First, use the differential rate multiplied by the remainder of the new moon day, just like in calculus, to calculate a value, then subtract this value from the remainder of the days calculated by the calendar. If it is still short by one day, subtract another day; if it is not enough to subtract, add the number of days in a month and then subtract. Then add the resulting whole days to the fractional part, simplify the fraction to obtain a decimal value, and this will yield the calendar date of the new moon at midnight.
Next, calculate the second day by adding one day; the remainder of the day is 31, and the decimal is also 31. If the decimal exceeds the integer, subtract the number of days in a month. Then add one more day and calculate through to the end of the calendar. If the remainder of the days exceeds the fractional part, subtract the fractional part; this marks the starting date of the calendar. If the remainder of the days does not reach the fractional part, retain it, then add 2720, resulting in a fractional part of 31; this is the date of the next calendar.
Multiply the total number by the late-night value of the historical calendar and the remainder; if it exceeds half a week, it will be considered a decimal. Add the surplus number to the reduced number, then subtract the remainder of the yin and yang days. If there is a surplus or shortage of days, adjust the days with the month and week. Multiply the determined remainder of the days by the profit-loss ratio; if the result is equal to the month and week, use the comprehensive value of profit and loss to determine the midnight value.
Multiply the profit-loss ratio by the latest solar terms, divide by 200 to obtain the sunrise time, then subtract this value from the profit-loss ratio to get the sunset time, and then use the midnight profit-loss figure to determine sunrise and sunset times. If the total time added equals the determined values for sunrise and sunset, divide by 12 to obtain the degree; multiply the remainder by one-third. If the result is less than one, it is considered weak; if more than one, it is strong; two weak values are also considered weak. This indicates the degree of the moon's departure from the ecliptic. For the solar calendar, add the day's degree to the ecliptic calendar; for the lunar calendar, subtract it, to determine the degree of the moon's departure from the ecliptic. A positive value indicates strength, while a negative value indicates weakness; combine the strong and weak values, add the same names, and subtract the different names. When subtracting, same names cancel each other out, different names add up; if there is no cancellation, combine two strong values and subtract one weak value.
From the Ji-Chou year of the Shangyuan era to the Bing-Xu year of the Jian'an era, a total of 7,378 years have elapsed. 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.
Five Planets: Wood (Jupiter), Fire (Mars), Earth (Saturn), Metal (Venus), Water (Mercury). Calculate the lunar and solar rates using each planet's orbital period and celestial longitude. Multiply the annual cycle by the lunar formula to obtain the lunar method, and multiply the lunar method by the daily rate to obtain the monthly division. Divide the monthly division by the lunar method to obtain the number of months. Multiply the total by the lunar method to obtain the daily method. Multiply the total by the lunar rate to obtain the division. (The daily method is multiplied by the solar cycle, so here we also use division to multiply.)
The five stars have both large and small remainders. (Multiply the total method by the number of months to get the large remainder, and the leftover is the small remainder. Subtract the large remainder from 60 to find the final value.)
The five stars enter the month and day remainders. (Multiply the total method by each month’s remainder, multiply the total lunar method by the large remainder of the new moon day, add them together, simplify the result with the total, then divide by the daily method to get the final result.)
This passage records data calculated by ancient astronomical calendars, which appears to be a record of some kind of celestial calculation. Let's go through it sentence by sentence and explain it in modern terms.
First is the calculation method for the degrees and degree remainders of the "Five Stars." "Degrees and degree remainders of the Five Stars. (Subtract the excess as degree remainders, multiply by the number of days in a week, round with the solar method, the result is the degree, the remainder is the degree remainder, exceeding the number of days in a week is subtracted, and add the division.)" This means: calculate the degrees and remaining degrees of the five stars, subtract the excess first, then multiply by the number of days in a week, then divide by the solar method, the result is the degree, the remainder is the degree remainder; if it exceeds the number of days in a week, subtract the number of days in a week, and then add the division. This part involves complex calendar calculations, and we only need to know that it is used to calculate the positions of celestial bodies.
Next is a series of numbers, which should be various astronomical parameters. "Lunar cycle, 7285. Leap months, 7. Record of the month, 235. Months in a year, 12. Total method, 43026. Solar method, 1457. Total, 47. Number of days in a week, 215130. Division, 145." These numbers represent: lunar cycle is 7285, leap months is 7, record of the month is 235, there are 12 months in a year, total method is 43026, solar method is 1457, total is 47, number of days in a week is 215130, division is 145. Understanding what these specific numbers represent requires knowledge of the historical calendar system of that era.
Continue reading; it is about the calculation data of Jupiter. "Jupiter: orbital period, 6722. Solar year, 7341. Total synodic months, 13. Remaining synodic months, 6481. Total lunar cycle count, 12718. Solar day count, 3959258. Major lunar surplus, 23. Minor lunar surplus, 1307. Day of entry month, 15. Daily surplus, 3484646. Phantom lunar division, 150. Dipper division count, 974690. Degree count, 33. Degree surplus, 2509956." This paragraph is about a series of parameters related to Jupiter's orbital period, solar year, total synodic months, etc., and the specific meanings require professional astronomical calendar knowledge to interpret. In short, it is all about calculations related to Jupiter's orbit and timekeeping.
Then there are similar data for Mars, Jupiter, and Venus, which consist of a series of astronomical parameters and calculation results. "Mars: orbital period, three thousand four hundred seventy. Solar rate, seven thousand two hundred seventy-one. Total lunar months, twenty-six. Remaining days, twenty-five thousand six hundred twenty-seven. Combined lunar calculation, sixty-four thousand seven hundred thirty-three. Solar degree method, two million six thousand seven hundred twenty-three. Major lunar remainder, forty-seven. Minor lunar remainder, one thousand one hundred fifty-seven. Days to the new moon, twelve. Remaining days until the new moon, nine hundred seventy-three thousand one hundred thirteen. New moon virtual division, three hundred. Dipper division, four hundred ninety-four thousand one hundred fifteen. Degrees, forty-eight. Degree remainder, one million nine hundred ninety-one thousand seven hundred six. Earth: orbital period, three thousand five hundred twenty-nine. Solar rate, three thousand six hundred fifty-three. Total lunar months, twelve. Remaining days, fifty-three thousand eight hundred forty-three. Combined lunar calculation, sixty-seven thousand fifty-one. Solar degree method, two hundred seventy-seven thousand eight hundred fifty-eight. Major lunar remainder, fifty-four. Minor lunar remainder, five hundred thirty-four. Days to the new moon, twenty-four. Remaining days until the new moon, one hundred sixty-six thousand two hundred seventy-two. New moon virtual division, nine hundred twenty-three. Dipper division, five hundred eleven thousand seven hundred five. Degrees, twelve. Degree remainder, one million seven hundred thirty-three thousand one hundred forty-eight. Venus: orbital period, nine thousand twenty-two. Solar rate, seven thousand two hundred thirteen. Total lunar months, nine." These numbers are similar to those of Jupiter, representing the calculated results of the orbital patterns of Mars, Jupiter, and Venus. Each planet has its own set of parameters.
In summary, this passage is a record of ancient astronomical calendar calculations, filled with various astronomical parameters and calculation results, which require specialized knowledge of astronomy and calendars to be understood by modern readers. We can only see these numbers but cannot fully grasp the meanings behind them.
A month has passed, and the value is one hundred fifty-two thousand two hundred ninety-three. According to the combined lunar calculation, the result is one hundred seventy-one thousand four hundred eighteen. Using the solar degree method, the result is five hundred thirty-one thousand three hundred ninety-eight. The major lunar remainder is twenty-five. The minor lunar remainder is one thousand one hundred twenty-nine. The days to the new moon are twenty-seven. The remaining days until the new moon are fifty-six thousand nine hundred fifty-four. The new moon virtual division is three hundred twenty-eight. The dipper division is one hundred thirty thousand eight hundred ninety. The degrees are two hundred ninety-two. The degree remainder is fifty-six thousand nine hundred fifty-four.
The lunar cycle of water is eleven thousand five hundred and sixty-one.
The lunar cycle of the sun is one thousand eight hundred and thirty-four.
The total number of lunar months is one.
Next, the resulting value after one month is two hundred and eleven thousand three hundred and thirty-one.
The result of the combined lunar month calculation is two hundred and nineteen thousand six hundred and fifty-nine.
The result of the solar day method calculation is six million eight hundred and ninety-four thousand two hundred and twenty-nine.
The major lunar remainder is twenty-nine.
The small moon remainder is seven hundred and seventy-three.
The entry month day is twenty-eight.
The daily remainder is six million four hundred and nineteen thousand six hundred and sixty-seven.
The lunar virtual part is six hundred and eighty-four.
The star part is one million six hundred and seventy-six thousand three hundred and forty-five.
The degree is fifty-seven.
The degree remainder is six million four hundred and nineteen thousand six hundred and sixty-seven.
First, substitute the value of the previous lunar year; multiply it by the lunar cycle. If it can be divided by the daily rate to get one, it is called the product, and the part that cannot be divided is the total remainder. Divide the lunar cycle by the total remainder; if it can be divided to get one, it is the star in the previous year; if it can be divided to get two, it is the star in the previous two years; if it cannot be divided, it is the total in this year. Subtract the total remainder from the lunar cycle to get the degree part. The combination of gold and water indicates that odd numbers refer to the morning and even numbers refer to the evening.
Multiply the month number and the month remainder by the product; if it can be divided by the total month method to get the value of one month, then use this value, and the part that cannot be divided is the month remainder. Subtract the accumulated month from the accumulated month, and the remaining is the entry month. Then multiply it by the leap month; if it can be divided by the leap month to get a leap month, subtract it from the entry month, and subtract the remaining value from the year; this represents the total lunar month outside the Tianzheng calculation. If there is a leap month transition, adjust it with the new moon.
Multiply the common method by the month remainder, multiply the total month method by the small moon remainder, and then divide by the meeting number. If the result can be divided by the daily method to get one, it is the entry month day of the star conjunction; the part that cannot be divided is the daily remainder, recorded outside the Tianzheng calculation.
Multiply the lunar cycle by the degree; if it can be divided by the daily method to get one degree, the part that cannot be divided is the remainder. Use the method of determining the degree with the first five cows to determine the degree.
The above outlines the method for calculating lunar star conjunctions.
Add the month numbers, add the month remainders; if it can be divided by the total month method to get a month, then use it. If it cannot be divided, use it to determine the year; subtract if it can be divided, consider the leap month, and the remaining is the value of the next year; if it can be divided again, it is the value of the next two years. The combination of gold and water means adding morning to evening and evening to morning.
First, let's calculate the size of the moon's phases. Add up the sizes of the new moon periods; if the total exceeds a month, then add another twenty-nine days (for a large remainder) or seven hundred seventy-three minutes (for a small remainder). When the small remainder is full, calculate using the same method as for the large remainder, and the rest remains the same as before.
Next, calculate the moon entry date and the remaining days. Add up the moon entry date and the remaining days; if the remainder is enough for a day, then add a day. If the small remainder just fills the gap at the time of the new moon, then subtract a day; if the small remainder exceeds seven hundred seventy-three, then subtract twenty-nine days; if not enough, then subtract thirty days, and the remainder will be calculated using the method for the subsequent new moon.
Finally, add the degrees together and their remainders; if it sums to a full day's worth of degrees, then add one degree.
Here are the movement details of the various planets:
Jupiter: In a dormant state for 32 days, moves 3484646 minutes; appeared for 366 days; dormant movement of 5 degrees, 2509956 minutes; visible movement of 40 degrees (retrograde 12 degrees, actual movement 28 degrees).
Mars: In a dormant state for 143 days, moves 973113 minutes; appeared for 636 days; dormant movement of 110 degrees, 478998 minutes; visible movement of 320 degrees (retrograde 17 degrees, actual movement 303 degrees).
Saturn: In a dormant state for 33 days, moves 166272 minutes; appeared for 345 days; dormant movement of 3 degrees, 1733148 minutes; visible movement of 15 degrees (retrograde 6 degrees, actual movement 9 degrees).
Venus: In a dormant state in the east in the early morning for 82 days, moves 113908 minutes; appeared in the west for 246 days (retrograde 6 degrees, actual movement 240 degrees); dormant movement of 100 degrees, 113908 minutes; appeared in the east (the daily and western positions are the same, dormant for 10 days, retrograde 8 degrees).
Mercury: In a dormant state in the east during the morning for 33 days, moves 612505 minutes; appeared in the west for 32 days (retrograde 1 degree, actual movement 31 degrees); dormant movement of 65 degrees, 612505 minutes; appeared in the east (the daily and western positions are the same, dormant for 18 days, retrograde 14 degrees).
Goodness, this text looks daunting; let's break it down sentence by sentence. The first sentence, “以法伏日度及余，加星合日度余，余满日度法得一，从全命之如前，得星见日及度也。” translates to plain language as: First, use the method of calculating solar degrees to find the remainder, then add the remainder from the planetary conjunction with the solar degrees; if the remaining degrees reach an integer multiple of the solar degrees, this indicates a complete cycle has been achieved, and according to the previous calculation method, you can determine the time and degrees of the planet's appearance.
Next, “以星行分母乘见度，余如日度法得一，分不尽半法以上亦得一；而日加所行分，分满其母得一度，逆顺母不同，以当行之母乘故分，如故母而一，当行分也。” This part is more complex, meaning: Multiply the denominator of the planet's motion by the observed degrees, and calculate the remaining portion using the solar degree method; if it cannot be divided evenly and the remainder exceeds half, treat it as an integer multiple; then add the degrees of the planetary motion to the solar degrees, and if the degrees reach an integer multiple of the denominator, increase by one degree. The denominators for direct and retrograde motion are different; you need to multiply the current motion's denominator by the previous degrees, and then divide by the previous denominator to get the current motion's degrees.
“留者承前，逆则减之，伏不尽度，经斗除分，以行母为率，分有损益，前后相御。” This sentence states: If the planet stops moving, use the previous degrees; if it is in retrograde, subtract the degrees; if the degrees fall short, use a specific method (经斗除分, which refers to a particular calculation method that is hard to translate directly) to calculate, using the running denominator as a ratio; the degrees will have increases and decreases, affecting each other.
The last sentence, “凡言如盈约满，皆求实之除也；去及除之，取尽之除也。” explains the calculation method: Any references to "盈," "约," or "满" indicate precise division, whereas "去" and "除" denote exhaustive division.
Next is the description of the movement of Jupiter: "Jupiter: in the morning it conjuncts with the sun, then continues in the same direction. Sixteen days later, it travels 174,223 arcminutes, the planet moves 2 degrees and 323,467 arcminutes, and is visible in the east after the sun. Continuing in the same direction, it moves rapidly, traversing 11/58 of the sun's path, moving 11 degrees in 58 days. Moving even faster, it traverses 9 arcminutes of the sun's path, moving 9 degrees in 58 days. It remains stationary for 25 days before changing direction. In retrograde motion, it traverses 1/7 of the sun's path, retreating 12 degrees in 84 days. It remains stationary again for 25 days and then moves in the same direction, traversing 9/58 of the sun's path, moving 9 degrees in 58 days. Moving in the same direction, it moves rapidly, traversing 11 arcminutes of the sun's path, moving 11 degrees in 58 days, and is visible in the evening sky. Sixteen days later, it travels 174,223 arcminutes, the planet moves 2 degrees and 323,467 arcminutes, and conjuncts with the sun. In total, one complete cycle lasts 398 days, during which it covers 348,646 arcminutes, and the planet travels 43 degrees and 250,956 arcminutes." This passage outlines a sophisticated method of astronomical calculation, which can be challenging to articulate in contemporary terms, but the core idea is to predict the trajectory and appearance time of planets through a series of calculations. The original text employs numerous ancient astronomical terms, requiring a significant amount of professional knowledge to fully understand.
Sun: In the morning, it appears with the sun and then hides away. Next, it enters a direct motion phase, lasting 71 days and totaling 1,489,868 minutes, indicating that the planet moved 55 degrees and 242,860.5 minutes. Then it can be seen in the east in the morning, behind the sun. During direct motion, it moves 14 degrees and 23 minutes each day, covering a total of 112 degrees over 184 days. The direct motion speed increases a bit, then slows down, moving 12 degrees and 23 minutes each day, covering 48 degrees in 92 days. Then it stops and remains motionless for eleven days. Next, it goes retrograde, moving 17 degrees and 62 minutes each day, retreating 17 degrees in 62 days. After stopping again, it starts direct motion after eleven days, moving 1 degree and 12 minutes each day, covering 48 degrees in 92 days. Once more in direct motion, the speed increases, moving 1 degree and 14 minutes each day, covering a total of 112 degrees over 184 days, at which point it is in front of the sun and then hides in the west by evening. For 71 days, it totals 1,489,868 minutes, meaning the planet moved 55 degrees and 242,860.5 minutes, and then it reappears alongside the sun. This completes one full cycle, totaling 779 days and 973,113 minutes, with the planet moving 414 degrees and 478,998 minutes.
Saturn: In the morning, it appears with the sun and then hides away. Next, it enters a direct motion phase, lasting 16 days and totaling 1,122,426.5 minutes, indicating that the planet moved 1 degree and 1,995,864.5 minutes. Then it can be seen in the east in the morning, behind the sun. During direct motion, it moves 3 degrees and 35 minutes each day, covering a total of 7.5 degrees over 87.5 days. Then it stops, remaining motionless for 34 days. Next, it goes retrograde, moving 1 degree and 17 minutes each day, retreating 6 degrees in 102 days. After 34 days, it starts direct motion again, moving 1 degree and 3 minutes each day, covering a total of 7.5 degrees over 87 days, at which point it is in front of the sun and hides in the west by evening. For 16 days, it totals 1,122,426.5 minutes, meaning the planet moved 1 degree and 1,995,864.5 minutes, and then it reappears alongside the sun. This completes one full cycle, totaling 378 days and 166,272 minutes, with the planet moving 12 degrees and 1,733,148 minutes.
Venus, when it meets the Sun in the morning, first hides (伏), then moves back, retreating four degrees in five days. After that, it can be seen in the east in the morning, behind the Sun. It continues to move retrograde, covering 0.6 degrees each day, retreating six degrees over ten days. Then it stops (留) for eight days without moving. Next, it begins to move direct, at a slow speed (迟), covering three degrees and 46 minutes each day, for a total of 33 degrees over 46 days. Then its speed increases (疾), covering one degree, 91 minutes, and 15 seconds each day, totaling one hundred six degrees in ninety-one days. It then speeds up even more (更顺，益疾), covering one degree, 91 minutes, and 22 seconds each day, totaling one hundred thirteen degrees in ninety-one days. At this point, it is behind the Sun, appearing in the east in the morning. Finally, it moves direct for forty-one days, covering 56,654 minutes, totaling fifty degrees and 56,654 minutes, before it meets the Sun again. One conjunction cycle lasts two hundred ninety-two days and 56,654 minutes, with the same number of degrees traveled by the planet.
In the evening, when Venus meets the Sun, it first hides (伏), then moves direct, covering fifty degrees and 56,654 minutes in forty-one days, before it shows up in the west in front of the Sun. It then continues to move direct, speeding up (疾), covering one degree, 91 minutes, and 22 seconds each day, totaling one hundred thirteen degrees in ninety-one days. After that, its speed decreases (更顺，减疾), covering one degree and fifteen minutes each day, totaling one hundred six degrees in ninety-one days, and then continues moving direct. Its speed slows down (迟), covering three degrees and 46 minutes each day, totaling thirty-three degrees over forty-six days. Then it stops (留) for eight days without moving. Next, it begins to move retrograde (旋，逆), covering 0.6 degrees each day, retreating six degrees over ten days. At this point, it's in front of the Sun, appearing in the west in the evening. It continues to move retrograde, speeding up (逆，疾), retreating four degrees in five days, before it meets the Sun again. Two conjunctions make up a complete cycle, totaling 584 days and 113,908 minutes, with the same number of degrees traveled by the planet.
Mercury, when it meets the sun in the morning, first hides (submerges), then retrogrades, moving back seven degrees over nine days. After that, it can be seen in the east in the morning behind the sun. Continuing to retrograde, its speed increases (faster, swift), moving back one degree each day. Then it stops (pauses), remaining still for two days. Then it starts to move forward, slowly (delayed), moving eight-ninths of a degree each day, moving eight degrees in nine days, and then moves forward. Speeding up (swiftly), moving one and a quarter degrees each day, moving twenty-five degrees in twenty days, at this point it is behind the sun, appearing in the east in the morning. Finally, moving forward, after sixteen days, it moves six million four hundred nineteen thousand six hundred sixty-seven arc minutes, the planet moves thirty-two degrees six hundred nineteen thousand six hundred sixty-seven arc minutes, and then meets the sun again. The cycle of one conjunction is fifty-seven days six hundred nineteen thousand six hundred sixty-seven arc minutes, and the degree of the planet's movement is the same.
Speaking of Mercury, it sets with the sun, then, as if it is lurking, its trajectory is forward. Specifically, in sixteen days, it will travel thirty-two degrees six hundred nineteen thousand six hundred sixty-seven arc minutes of a circle (original text: sixteen days six hundred nineteen thousand six hundred sixty-seven parts planet thirty-two degrees six hundred nineteen thousand six hundred sixty-seven parts). In the evening, you can see it in the west, and it is always in front of the sun.
When moving forward, it moves quite quickly, moving one and a quarter degrees per day, able to move twenty-five degrees in twenty days. If it moves slowly, it only moves seven-eighths of a degree per day, taking nine days to move eight degrees. If it stagnates, it remains still for two days. Then, it will retrograde, moving backwards, retreating one degree per day, still in front of the sun, lurking in the west in the evening. When retrograding, it moves slower, taking nine days to retreat seven degrees, and finally meets the sun again.
From its conjunction with the sun to the next conjunction, this cycle lasts 115 days and roughly 601,205 parts of a day. Mercury's movement follows this same pattern.
