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.
