Moon phases for the Five Stars. Multiply the common method by the month, then divide by the daily method; the quotient is the old moon, and the remainder is the small moon. Then subtract 60 from the old moon. Calculate the entry and exit of the five stars in the month and the day. Multiply the common method by the month remainder, then multiply by the combined month method with the new moon, add these two results, simplify, and then divide by the daily method to obtain the result. Finally, calculate the degree and degree remainder of the five stars. Subtract the excess degrees; the remaining value is the degree remainder in minutes. Then multiply by the week and divide by the daily method; the quotient is the degree, and the remainder is the degree remainder. If the degree exceeds the weekly total, subtract both the weekly total and the Dipper value.

Key parameters include: Record Month: 7285, Leap Month: 7, Chapter Month: 235, Year Middle: 12, Common Method: 43026, Daily Method: 1457, Meeting Count: 47, Weekly Total: 215130, Dipper Value: 145.

Here are the specific data for Jupiter: the orbital period is 6722, the daily circumference is 7341, the total number of months is 13, the lunar excess is 64810, the total lunar cycle is 127718, the daily method is 3959258, the major lunar surplus is 23, the minor lunar surplus is 1370, the day the month begins is 15, the day surplus is 3484646, the lunar virtual part is 150, the lunar part is 974690, the degree is 33, and the degree surplus is 2509956.

Data for Mars: the orbital period is 3447, the daily circumference is 7271, the total number of months is 26, the lunar excess is 25627, the total lunar cycle is 64733, the daily method is 2006723, the major lunar surplus is 47, the minor lunar surplus is 1157, the day the month begins is 12, the day surplus is 973113, the lunar virtual part is 300, the lunar part is 494115, the degree is 48.

Wow, all these numbers are making my head spin! Is this some kind of astronomical calendar calculation? Let me translate it into plain language for you sentence by sentence.

First paragraph: The total is 1991760. Saturn: It makes 3529 orbits in a year. It makes 3653 orbits per day. A year is calculated as 12 months. There are 53843 extra orbits in a month. In total, that's 6751 orbits for the month. Calculated per day, it is a total of 278581 orbits. Major lunar surplus 54, minor lunar surplus 534. On the 24th of each month. There are 166272 extra orbits per day. Lunar virtual part 923. Lunar part 51175. Degree 12. The total is 1733148.

Second paragraph: Venus: It makes 9022 orbits in a year. It makes 7213 orbits per day. A year is calculated as 9 months. There are 152293 extra orbits in a month. In total, that's 171418 orbits for the month. Calculated per day, it is a total of 5313958 orbits. Major lunar surplus 25, minor lunar surplus 1129. On the 27th of each month. There are 56954 extra orbits per day. Lunar virtual part 328. Lunar part 1308190. Degree 292. The total is 56954.

Third paragraph: Mercury: It makes 11561 orbits in a year. It makes 1834 orbits per day. A year is calculated as 1 month. There are 211331 extra orbits in a month. In total, that's 219659 orbits for the month. Calculated per day, it is a total of 6809429 orbits. Major lunar surplus 29, minor lunar surplus 773. On the 28th of each month. There are 6419967 extra orbits per day. Lunar virtual part 684. Lunar part 1676345. Degree 57. The total is 6419967.

Last paragraph: Substitute the starting point of the year you want to calculate, multiply it by the orbital period; the portion that divides evenly by the daily circumference is called "integral sum," and the part that cannot be divided is called "total surplus." Divide the total surplus by the orbital period, and you will get an integer, representing which year of the past it is; if you cannot get an integer, then it is the current year. Subtract the orbital period from the total surplus to get the degree fraction. For Venus and Mercury, odd integral sums indicate morning, while even sums indicate evening.

In summary, this passage describes a complex method of calendar calculation, with a multitude of numbers used to calculate the orbits and times of the planets. Specific details require professional astronomical knowledge to fully understand. This is simply an ancient astronomical calculation program! Let's first calculate the conjunction date of the constellations. Multiply the number of months by the remainder of the month, add them together; if it exceeds the standard for a month, count it as one month; if not enough, note the remaining month remainder. Then subtract the total months from this month remainder; the remainder is the month of entry. Next, consider the impact of leap months; if the number of months is enough for a leap month, subtract it from the month of entry, and the remaining part is then deducted from the year. The remaining is the conjunction month outside the Tianzheng calculation. If a leap month transition occurs, adjust it using the new moon. Next, multiply the month remainder by the common method, then multiply the small remainder of the conjunction day, add these two results together, and then simplify the result. If the result is just enough for a daily method, that is the conjunction date of the constellations. If not enough, the remaining is the remainder of the day, recorded outside the Tianzheng calculation. Then, multiply the degrees and minutes by the day; if the result is enough for a daily method, record one degree; if not enough, record the remaining degrees and minutes, starting from the first five degrees before the Ox. The above outlines the method for calculating the conjunction date of the constellations. Next, calculate the conjunction cycle of the planets. Add the number of months; also add the remainder of the month; if it exceeds the standard for a month, record one month; if it does not exceed one year, record it in this year; if it exceeds one year, subtract it; if there is a leap month, consider it; the remaining is recorded in the next year; if it exceeds again, record it in the next two years. For Venus and Mercury, if added in the morning, it becomes evening; if added in the evening, it becomes morning. (This refers to the conversion between the morning star and the evening star of Venus and Mercury). Then, combine the large and small remainders of the new moon and the conjunction. If it exceeds a standard month, add the large and small remainders, adding 29 to the large remainder and 773 to the small remainder. If the small remainder exceeds the daily method, subtract it from the large remainder, using the same method as outlined earlier. Add the entry date of the month and the remainder of the day; if it exceeds the daily method, record one day. If the small remainder of the previous conjunction is just enough for its virtual division, subtract one day. If the small remainder exceeds 773, subtract 29 days; if not enough, subtract 30 days; the remainder is the entry date of the subsequent conjunction. Finally, add up the degrees; also add up the remainder of the degrees; if it exceeds the daily method, record one degree. The following are the operational data for Jupiter, Mars, Saturn, and Venus:

**Jupiter:** Retrograde for 32 days, 3,484,646 minutes; Direct for 366 days; Retrograde 5 degrees, 2,509,956 minutes; Direct 40 degrees. (Retrograde 12 degrees, actual movement 28 degrees.)

**Mars:** Retrograde for 143 days, 973,113 minutes; Direct for 636 days; Retrograde 110 degrees, 478,998 minutes; Direct 320 degrees. (Retrograde 17 degrees, actual movement 303 degrees.)

**Saturn:** Retrograde for 33 days, 166,272 minutes; Direct for 345 days; Retrograde 3 degrees, 1,733,148 minutes; Direct 15 degrees. (Retrograde 6 degrees, actual movement 9 degrees.)

**Venus:** Morning retrograde in the eastern sky for 82 days, 113,908 minutes; Direct in the west for 246 days. (Retrograde 6 degrees, actual movement 240 degrees.) Morning retrograde 100 degrees, 113,908 minutes; appearing in the east. (The number of days in the west is the same; retrograde for 10 days, retrograde 8 degrees.)

Mercury, in the morning it hides, having traveled a total of 6,122,555 minutes over 33 days. Then it appears in the west for a total of 32 days. (First, subtract one degree; ultimately, it is calculated as having moved 32 degrees.) It travels underground for a total of 65 degrees, also 6,122,555 minutes. After that, it appears in the east. The degrees it appears in the east are the same as those in the west, hiding for 18 days, retrograding 14 degrees.

Calculate the days and degrees that Mercury hides, then add the remaining degrees after it conjoins with the sun. If the remaining degrees are enough for a cycle, calculate it as previously done to determine the days and degrees Mercury is visible. Multiply the denominator of Mercury's movement by the degrees it appears; if the remaining degrees are enough for a cycle, count it as one cycle; if not enough but more than half, also count it as one cycle; then add the calculated degrees to its movement degrees; if the degrees are enough for a cycle, add one degree. The calculation methods for direct and retrograde are different; multiply the current running denominator by the remaining degrees; if the result equals the original denominator, that is its current running degrees. The remaining (referring to the remaining degrees) continues to use the previous result for calculation, and for retrograde, it needs to be subtracted. If the degrees it moves underground are not enough for a cycle, divide the remaining degrees using a 'dou' (a traditional Chinese measurement unit), using the running denominator as a ratio; the degrees will increase or decrease, and the increases and decreases must balance each other out. Any reference to "like fullness approaching fullness" refers to division seeking exact values; "go and divide it, take the complete value" refers to division for taking the complete value.

Jupiter, in the morning, appeared alongside the sun and then concealed itself. It was in direct motion, concealing itself for 16 days, traversing 1,742,323 minutes, and the planet moved 2 degrees and 3,234,467 minutes. Then it appeared in the eastern sky behind the sun. In direct motion, it moved quickly, covering 11 degrees in 58 days. Then it continued in direct motion, but at a slower speed, moving 9 degrees each day, covering 9 degrees in 58 days. After that, it stopped for 25 days before turning. In retrograde motion, it retreated 1/7 of a degree each day, moving back 12 degrees over 84 days. It stopped again and after 25 days resumed direct motion, moving 9/58 of a degree each day, covering 9 degrees in 58 days. In direct motion, it moved quickly, covering 11 degrees each day, traveling 11 degrees in 58 days, appearing in front of the sun and concealing itself in the western sky at night. It concealed itself for 16 days, traversed 1,742,323 minutes, and the planet moved 2 degrees and 3,234,467 minutes, then it reappeared alongside the sun. One cycle ended, totaling 398 days, traversing 3,484,646 minutes, and the planet moved 43 degrees and 2,509,956 minutes.

The sun: In the morning, it appeared with the sun and then concealed itself. Next was direct motion, lasting 71 days, traversing 1,489,868 minutes, meaning the planet moved 55 degrees and 242,860.5 minutes. Then it could be seen in the east in the morning, behind the sun. During direct motion, it moved 14/23 of a degree each day, covering 112 degrees in 184 days. Continuing in direct motion, but at a slower speed, it moved 12/23 of a degree each day, covering 48 degrees in 92 days. Then it stopped for 11 days. After that, it moved in retrograde, retreating 17/62 of a degree each day, moving back 17 degrees over 62 days. It stopped again for 11 days and then resumed direct motion, moving 12 degrees each day, covering 48 degrees in 92 days. It continued in direct motion, moving faster, covering 14 degrees each day, traveling 112 degrees in 184 days, at which point it was in front of the sun, concealing itself in the western sky at night. After 71 days, it traversed 1,489,868 minutes, and the planet moved 55 degrees and 242,860.5 minutes, then it reappeared alongside the sun. One cycle ended, totaling 779 days and 973,113 minutes, and the planet moved 414 degrees and 478,998 minutes.

Mars: It appears in the morning with the sun, then it disappears. Next is direct motion, lasting 16 days, traversing a distance equivalent to 1,122,426.5 minutes, which corresponds to 1 degree and 1,199,864.5 minutes of planetary motion. Then it can be seen in the eastern sky in the morning, just behind the sun. During direct motion, it travels 35 minutes of arc per day, covering 7.5 degrees in 87.5 days. After 34 days, it resumes direct motion. It then goes into retrograde, moving 17 minutes per day, retreating 6 degrees after 102 days. After another 34 days, it starts direct motion again, moving 3 minutes per day, covering 7.5 degrees in 87 days; at this point, it is in front of the sun and disappears in the western sky at night. After 16 days, having traversed 1,122,426.5 minutes of distance, the planet has moved 1 degree and 1,199,864.5 minutes, and then it appears with the sun again. One cycle is completed, totaling 378 days and 166,272 minutes, with the planet traveling 12 degrees and 1,733,148 minutes.

Venus, when it meets with the sun in the morning, first disappears (goes into retrograde motion), then goes into retrograde, retreating 4 degrees in 5 days, after which it can be seen in the east in the morning, behind the sun. It continues to retrograde, moving 1.67 degrees per day, retreating 6 degrees in 10 days. Then it remains stationary for 8 days. It then turns (rotates), starting direct motion, moving slowly, 46/33 degrees per day, covering 33 degrees in 46 days. As it speeds up, it moves 6.07 degrees per day, covering 160 degrees in 91 days. Then it accelerates further during direct motion, moving 4.14 degrees per day, covering 113 degrees in 91 days; at this point, it is behind the sun and appears in the east in the morning. Continuing direct motion, it traverses 1/56 of a circle in 41 days, with the planet also covering 50 degrees in 41 days, then it meets with the sun again. One conjunction is completed, totaling 292 days and 1/56 of a circle, with the planet traveling the same distance.

When Venus conjoins with the Sun in the evening, it first conceals itself (伏) and then moves forward. In forty-one days, it travels one fifty-six-thousand nine hundred fifty-fourth of a complete circle, moving a total of fifty degrees and one fifty-six-thousand nine hundred fifty-fourth of a complete circle, and then can be seen in the western sky in front of the Sun at night. Continuing to move forward, it speeds up (顺，疾), traveling two twenty-second degrees each day, and in ninety-one days, it covers one hundred thirteen degrees. Then it slows down (更顺，减疾), traveling one-fifteenth of a degree each day, and in ninety-one days, it travels one hundred six degrees, then continues moving forward. The speed decreases (迟), covering three thirty-sixths of a degree each day, and in forty-six days, it travels thirty-three degrees. After that, it stops (留) for eight days, remaining stationary. Then it turns (旋) and begins its retrograde motion, moving backward three-fifths of a degree each day, and in ten days, it moves backward six degrees; at this point, it is in front of the Sun, appearing in the western sky at night. Continuing to move backward, it speeds up (逆，疾), moving backward four degrees in five days, and then it conjoins with the Sun again. Two conjunctions constitute one cycle, totaling five hundred eighty-four days and one hundred thirteen thousand nine hundred eight one-hundredth of a complete circle; the distance covered by the planet is the same.

As for Mercury, when it conjoins with the Sun in the morning, it first conceals itself (伏) and then moves backward, retreating seven degrees in nine days, after which it becomes visible in the eastern sky, positioned behind the Sun in the morning. Continuing to move backward, it speeds up (更逆，疾), moving backward one degree each day. Then it stops (留) for two days, remaining stationary. Next, it turns (旋) and begins to move forward, moving slowly (迟), covering eight-ninths of a degree each day, and in nine days, it covers eight degrees. The speed increases (疾), moving one and one-fourth degrees each day, and in twenty days, it travels twenty-five degrees; at this point, it is behind the Sun, appearing in the eastern sky in the morning. Continuing to move forward (顺), it travels one six hundred forty-one million nine hundred sixty-seven thousandth of a complete circle in sixteen days, and the planet also covers thirty-two degrees one six hundred forty-one million nine hundred sixty-seven thousandth of a complete circle, before conjoining with the Sun again. One conjunction totals fifty-seven days and six hundred forty-one million nine hundred sixty-seven thousandth of a complete circle; the distance covered by the planet is equivalent.

The sun has set, and it has encountered Mercury. The pattern of Mercury's movement is: when moving forward, it can cover 32 degrees and 641,960,667/1 of a degree in sixteen days; at this time, it can be seen in the evening to the west, positioned ahead of the sun. When moving forward quickly, it can cover 1.25 degrees in a day, and 25 degrees in twenty days. When moving forward slowly, it covers 8/7 degrees in a day, and 8 degrees in nine days. If Mercury stops, it remains stationary for two days. If Mercury moves backward, it will retreat one degree each day, and it can be seen in the west in the evening, positioned ahead of the sun. When moving backward slowly, it retreats seven degrees over nine days, and then it encounters the sun again. From one encounter to the next, it takes a total of 115 days and 601,255,005/1 of a day, and Mercury's movement is like this.

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Written by: Realhistories

Category: Book of Jin （晉書）

Published: 02 January 2025

Created: 02 January 2025

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This text describes ancient calendar calculation methods, which are quite professional and complex. Let's explain it in modern language, sentence by sentence.

First, "The lunar calendar has four tables, three trajectories, intersecting and dividing the celestial sphere, dividing by the lunar rate to obtain the days of the calendar." This means: based on the four tables of the moon's movement (possibly referring to different observational data), as well as the three trajectories of the moon's movement (possibly referring to different calculation methods), combining them and calculating according to the speed of the moon's movement, you can obtain the date of each day. This part explains the basics of calendar calculation.

Next, "Multiply the lunar cycles and the new moons together, like a conjunction of the new moons. Multiply the common factors by the meeting factor; the remainder is like the meeting factor, the retreat is the separation. Proceed with the month, for the daily progress. The meeting factor is one, the difference rate is also." These sentences are more abstract. "Weeks" refer to a year, "weeks and new moons together" refer to the cycle from new moon to full moon and back to new moon, "meeting of the months" may refer to a specific point in time. In general, this passage describes a calculation method, through some multiplication and division operations, to obtain a "difference rate" for subsequent calculations. This part is the core step of the calculation method.

Then there is the table section, which lists the "gain and loss ratio" and "joint number" for each day. This part involves specific numerical calculations, which are difficult to directly translate into modern language because they depend on specific parameters and calculation methods of ancient calendars. In simple terms, every day is adjusted based on specific rules to more accurately reflect the moon's movement. For example, "on the first day, reduce by one day and increase by seventeen units" means reducing one day on the first day and adding seventeen of some unit (unit not specified). The following "second day," "third day," and so on, all follow similar adjustment rules. The meaning of these numbers needs to be understood in conjunction with the knowledge of ancient calendars at that time, which will not be explained here.

"At the limit, a remainder of three thousand nine hundred twelve, with a correction factor of one thousand seven hundred fifty-two. This is referred to as the later limit." This sentence means: after calculating to a certain stage, a remainder of three thousand nine hundred twelve is obtained, along with a correction factor of one thousand seven hundred fifty-two, and this stage is referred to as the "later limit." This part is a record of the calculation results.

"The method for determining the final result involves taking the month of the conjunction and adding it to the month of the full moon phase, then multiplying the remaining new moon and its fractional parts by each other. The fractional part is adjusted according to the method derived from the new moon, and the full moon phase is subtracted from the complete cycle, leaving the remainder that does not complete the cycle is entered into the solar calendar; the remainder is then entered into the lunar calendar. The remainder, calculated as one day for each lunar cycle, is sought to determine the new moon's entry into the calendar, which does not completely account for the extra days." This passage describes the final step in calendar calculation. Through a series of computations, it ultimately determines whether the day belongs to the solar or lunar calendar and calculates the remaining date. This part is the final step of the calendar calculation and is also the most difficult to understand.

Finally, by adding two days, the total remaining days becomes two thousand five hundred eighty, with a fractional part of nine hundred fourteen. Using the method, this forms a day; subtracting thirteen gives a remainder that is divided as fractional days. The solar and lunar calendars interweave, with the entry into the calendar occurring before the remainder reaches the limit, and the remainder after the limit being the midpoint of the lunar cycle. Each is placed into the slow and fast calendars, with the full and reduced cycles calculated; the total days multiplied by the small fractions form the fractional parts, with the increase and decrease adjusting the solar and lunar days. The excess days are adjusted by either advancing or retreating the days. The remaining days are defined by multiplying the gain and loss rate, as one day corresponds to each lunar cycle, with the gain and loss combined to define the time.

In summary, this text describes a very complex ancient calendar calculation method, with underlying mathematical principles and astronomical observation knowledge that are quite profound. Explaining it in modern vernacular is relatively difficult and requires an in-depth understanding of ancient calendars for complete comprehension.

First, multiplying the difference rate by the decimal part of the remaining new moon day, similar to calculus, yields a numerical value, which is then subtracted from the remaining days calculated in the calendar. If it is insufficient to subtract, a full month’s cycle is added before subtracting, and then one day is retreated. The resulting remaining days are then added to their fractional part, simplifying the small fractions to obtain the moment when the new moon phase is recorded in the calendar at midnight.

Next, to find the next day, add one day; the remaining days total 31, and the small fraction is also 31. If the small fraction exceeds the total number of days, subtract a full month’s cycle. Then add one more day; the calendar calculation continues until the end. If the remaining days exceed the fractional days, subtract the fractional days, which gives the starting moment of entry into the calendar. If the remaining days do not complete the fractional days, keep them, adding 2720, with the small fraction being 31, thereby determining the moment of transition into the next calendar."

Using the total number multiplied by the lunar phases of fullness and contraction in the calendar, as well as the remaining days, if the remaining days exceed half a week, it is considered a subdivision. Adding the full number to the contraction value, then subtracting the Yin and Yang residual days, if there is either a surplus or a deficiency in the residual days, adjust the days using the lunar month and the solar week. Multiply the established residual days by the profit and loss rate; if it equals the lunar month and solar week, use the overall profit and loss value as the established value for midnight.

Multiply the profit and loss rate by the time of the nearest solar term at night, divide by 200 to determine the sunrise time, subtract this value from the profit and loss rate to determine the sunset time, and use the profit and loss midnight number as the established value for sunrise and sunset.

If adding the time equals the established value for sunrise and sunset, divide by 12 to get the degree; one-third of the remainder indicates a deficiency, less than one minute indicates strength, and two deficiencies indicate weakness. The resulting value is the angle of the moon's deviation from the ecliptic. For the solar calendar, subtract the extreme from the ecliptic position of the day; for the lunar calendar, add the extreme to the ecliptic position of the day to get the degree of the moon leaving the extreme. Positive for strength, negative for weakness, combine the strengths and weaknesses, subtract items of the same type and add items of different types. When subtracting, subtract items of the same type, add items of different types; if there is no complement, add two strengths and subtract one weakness.

Beginning from the Ji-Chou year of the Yuan Dynasty to the Bing-Xu year in the eleventh year of Jian'an, a total of 7378 years:

- Ji-Chou

- Wu-Yin

- Ding-Mao

- Bing-Chen

- Yi-Si

- Jia-Wu

- Gui-Wei

- Ren-Shen

- Xin-You

- Geng-Xu

- Ji-Hai

- Wu-Zi

- Ding-Chou

- Bing-Yin

The Five Elements: Wood (Jupiter), Fire (Mars), Earth (Saturn), Metal (Venus), and Water (Mercury). Use their respective end days and celestial degrees to obtain the weekly rate and daily rate. Multiply the chapter by the weekly rate to obtain the monthly method. Multiply the chapter by the monthly method to obtain the monthly division. Divide the monthly division by the monthly method to get the month number. Multiply the total number by the monthly method to get the daily method. Multiply the Dipper by the weekly rate to get the Dipper. (The daily method uses the record method multiplied by the weekly rate, so here we also use division to multiply.)

The large and small remainders of the five stars on the first day of the lunar month. (Multiply the total method by the month number separately, divide by the daily method separately, get the large remainder, and the remainder that cannot be divided is the small remainder. Subtract the large remainder from 60.)

The five stars enter the month and the daily remainder. (Multiply the total method by the month remainder separately, multiply the combined month method by the first day of the month small remainder, add the two, simplify with the total, divide by the daily method, and you will get the results.)

This passage records some data calculated by ancient astronomical calendars; let's break it down sentence by sentence in contemporary language.

First, "the degrees and remaining degrees of the five stars. (Subtract the integer part for the remaining degree part, multiply by the number of weeks, and approximate using the daily degree method; the result gives the degree, while the remainder indicates the remaining degree. If it exceeds the number of weeks, subtract the weeks and add the 斗分.)" This sentence means to calculate the degrees and remaining degrees of the five stars. The calculation method is: first subtract the integer part, what remains is the remaining degree part, then multiply the remaining degree part by the number of weeks, and divide by the daily degree method; the quotient gives the degree, while the remainder indicates the remaining degree. If the remaining degree exceeds the number of weeks, subtract the weeks and add the 斗分. In short, this is a complex astronomical calculation method that involves many astronomical units.

Next, "the month count totals seven thousand two hundred eighty-five; the leap month count is seven; the month total is two hundred thirty-five; there are twelve in a year; the total method is forty-three thousand twenty-six; the daily method is one thousand four hundred fifty-seven; the meeting number is forty-seven; the number of weeks is two hundred fifteen thousand one hundred thirty; the 斗分 is one hundred forty-five." These are some basic parameters in astronomical calendars, such as how many months are in a year (month count), how many leap months there are (leap month count), etc. These numbers represent different astronomical cycles and units, serving as the basis for subsequent calculations.

"Jupiter: the weekly rate totals six thousand seven hundred twenty-two; the daily rate is seven thousand three hundred forty-one; the total month count is thirteen; the remaining month count is sixty-four thousand eight hundred one; the total month method is one hundred twenty-seven thousand seven hundred eighteen; the daily degree method is three million nine hundred fifty-nine thousand two hundred fifty-eight; the large new moon remainder is twenty-three; the small new moon remainder is one thousand three hundred seven; the day of entry into the new month is fifteen; the daily remainder is three million four hundred eighty-four thousand six hundred forty-six; the new moon 虚分 is one hundred fifty; the 斗分 is nine hundred seventy-four thousand six hundred ninety; the degree is thirty-three; the remaining degree is two million nine hundred ninety-nine thousand five hundred fifty-six." This section is about specific data for Jupiter, including its weekly rate, daily rate, etc. These data are related to Jupiter's orbital period and position, used for more accurate calculations of Jupiter's trajectory. Behind these numbers are complex calculations; we only need to understand that these represent specific parameters of Jupiter's movement.

"Fire: Orbital circumference, three thousand four hundred seventy; orbital daily rate, seven thousand two hundred seventy-one; total lunar cycles, twenty-six; remaining lunar months, twenty-five thousand six hundred twenty-seven; combined lunar calculation method, sixty-four thousand seven hundred thirty-three; daily degree calculation method, two million six thousand seven hundred twenty-three; large new moon excess, forty-seven; small new moon excess, one thousand one hundred fifty-seven; day of entry into the lunar month, twelve; remaining days, nine hundred seventy-three thousand one hundred thirteen; virtual division of the new moon, three hundred; 斗分, four hundred ninety-four thousand one hundred fifteen; number of degrees, forty-eight; remaining degrees, one hundred ninety-nine million one thousand seven hundred six." This section is about Mars, similar data describing the parameters of Mars' trajectory.

"Earth: Orbital circumference, three thousand five hundred twenty-nine; orbital daily rate, three thousand six hundred fifty-three; total lunar cycles, twelve; remaining lunar months, fifty-three thousand eight hundred forty-three; combined lunar calculation method, sixty-seven thousand fifty-one; daily degree calculation method, two hundred seventy-eight thousand five hundred eighty-one; large new moon excess, fifty-four; small new moon excess, five hundred thirty-four; day of entry into the lunar month, twenty-four; remaining days, one hundred sixty-six thousand two hundred seventy-two; virtual division of the new moon, nine hundred twenty-three; 斗分, five hundred eleven thousand seven hundred five; number of degrees, twelve; remaining degrees, one hundred seventy-three thousand three hundred forty-eight." This section contains the calculation data for Saturn, similar to the previous data for Jupiter and Mars.

"Metal: Orbital circumference, nine thousand twenty-two; orbital daily rate, seven thousand two hundred thirteen; total lunar cycles, nine." Finally, this is the data for Venus, with only part of the parameters listed. These data were derived by ancient astronomers through long-term observation and complex calculations, reflecting the exquisite level of ancient astronomical calendars. Although we now find it difficult to understand the specific processes of these calculations, these numbers themselves record the achievements of ancient astronomical research.

A month has passed, and the value is one hundred fifty-two thousand two hundred ninety-three.

According to the combined lunar calculation method, the result is one hundred seventy-one thousand four hundred eighteen.

Using the daily degree calculation method, the result is five hundred thirty-one thousand three hundred ninety-eight.

The large new moon excess is twenty-five.

The small new moon excess is one thousand one hundred twenty-nine.

The day of entry into the lunar month is twenty-seven.

The remaining days are fifty-six thousand nine hundred fifty-four.

The virtual division of the new moon is three hundred twenty-eight.

斗分 is one hundred thirty thousand eight hundred ninety.

The number of degrees is two hundred ninety-two.

The remaining degrees are fifty-six thousand nine hundred fifty-four.

The circumference of water (水) is eleven thousand five hundred and sixty-one. The circumference of the sun (日) is one thousand eight hundred and thirty-four. The total number of months is one (合月数是一). Next, after one month, the value is two hundred and eleven thousand three hundred and thirty-one (数值是二十一万一千三百三十一). According to the combined month method (合月法), the result is two hundred and nineteen thousand six hundred and fifty-nine. Using the daily method (日度法), the result is six hundred and eighty thousand nine hundred and forty-nine (六百八十万九千四百二十九). The large remainder of the new moon (朔大余) is twenty-nine. The small remainder of the new moon (朔小余) is seven hundred and seventy-three. The day of entering the month (入月日) is twenty-eight. The day's remainder (日余) is six hundred and forty-one thousand nine hundred and sixty-seven. The moon's virtual fraction is six hundred and eighty-four. The Dipper fraction is one hundred and sixty-seven thousand six hundred and forty-five. The degree is fifty-seven. The degree's remainder is six hundred and forty-one thousand nine hundred and sixty-seven.

Okay, next is the calculation method: first, multiply the total number of years by the circumference; if it can be evenly divided by the daily rate to get one, it is called the integrated total, and the remaining part that cannot be divided is called the total remainder. Divide the total remainder by the circumference; if it can be evenly divided by one, it is the star-integrated previous year; if it can be evenly divided by two, it is the total previous year; if it cannot be divided, it is the total year. Subtract the circumference from the total remainder to get the minute fraction. The combination of gold and water, odd numbers are morning, even numbers are evening.

Then, multiply the number of months and the month's remainder separately by the integrated total; if the result can be evenly divided by the total month method, you get the month, and the remaining part that cannot be divided is the month's remainder. Subtract the integrated month from the accumulated month; the remaining is the entry month. Multiply it by the chapter leap; if it can be evenly divided by the chapter month to get a leap, subtract the entry month, and subtract the remaining part in the year; this is called the correct calculation, total month. If it is a leap year, use the moon to control it.

Multiply the common method by the month's remainder, multiply the total month method by the moon's small remainder, then divide by the number of meetings; if the result can be evenly divided by the daily method to get one, it is the star-integrated day of the month. The remaining part is the day's remainder; this is called the outer calculation of the moon. Multiply the circumference by the degree fraction; if it can be evenly divided by the daily method to get one degree, the remaining part is the remainder; this degree is calculated from the five in front of the ox.

The above is the method for calculating the star conjunction (求星合的方法). Finally, add the number of months and their remainders together; if it can be evenly divided by the total month method to get one month, then it is the total year; if it can be evenly divided, subtract it; if it is a leap year, calculate the leap year; the remaining is the later year; if it is full, then it is the next two years. Gold and water add morning to get evening, add evening to get morning.

First, let's talk about the method for calculating the lunar phase and remainder. Add the phase and remainder of the new moon together; if the total exceeds a month, add either twenty-nine days (large remainder) or seven hundred seventy-three parts (small remainder). When the small remainder reaches its maximum, calculate it using the method for large remainders, and the remaining steps follow the same procedure. Next, calculate the entry date and daily remainder. Add the entry date and daily remainder together; if the remainder is sufficient for a full day, add one day. If the small remainder exactly fills the gap from the previous new moon, then subtract one day; if the small remainder exceeds seven hundred seventy-three, subtract twenty-nine days; if it is insufficient, subtract thirty days, and the remaining amount will be calculated using the following method for determining the entry date. Finally, calculate the angles. Add the angles together, including their remainders; if the total is equivalent to a full day's worth of angles, add one angle. Below are the operational data for Jupiter, Mars, Saturn, Venus, and Mercury:

Jupiter: observed for 32 days, 3,484,646 minutes; visible for 366 days; observed running 5 degrees, 2,509,956 minutes; visible running 40 degrees (accounting for retrograde motion of 12 degrees, actual running 28 degrees).

Mars: observed for 143 days, 973,113 minutes; visible for 636 days; observed running 110 degrees, 478,998 minutes; visible running 320 degrees (accounting for retrograde motion of 17 degrees, actual running 303 degrees).

Saturn: observed for 33 days, 166,272 minutes; visible for 345 days; observed running 3 degrees, 1,733,148 minutes; visible running 15 degrees (accounting for retrograde motion of 6 degrees, actual running 9 degrees).

Venus: observed in the east in the morning for 82 days, 113,908 minutes; visible in the west for 246 days (accounting for retrograde motion of 6 degrees, actual running 240 degrees); observed in the east in the morning running 100 degrees, 113,908 minutes; visible in the east (the daily angle is the same as that in the west, observed for 10 days, with 8 degrees of retrograde motion).

Mercury: observed in the east in the morning for 33 days, 612,505 minutes; visible in the west for 32 days (accounting for retrograde motion of 1 degree, actual running 31 degrees); observed running 65 degrees, 612,505 minutes; visible in the east (the daily angle is the same as that in the west, observed for 18 days, with 14 degrees of retrograde motion).

First, let's talk about how to calculate the movement of this celestial body. Start by subtracting the daily movement of the celestial body from that of the sun. If the remaining degrees exactly equal a standard value we calculated earlier (the solar day method), then it means we've calculated correctly, and the celestial body has appeared! Next, multiply the celestial body's movement degrees by a denominator (which is a ratio coefficient we set earlier), and then divide the remaining degrees by the solar day method. If it exceeds half, treat it as fully divided. Then, add the calculated degrees to the sun's daily movement degrees; if the total is an integer, it means the celestial body has moved one degree. The calculation methods differ for direct and retrograde motion, and you need to choose the corresponding denominator based on the direction of motion at that time. During the calculation process, if there are remaining degrees, carry them into the next calculation; subtract them for retrograde motion. If there aren't enough degrees to subtract, use the ratio method, using the degrees of movement as the ratio coefficient for adjustment. Basically, terms like "excess," "approximate," and "full" are all about getting precise division results; while "remove," "and," and "divide" help us get the final precise degree.

Next, let's take a look at the details about this planet (Jupiter). This planet appears in the morning with the sun, then it "hides" (or conceals itself). During its direct motion, it moves 1,742,323 minutes in 16 days, while the planet itself moves for 2,323,467 minutes. Then it appears in the east, behind the sun. During its direct motion, it moves quickly, covering eleven-fifty-eighths of its path each day, completing eleven degrees over 58 days. As it continues its direct motion, it slows down, moving nine minutes each day, covering nine degrees over 58 days. Sometimes it stops and remains stationary for 25 days. During retrograde motion, it moves back one-seventh of a degree each day, covering twelve degrees overall in 84 days. After stopping again for 25 days, it resumes its forward motion, moving nine-fifty-eighths each day and covering nine degrees over 58 days. It speeds up again during direct motion, moving eleven minutes each day and completing eleven degrees in 58 days, appearing in front of the sun and setting in the west in the evening. Sixteen days later, it appears again with the sun, covering 1,742,323 minutes, while the planet itself moves for 2,323,467 minutes. In total, one cycle is 398 days, covering 3,484,646 minutes, while the planet itself moves for 43,250,956 minutes.

In the morning, the sun and Mars align, causing Mars to disappear. Then it starts its direct motion, lasting 71 days, moving 1,489,868 minutes, equivalent to 55 degrees and 242,860.5 minutes. After that, we can see it in the east in the morning, behind the sun. During its direct motion, Mars covers fourteen-thirds of a degree each day, completing 112 degrees in 184 days. Its direct motion speed then slows down, moving twelve-thirds of a degree each day and covering 48 degrees over 92 days. Then it stops and remains stationary for 11 days. It then goes into retrograde, moving back seventeen sixty-thirds of a degree each day and covering 17 degrees in 62 days. It stops again for 11 days, then starts moving forward, moving 12 minutes each day and covering 48 degrees over 92 days. During the next phase of direct motion, it speeds up, moving 14 minutes each day and covering 112 degrees in 184 days, appearing in front of the sun and setting in the west in the evening. After 71 days, moving 1,489,868 minutes, equivalent to 55 degrees and 242,860.5 minutes, it meets the sun again. Throughout one complete cycle, it totals 779 days and 973,113 minutes, covering a total of 414 degrees and 478,998 minutes.

In the morning, the Sun encountered Saturn, and Saturn went into conjunction. Then it began to move forward for 16 days, traveling 1,122,426.5 minutes, equating to 1 degree and 1,995,864.5 minutes. After that, we could see it in the eastern sky in the morning, behind the Sun. While moving forward, Saturn traveled 3/35 of a degree each day, covering 7.5 degrees in 87.5 days. Then it stopped and remained still for 34 days. After another 34 days, it resumed direct motion, moving 3 minutes each day, covering 7.5 degrees in 87 days, at which point it was in front of the Sun, setting in the western sky in the evening. After 16 days, it had traveled 1,122,426.5 minutes, equating to 1 degree and 1,995,864.5 minutes, before meeting the Sun again. Over the entire cycle, it totaled 378 days and 166,272 minutes, covering 12 degrees and 1,733,148 minutes.

As for Venus, when it conjoined with the Sun in the morning, it went into hiding and went retrograde, receding by 4 degrees over 5 days, and then it could be seen in the eastern sky in the morning, behind the Sun. While retrograde, it moved 3/5 of a degree each day, retreating 6 degrees in 10 days. Then it stopped for 8 days. After that, it changed to direct motion, moving slowly at a rate of 3/33 of a degree each day, covering 33 degrees in 46 days to resume direct motion. When moving faster, it traveled 1 degree and 91/15 minutes each day, covering 106 degrees in 91 days. Then it moved forward even faster, traveling 1 degree and 91/22 minutes each day, covering 113 degrees in 91 days, at which point it was behind the Sun, appearing in the eastern sky in the morning. During direct motion, it traveled 1/56,954 of a circle in 41 days, covering 50 degrees plus 1/56,954 of a circle, and then it conjoined with the Sun. The total time for one conjunction was 292 days plus 1/56,954 of a circle, following the same pattern as the planet.

Venus, when it aligns with the sun at night, moves quietly, following its orbit and traces one fifty-fourth of its orbit in forty-one days, tracing fifty degrees and one fifty-fourth of its orbit, then it can be seen in the west in front of the sun at night. It moves swiftly, covering one degree and two twenty-second of a degree per day, and one hundred and thirteen degrees in ninety-one days. Then it slows down, moving at a rate of one fifteenth of a degree per day, and one hundred and sixty degrees over ninety-one days. While moving slowly, it covers thirty-three degrees over forty-six days. Then it comes to a stop for eight days. After that, it starts moving in retrograde, moving three-fifths of a degree each day, receding six degrees over ten days, appearing in the west in front of the sun at night, moving swiftly in retrograde, receding four degrees over five days, and then aligning with the sun once more. The two alignments together span five hundred eighty-four days and one eleventh of an orbit, and the planet follows this pattern.

Mercury, when it aligns with the sun in the morning, moves quietly, receding seven degrees in nine days, then it appears in the east behind the sun in the morning. Continuing its swift retrograde motion, it recedes one degree each day. Then it stops and remains still for two days. After that, it begins its direct motion, moving slowly, tracing eight-ninths of a degree each day, and eight degrees in nine days. When moving quickly, it travels one and a quarter degrees each day, and twenty-five degrees in twenty days, appearing in the east behind the sun in the morning. It moves directly, tracing thirty-two degrees and one sixty-seventh of its orbit in sixteen days, and aligns with the sun once more. The alignment spans fifty-seven days and one sixty-seventh of an orbit, and the planet follows this pattern.

It is said that Mercury, when it sets with the sun, appears to hide. Its orbit is varied, sometimes direct, sometimes retrograde. Specifically, in sixteen days, it can cover 32 degrees and 641,967/1 minutes. In the evening, it can be seen in the western sky at dusk, always in front of the sun.

When moving forward, it moves quickly, covering 1.25 degrees in a day, so in twenty days it covers 25 degrees. When it slows down, it only covers 7/8 degrees in a day, taking nine days to cover just 8 degrees. At times, it even pauses for two days. When retrograde, it moves backwards, retreating 1 degree a day, still in front of the sun, lurking in the west in the evening. Retrograde movement is slow, taking nine days to retreat 7 degrees, eventually reuniting with the sun.

From its conjunction with the sun to the next conjunction, the entire cycle lasts 115 days and 61,255 minutes; that's how Mercury moves. "Mercury: In the evening, it sets with the sun, lurking, direct, sixteen days 641,967/1 minutes, appearing in the west in front of the sun. Direct, swift, covering 1.25 degrees a day, in twenty days covering 25 degrees. Slow, covering 7/8 degrees in a day, taking nine days to cover just 8 degrees. Pause, not moving for two days. Rotate, retrograde, retreating 1 degree a day, in front of the sun, lurking in the west in the evening. Retrograde, slow, taking nine days to retreat 7 degrees, eventually reuniting with the sun. Conjunction, repeating, 115 days and 61,255 minutes; the planet moves in this way."

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Written by: Realhistories

Category: Book of Jin （晉書）

Published: 02 January 2025

Created: 02 January 2025

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This text describes the methods of calculating ancient calendars, which are quite technical and intricate. Let's explain it sentence by sentence in modern colloquial language.

First, it talks about how to calculate the lengths of day and night. "Based on the distance the moon travels, multiply it by the number of nighttime hours of the closest solar term, and then divide by 200 to get the daytime duration (mingfen); subtract this value from the moon's travel distance to determine the nighttime duration (hunfen)."

The meaning is to calculate the daytime and nighttime durations based on the moon's movement and the number of night hours during the recent solar term. "Convert the lengths of day and night into units, multiply the total units by the daytime and nighttime durations, and add the units for midnight to determine the units for daytime and nighttime. If the remainder exceeds half, round it up; otherwise, discard it."

Next, it discusses how to calculate the "day" in the calendar. "The moon's movement is tracked by four tables and three paths, and by dividing the moon's speed by these values, you can determine the calendar day."

"The number of weeks multiplied by the synodic month gives the synodic division; the total units multiplied by the synodic number, with the remainder divided by the synodic month, gives the retrograde calculation; based on the moon's cycle, calculate the degrees added each day, then divide by the synodic month to find the difference rate."

This section explains that by multiplying the number of weeks by the synodic month (which refers to the moon's full cycle), and then dividing by the number of synodic months, one can obtain the synodic division; multiplying the total units by the synodic number, and dividing the remainder by the synodic month gives the retrograde calculation; finally, calculate the degrees added each day, then divide by the synodic month to find the difference rate.

Next is the profit and loss rate table of the lunar-solar calendar, which can be quite complex. It lists the daily increases and decreases for the lunar-solar calendar, specifically indicating that adjustments need to be made based on the date. Some days require a reduction in value, while others require an increase. The table lists the specific daily increases and decreases. "Reduce one and increase seventeen on the first day" means to reduce one on the first day and increase seventeen, and so on. The explanation in parentheses clarifies some special cases, such as "if the reduction is insufficient, it should be treated as an increase," meaning that if the result of the reduction is insufficient, it should be reversed and treated as an increase. "Remaining value" and "differential" represent some remainders and fine adjustments in the calculations.

Finally, it explains how to determine whether it is the lunar calendar or the solar calendar based on the calculation results. "Subtract the accumulated months from the starting point, and multiply the remaining by the lunar-solar fraction and the differential respectively. If the differential reaches a certain threshold, it is added to the lunar-solar fraction. If the lunar-solar fraction reaches the total weeks, it is subtracted from the week count. The remaining value, if less than the calendar cycle, is the solar calendar; if it is equal to or exceeds the calendar cycle, it is classified as the lunar calendar." This means that the accumulated number of combined months is subtracted from the accumulated number of the starting point (where the starting point refers to the beginning of a calendar), and the remaining values are multiplied by the lunar-solar fraction and the differential. If the differential reaches a certain threshold, it is added to the lunar-solar fraction; if the lunar-solar fraction reaches the total weeks, the week count is subtracted. Finally, if the remainder is less than the calendar cycle, it is the solar calendar; if it is equal to or exceeds the calendar cycle, it is the lunar calendar. "All remaining values are like the monthly cycle yielding one day, excluding the calculation, the desired month is combined with the lunar-solar calendar, with not all values being day remainders. Add two days; the day remainder is two thousand five hundred eighty, and the differential is nine hundred fourteen. Using this method to calculate days, subtract thirteen when complete and divide the remainder into fractions of days. The lunar and solar calendars ultimately interconvert, with the entry into the calendar happening before the remaining value, and the latter remaining after, indicating the midpoint of the monthly cycle." This passage summarizes the calculation method and explains the meaning of the final results, as well as the interconversion relationship between the lunar and solar calendars. Overall, this text outlines a very complex ancient calendar calculation system that involves a lot of astronomical and mathematical knowledge.

First, let’s explain this ancient text, which outlines a complex method for calculating calendars. This text describes an algorithm for calendar calculation, involving many technical terms such as "day remainder," "small fraction," "combined number," "profit and loss rate," etc., which all require an understanding of the calendar knowledge of that time to comprehend. We will translate it sentence by sentence into modern vernacular, aiming for clarity and ease of understanding.

First, calculate all the time differences (speed and duration), expressed in smaller units of time, and represent the surplus and deficit using Yin-Yang surplus. Adjust the number of days based on the surplus and deficit situation. Then, multiply the adjusted number of days by the profit and loss rate (a correction coefficient). For example, the number of days in a month is fixed; use this method to calculate the overtime constant (adjusted number of days).

Next, multiply the difference rate by the small remainder of the new moon day (the first day of each lunar month), calculating a value similarly to calculus. Then subtract this value from the remaining days in the calendar. If it is not enough, add the number of days in a month and then subtract. If there is still one day short, subtract an additional day. Then add the fractional part to this result, and use the sum to approximate the differential, yielding small units of time, thus obtaining the calendar data for midnight on the new moon day.

To calculate the next day, simply add one day. If the day surplus is 31, the small units of time are also 31. If the small units of time exceed the total, subtract a month's days and add one day. If the calendar calculation reaches the end and the day surplus is full, subtract it to get the starting point of the calendar. If the day surplus is not full, retain it and add 2720; if the small units of time are 31, this gives the starting point of the next calendar.

Multiply the total number of days by the midnight surplus and remainder of the late-early calendar. If the remainder fills half a cycle, it is used to represent small units of time. Adjust the Yin-Yang surplus based on the surplus and deficit situation. Then multiply the adjusted number of days by the profit and loss rate to obtain the midnight constant.

Multiply the profit and loss rate by the number of water clocks at night during the recent solar terms (ancient timing devices); 200 water clocks equal one day's bright time. Subtract this value from the profit and loss rate to obtain the dim time, using the midnight value to determine the times of brightness and darkness.

Add the overtime number and dim constant together, then divide by 12 to get the degrees. One-third of the remainder indicates a deficiency, while a value of less than one minute signifies strength, and two deficiencies indicate weakness. This result represents the degrees the moon has moved away from the ecliptic. For the solar calendar, subtract the extreme from the calendar data at the ecliptic where the added day is located; for the lunar calendar, add it to get the degrees the moon has moved away from the extreme. Strength is positive, weakness is negative; same names add, different names subtract. When subtracting, same names cancel, different names add; there are no opposites—two strengths add to one deficiency and subtract one weakness.

From the year of the Metal Ox in the Yuan Dynasty to the year of the Metal Dog in the eleventh year of Jian'an, a total of 7378 years have passed.

己丑, 戊寅, 丁卯, 丙辰, 乙巳, 甲午, 癸未, 壬申, 辛酉, 庚戌, 己亥, 戊子, 丁丑, 丙寅

Five Elements: Wood (Year Star), Fire (Mars), Earth (Filling Star), Metal (Venus), Water (Chen Star). Calculate the weekly and daily rates using the degrees and celestial positions of their daily movements. Multiply the chapter age by the weekly rate to obtain the monthly calculation method, multiply the chapter month by the daily rate to get the month point, and divide the month point by the monthly calculation method to obtain the month number. Multiply the total number of days by the monthly calculation method to obtain the daily calculation method. Multiply the Dipper by the weekly rate to get the Dipper. (The daily calculation method is obtained by multiplying the record method by the weekly rate, so it is also multiplied by the minute here).

The method for calculating the calendar described in this text is quite complex, involving a wealth of astronomical and mathematical knowledge, and it is difficult for us to completely restore its calculation process now. It reflects the outstanding achievements of the ancient Chinese people in the field of astronomical calendar.

First, we need to understand what these numbers signify. This text discusses calculating the positions of planets using ancient algorithms, which can seem quite complicated. "Five Stars" refers to the five planets of metal, wood, water, fire, and earth. "Shuo" refers to the first day of the lunar calendar. "Da Yu," "Xiao Yu," "degrees," "degree residue," and so on, are intermediate results generated in the calculation process, the specific meanings of which we will not delve into, as long as we recognize that they are used for calculating planetary positions. "Ji Yue," "Zhang Run," "Zhang Yue," "Sui Zhong," and so on should all be constants related to the calendar. "Tong Fa," "Ri Fa," "Hui Shu," "Zhou Tian," and "Dou Fen" are also some fixed parameters. The numbers after Jupiter, Mars, and Saturn represent their respective calculation parameters.

Now, let's break it down into simpler language, sentence by sentence:

1. First, calculate the "Da Yu" and "Xiao Yu" values for the planets. (The calculation method is as follows: multiply "Tong Fa" by the month, and then divide by "Ri Fa"; the quotient is "Da Yu," and the remainder is "Xiao Yu." Then subtract 60 from "Da Yu.")

2. Then calculate the "Ru Yue Ri" and "Ri Yu" values for the planets. (The calculation method is: multiply "Tong Fa" by "Yue Yu," multiply "He Yue Fa" by "Shuo Xiao Yu," add these two results together, simplify if necessary, and finally divide by the "Ri Du Fa"; the result will yield "Ru Yue Ri" and "Ri Yu.")

3. Finally, calculate the "degree" and "remainder" of the planet. (The calculation method is: subtract the excess to get the "remainder," then multiply the "orbital period" by the "remainder," and then simplify using the "solar degree method," resulting in the "degree," with the remainder being the "remainder." If it exceeds the "orbital period," subtract the "orbital period" and the "Doufen" (斗分).)

4. The total number of months in the calendar year is 7285.

5. The total number of leap months is 7.

6. The total number of chapters (months) is 235.

7. There are 12 months in a year.

8. The common method is 43026.

9. The solar method is 1457.

10. The total number of cycles is 47.

11. The orbital period is 215130.

12. The Doufen is 145.

13. The following are the parameters for Jupiter: the orbital period is 6722, the solar rate is 7341, the total number of lunar months is 13, the lunar remainder is 64810, the combined lunar method is 127718, and the solar degree method is 3959258.

14. Jupiter's large lunar remainder is 23, small lunar remainder is 1370, the day of the new moon is 15, and the solar remainder is 3484646. The lunar virtual fraction is 150, Doufen is 974690, the degree is 33, and the remainder is 2509956.

15. The following are the parameters for Mars: the orbital period is 3447, the solar rate is 7271, the total number of lunar months is 26, the lunar remainder is 25627, the combined lunar method is 64733, and the solar degree method is 2006723.

16. Mars's large lunar remainder is 47, small lunar remainder is 1157, the day of the new moon is 12, and the solar remainder is 973113. The lunar virtual fraction is 300, Doufen is 494115, the degree is 48, and the remainder is 1991760.

17. The following are the parameters for Saturn: the orbital period is 3529, the solar rate is 3653, the total number of lunar months is 12, the lunar remainder is 53843, the combined lunar method is 6751, and the solar degree method is 278581.

18. Saturn's large lunar remainder is 54, small lunar remainder is 534, and the day of the new moon is 24.

This passage describes the methods used by ancient astronomers to calculate the positions of planets. Although it seems complex, it is actually a series of mathematical operations to determine the position of planets in the sky. These numbers and formulas reflect the ancient astronomers' observations and understanding of the universe, as well as their excellent mathematical calculation skills. Unfortunately, we no longer use this method; modern astronomical calculations have become more accurate and convenient.

Wow, these numbers are a lot to take in! Let me break it down for you in simpler terms.

Firstly, this first paragraph discusses the calculation results, various remainders and fractional values, such as "the remainder for the day of 166,272" and "the fraction for the new moon of 923," which are intermediate results in astronomical calculations. We don't need to get too caught up in the specific meanings; after all, it's just a bunch of numbers. "The Doufen of 511,750," "the angle of 12 degrees," and "the degree remainder of 1,733,148" are the same; they are all data from the calculation process. "Gold: the orbital period of 9,022, the daily orbital rate of 7,213, the combined lunar count of 9, and the lunar remainder of 152,293..." These numbers are also similar intermediate calculation results, related to the operational patterns of the "Gold" star. Later, there are terms like "combined lunar method," "daily degree method," "new moon large remainder," "new moon small remainder," "entry lunar day," etc.; all are technical terms in astronomical calculations, and we just need to know they are data from the calculation process.

Next, this section continues to list calculation results, similar to the previous ones, consisting of various remainders and fractional values, such as "the remainder for the day of 56,954," "the fraction for the new moon of 328," and "the Doufen of 1,308,190," etc. These numbers represent the operational law calculation results of the "Water" star. "Water: the orbital period of 11,561, the daily orbital rate of 1,834, the combined lunar count of 1, and the lunar remainder of 211,331..." Later, there are also "combined lunar method," "daily degree method," "new moon large remainder," "new moon small remainder," "entry lunar day," along with a long string of numbers, all of which are data from the calculation process. Finally, there are "the remainder for the day of 6,419,967," "the fraction for the new moon of 684," "the Doufen of 1,676,345," "the angle of 57 degrees," and "the degree remainder of 6,419,967."

The final paragraph begins to explain the calculation method. "Take the Shangyuan of the desired year, multiply by the cycle rate; if the result is exactly an integer multiple of the daily rate, it is called Jihe; if it is not an integer multiple, the remainder is called Heyu." This means that multiplying the Shangyuan of the year you want to calculate by the cycle rate, if the result is an integer multiple of the daily rate, it is called Jihe; if not, the remainder is Heyu. "Divide by the cycle rate; if the result is 1, it is the previous year; if it is 2, it is the year before the previous year; if there is no result, it is the current year." This means that when you divide the Jihe by the cycle rate, if the result is 1, it is calculated as the previous year; if it is 2, it is calculated as two years ago; if it does not divide evenly, it is calculated as the current year. "Subtract the cycle rate from the Heyu to obtain the degree." This means that you subtract the cycle rate from the Heyu to obtain the degree. "The accumulation of gold and water; if the result is an odd number, it is referred to as 'morning'; if it is an even number, it is called 'evening'." This means that the accumulation of gold and water, if it is an odd number, is referred to as 'morning'; if it is an even number, it is called 'evening'. "Multiply the accumulation of the number of months and the remainder of the month; the full month law starts from the month; if it is not an integer multiple, the remainder is the new month remainder." This means that you multiply the number of months and the remainder of the month by the accumulation; if the result is an integer multiple of the full month law, it represents the month; if it is not an integer multiple, the remainder is the new month remainder. "Subtract the accumulated month from the accumulated month; the remainder is the entry month. Multiply by the leap month; the full chapter month gets one leap; subtract from the entry month; subtract from the middle of the year; calculate according to the Tianzheng method, which also applies to the month. If it is at the leap intersection, use the new moon." This paragraph explains how to calculate the month and leap month, using many professional terms, which may be difficult to understand. In simple terms, it determines the month and leap month based on the calculation results. "Multiply the month remainder by the common method; multiply the full month law by the small remainder, and round off the number of meetings. If the result is an integer multiple of the full day law, it is the entry date of the star in the month. If it does not yield an integer multiple, it becomes the day remainder; calculate according to the new moon." This explains how to calculate the entry date of the star in the month, which is the specific date of a planet in a certain month. "Multiply the degree by the cycle day; if the full day law gets one degree, if it does not yield an integer multiple, it becomes the remainder; calculate the degree from the front five of the ox." This explains how to calculate degrees. "Seek the star conjunction." The last sentence means that the above outlines the method for determining the star conjunction.

In summary, this text describes the calculation methods for the rules of planetary motion in ancient astronomical calendars, involving a large number of technical terms and complex calculation processes, which can indeed be quite difficult for modern people to understand. Overall, it describes a complex mathematical model used to predict the positions of planets.

Let's calculate the days by first adding the months together, and also adding any surplus months. If the total is exactly one month, it indicates that there is no intercalary month that year; if it is not a full year, then carry over the excess to the next year; if the next year also completes, carry it over to the following year. For Venus and Mercury, if they appear in the morning, add them to the evening, and if they appear in the evening, add them to the morning.

Next, calculate the size of the new moon day (the first day of each lunar month) and the excess days, and add them together. If the total equals one month, then add 29 days (for a short month) or 30 days (for a long month), deducting the days of the short month from the long month. The calculation method is the same as before.

Then calculate the new moon days and the excess days, adding them together. If the excess days total one day, count it; if the excess days on the first day of the lunar month are insufficient, subtract one day; if the excess days exceed 29, subtract 29 days, and if not enough, subtract 30 days, leaving the remainder for the next month.

Finally, add the degrees together, and also add the excess degrees; if the degrees amount to one day, count it as one degree.

Here is the motion of Jupiter:

Jupiter: Concealed for 32 days and 3,484,646 minutes; appeared for 366 days; concealed running 5 degrees, 2,509,956 minutes; appeared running 40 degrees. (Retrograde 12 degrees, actual running 28 degrees.)

Mars: Concealed for 143 days and 973,113 minutes; appeared for 636 days; concealed running 110 degrees, 478,998 minutes; appeared running 320 degrees. (Retrograde 17 degrees, actual running 303 degrees.)

Saturn: Concealed for 33 days and 166,272 minutes; appeared for 345 days; concealed running 3 degrees, 1,733,148 minutes; appeared running 15 degrees. (Retrograde 6 degrees, actual running 9 degrees.)

Venus: Concealed in the east for 82 days and 113,908 minutes; appeared in the west for 246 days. (Retrograde 6 degrees, actual running 240 degrees.) Concealed in the morning running 100 degrees, 113,908 minutes; appeared in the east. (The solar position matches that of the west, concealed for 10 days and retrograde 8 degrees.)

Mercury: In the morning, it is concealed in the east for 33 days, 612,505 minutes; it appears in the west for 32 days. (Retrograde motion 1 degree, actual motion 31 degrees.) Concealed motion 65 degrees, 612,505 minutes; appears in the east. (The solar degree is the same as the west, concealed for 18 days, retrograde 14 degrees.)

First, let's talk about how to calculate the motion of planets. First, calculate the degrees the planet moves each day, then add the daily degree difference between it and the sun. If this difference equals the degree of one day, it indicates that the conjunction period between the planet and the sun has arrived, and as calculated before, we can know when the planet can be seen. Then, multiply the number of increments the planet moves (denominator) by the degrees it appears, and for the remaining part, divide by one day's degree. If it cannot be divided evenly, treat it as a whole day if it exceeds half. Next, sum the degrees the planet moves each day; if the total equals the number of increments it moves, it is considered to have completed a full orbit. The calculation methods for direct and retrograde motion are different and should be based on the planet's current increments (denominator). For the remaining part, based on previous calculations, if it is retrograde, it should be subtracted. If it cannot be divided evenly, use a unit for division, using the planet's increments (denominator) as a ratio, thus allowing the calculation of the increase or decrease in the degrees of the planet's motion, with mutual correction before and after. In summary, terms like “full,” “about,” and “complete” are used to achieve precise division results; while “go,” “reach,” and “divide” are all aimed at obtaining the final division results.

Next, let's look at the motion of Jupiter. Jupiter aligns with the sun in the morning, then goes retrograde, followed by direct motion, running for 16 days at 1,742,323 minutes, with the planet moving 2,323,467 minutes. At this time, Jupiter appears in the east, lagging behind the sun. During direct motion, it moves quickly, covering 11 minutes out of 58 daily, which translates to 11 degrees in 58 days. Then continuing direct motion, the speed slows down, moving 9 minutes daily, which results in 9 degrees in 58 days. Then it stops moving and resumes after 25 days. In retrograde, it moves 1 minute out of 7 daily, covering 12 degrees in 84 days. It stops moving again, and after 25 days resumes direct motion, moving 9 minutes out of 58 daily, which results in 9 degrees in 58 days. The direct motion speed is fast again, moving 11 minutes daily, covering 11 degrees in 58 days. At this time, Jupiter appears in front of the sun, setting in the west in the evening. After 16 days of running 1,742,323 minutes, with the planet moving 2,323,467 minutes, it aligns with the sun again. A complete cycle lasts 398 days, during which it runs for 3,484,646 minutes, covering a distance of 43 degrees and 2,509,956 minutes.

When the sun rises in the morning, Mars and the sun align, and then Mars becomes obscured. Next, it starts moving forward for a total of 71 days, covering a distance of 1,489,868 minutes, equivalent to 55 degrees and 242,860.5 minutes. Then, it becomes visible in the east behind the sun. During its forward motion, it travels 23 minutes and 14 seconds daily, covering 112 degrees in 184 days. Its speed then decreases, moving 23 minutes and 12 seconds each day, covering 48 degrees in 92 days. It then remains stationary for 11 days. Subsequently, it begins its retrograde motion, traversing 62 minutes and 17 seconds each day, moving back 17 degrees in 62 days. It stops for another 11 days, then resumes its forward motion, covering 12 minutes each day, covering 48 degrees in 92 days. Afterwards, the forward speed increases, covering 14 minutes each day, traversing 112 degrees in 184 days. At this point, it passes in front of the sun and becomes visible in the west during the evening. Over the course of this entire cycle, the total duration is 779 days and 973,113 minutes, covering 414 degrees and 478,998 minutes.

As for Saturn, it also aligns with the sun in the morning before becoming obscured. It then moves forward for 16 days, covering a distance of 1,122,426.5 minutes, equivalent to 1 degree and 1,995,864.5 minutes. Then, in the morning, it can be seen in the east, behind the sun. While moving forward, it covers 35 minutes and 3 seconds each day, covering 7.5 degrees in 87.5 days. It stops for 34 days. It then starts moving backward, covering 17 minutes and 1 second each day, moving back 6 degrees in 102 days. After an additional 34 days, it resumes its forward motion, covering 3 minutes each day, covering 7.5 degrees in 87 days. At this point, it passes in front of the sun and becomes visible in the west during the evening. Over the course of this entire cycle, the total duration is 378 days and 166,272 minutes, traversing 12 degrees and 1,733,148 minutes.

Venus, when it conjuncts the sun in the morning, first "submerges," which means it retrogrades. Over five days, it will move back four degrees, and then it can be seen in the east in the morning; at this point, it is positioned behind the sun. Continuing to retrograde, it retrogrades three-fifths of a degree each day, totaling six degrees over ten days. Next comes "stationary," meaning it remains still for eight days. Then it begins to "rotate," which means it moves forward, but at a slower speed, covering three thirty-sixths of a degree each day, totaling thirty-three degrees over the course of forty-six days. After that, the speed increases to one degree and fifteen ninety-firsts each day, totaling one hundred six degrees over the course of ninety-one days. The speed continues to increase, moving one degree and twenty-two ninety-firsts each day, totaling one hundred thirteen degrees over the course of ninety-one days, at which point it has moved back behind the sun again, which can be seen in the east in the morning. Finally, moving forward, it covers one fifty-six thousand nine hundred fifty-fourth of a complete circle in forty-one days, while the planet also travels fifty degrees one fifty-six thousand nine hundred fifty-fourth of a complete circle, ultimately meeting the sun again. The cycle for one conjunction is two hundred ninety-two days and one fifty-six thousand nine hundred fifty-fourth of a complete circle, and the planet's movement follows this pattern.

In the evening, when Venus conjuncts the sun, it first "submerges," this time moving forward. It covers one fifty-six thousand nine hundred fifty-fourth of a complete circle in forty-one days, while the planet travels fifty degrees one fifty-six thousand nine hundred fifty-fourth of a complete circle, and it can be seen in the west during the evening; at this point, it is in front of the sun. Then it continues to move forward, with increasing speed, moving one degree and twenty-two ninety-firsts each day, totaling one hundred thirteen degrees over the course of ninety-one days. The speed then starts to decrease, moving one degree and fifteen hundredths each day, totaling one hundred six degrees over the course of ninety-one days, and then continues to move forward. The speed decreases, moving three thirty-sixths of a degree each day, totaling thirty-three degrees over the course of forty-six days. Next comes "stationary," remaining still for eight days. Then it "rotates," this time retrograding, retrograding three-fifths of a degree daily, which totals six degrees over ten days. At this point, it has moved in front of the sun, and can be seen in the west in the evening, continuing to retrograde, with increasing speed, moving back four degrees in five days, ultimately meeting the sun again. The two conjunctions complete one cycle, spanning five hundred eighty-four days and eleven thousand three hundred ninety-eight hundredths of a complete circle, and the planet's movement follows this pattern as well.

Mercury, when it meets the sun in the morning, first "伏" (lies low), which means retrograde. It retreats seven degrees after a period of nine days, and then in the morning, it can be seen in the east, positioned behind the sun. Continuing to retrograde, the speed increases, retreating one degree after one day. It then "pauses" for two days. Then it "turns," which means direct motion, at a slower speed, moving eight-ninths of a degree each day, covering eight degrees in nine days. The speed increases, moving one and a quarter degrees each day, covering twenty-five degrees in twenty days, at which point it is once more behind the sun and visible in the eastern sky in the morning. Finally, it moves direct, traversing 641,009,067 parts of a circle in sixteen days, with the planet moving thirty-two degrees 641,009,067 parts of a circle, ultimately meeting the sun. The cycle of one conjunction is fifty-seven days 641,009,067 parts of a circle, and the planet's movement follows this pattern.

Speaking of Mercury, when it appears at the same time as the sun, it is called a "conjunction." The patterns of Mercury's movement are as follows: sometimes it moves quickly, covering three hundred twenty-six degrees forty-one minutes nine hundred sixty-seven seconds in sixteen days (this is precise to the second!), at which point it can be seen in the evening sky to the west, positioned ahead of the sun. When moving quickly, it covers one and a quarter degrees in one day, completing twenty-five degrees in twenty days.

Sometimes it moves slowly, covering only seven-eighths of a degree in one day, taking nine days to cover eight degrees. Sometimes it simply comes to a halt for two days. Even more astonishing is that it can retrograde! It can retreat one degree in a day, at which point it can be seen in the evening sky to the west, positioned ahead of the sun. During retrograde, its speed is also slow, taking nine days to retreat seven degrees, and then it "conjoins" with the sun again.

From one "conjunction" to the next, taking into account all its movement phases, it takes a total of one hundred fifteen days six hundred one million two thousand five hundred five seconds, and this constitutes the complete cycle of Mercury's movement.