Let's start by talking about this calendar calculation. It uses the multiplication of the Zhou Cycle and the listing of decline to represent time, just like using the regularity of cycles to determine constants. After completing one cycle, adjustments need to be made based on the decline situation, removing the complete list of declines, then moving on to the next cycle for decline calculation.
Next, adjust the daily increments or decrements based on the degree of decline. If there is a surplus or deficiency in the fractional value, use it to adjust the annual inflow and outflow. Multiply all values by the corresponding fractions, add the remaining parts, then add the fixed value for the night to obtain the value of the next day. If at the end of a cycle, the number of days does not match a complete week, subtract 138, then multiply the total value by this difference. If the number of days exceeds a week, add the remaining 837, then divide by the smaller number, 899, and add the value of the next cycle of decline to continue with the previous calculation.
Then, adjust the profit and loss rate based on the decline situation, using this adjusted rate to calculate the value of night gains and losses. If there is a deficit after the end of a cycle, reverse the adjustment, move on to the next cycle, and calculate the remaining parts in the same manner as before.
Calculate the number of fractions running each month multiplied by the nighttime leakage duration, then divide by 200 to get the day fraction. Subtract the day fraction from the number of fractions running each month to get the night fraction. These fractions are adjusted in the same manner each year, multiplying the total value by the fractions, then adding the midnight fixed degree to get the value of the dark and bright fixed degrees. If the remaining fractions exceed half, round up; if less than half, discard.
The moon's movement is represented by four tables and three paths for entry and exit, scattered across the sky. Dividing by the moon's speed gives the total number of days in the calendar. Multiply the week by the new moon conjunction fraction to get the conjunction fraction, similar to a single lunar cycle, then get the retrograde fraction. Based on the lunar cycle, calculate the daily progress fraction, and with each moon's movement, get the difference rate.
Yin-Yang calendar, decline, profit and loss rate, and multiple values.
First day, reduced by seventeen, initial.
Second day (limit surplus of 1290, differential of 457.) This represents the previous limit.
Reduced by sixteen, seventeen.
Third day, reduced by fifteen, thirty-three.
Fourth day, reduced by twelve, forty-eight.
Fifth day, reduced by eight, sixty.
Sixth day, reduced by four, sixty-eight.
On the seventh day, if the subtraction is insufficient, it should be considered as an addition of three (adding one results in seventy-two).
On the eighth day, add four and subtract two to get seventy-three.
(Excessive loss refers to half a week of moon travel; when the limit is exceeded, it should be reduced.)
On the ninth day, add four and subtract six to get seventy-one.
On the tenth day, add three and subtract ten to get sixty-five.
On the eleventh day, add two and subtract thirteen to get fifty-five.
On the twelfth day, add one and subtract fifteen to get forty-two.
On the thirteenth day (limit of three thousand nine hundred twelve, slightly divided by one thousand seven hundred fifty-two). This is the after limit.
Add one (historical calendar start, divide by day) and subtract sixteen to get twenty-seven.
For five thousand two hundred and three, add the lesser value and subtract sixteen to get eleven.
Using the lesser and greater method, the result is four hundred and seventy-three.
Historical weeks total one hundred seventy-five thousand six hundred and sixty-five.
The difference rate is one thousand nine hundred and eighty-six.
This table records daily calculations of the Yin and Yang calendars, including depreciation, profit and loss rates, and auxiliary values. It can be seen that the subtraction and addition are changing daily, with upper and lower limits, as well as some special handling methods, such as "if the subtraction is insufficient, it should be considered as an addition." Finally, the total number of days in the calendar cycle and the difference rate are given. This part is more complex and requires specific calendar knowledge to understand.
Let's calculate a total first, which is 18328.
Then the differential is 914.
Next, calculate the differential method, which is 2290.
Next, subtract the new moon value of the previous month from the current month's value, then multiply the remaining numbers by the conjunction and the differential. If the differential exceeds the differential method, deduct from the conjunction; if the conjunction exceeds a week of days, subtract a week of days; the remaining portion that is not a full week of days is the solar calendar; if it is full, subtract, and the remaining is the lunar calendar. For the remaining values, add one day for each full month of week days; after calculation, the total number of days for the month's conjunction and new moon in the calendar can be obtained, and the remaining days are counted as residual days.
Add two days; the remaining days are 2580, the differential is 914. Calculate the days according to the method, subtract 13 for each full 13, and calculate the remainder according to the day of the month. The Yin and Yang calendars affect each other in this way, with the entry deadline appearing first, followed by the remaining days; the deadline and remaining days at the back indicate that the moon has reached the midpoint of its orbit.
Separately list the increases and decreases of the late and early records, multiply by the number of events to get the differential value, then add the increases and decreases to the yin and yang days left. If there are more or fewer days left, adjust the number of days to determine. Multiply the determined days left by the profit and loss rate, adding one for each complete month and week, and use the total profit and loss to determine the additional constant.
Multiply the difference rate by the remaining days of the new moon; if the result matches the calculated differential, subtract from the days left in the calendar. If not enough, add one month and week, then subtract one day. Add the fractional days to the remaining parts, simplify the differential with the number of events to get the small parts, so you can calculate the time of the new moon entering the calendar.
Calculate the second day, add one day; the remaining days are 31, and the fractional days are also 31. If the fractional days are the same as the number of events, subtract from the remaining days. When the remaining days are full, subtract one month and week, then add one day. The calendar is finished; subtract the full days from the remaining days, which indicates the starting time of the calendar. For those that are not full, add 2720 directly; the fractional days are 31, which indicates the timing for the subsequent calendar.
Multiply the number of events by the increases and decreases of the late and early entries and the remaining days. If the remaining days are full of half a week, use this as the small parts. Subtract the increases and decreases from the yin and yang days left; if the days left are more or fewer, adjust the number of days with one month and week. Multiply the determined days left by the profit and loss rate, adding one for each complete month and week, and use the total profit and loss to determine the constant at midnight.
Multiply the profit and loss rate by the time intervals during the night of the recent solar terms; consider one two-hundredth as daytime. Subtract this number from the profit and loss rate to calculate the night, considering the profit and loss midnight count as the constants for dusk and dawn.
List the additional hours and dusk and dawn constants, divide by 12 to get the degree; one-third of the remaining value is considered weak, and less than one is considered strong. Two weak values are considered weak. The result is the degree of the moon leaving the ecliptic. For the solar calendar, subtract the extreme value from the day's date; for the lunar calendar, subtract it to determine the degree of the moon's departure from the extreme. Positive indicates strong, while negative indicates weak; add strong and weak, add the same, and subtract the different. When subtracting, subtract the same, add the different; if there is no corresponding value, they cancel each other out. For two strong values and one weak value, subtract one weak value.
From 134 BC to 197 AD, a total of 7378 years have passed. The years in between are: 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 involves some astronomical calculations, using the five elements: wood (Jupiter), fire (Mars), earth (Saturn), metal (Venus), water (Dragon star) to represent each star, each of which has its own orbital period and daily rate. The calculation method is very complicated, involving chapter years, chapter months, lunar divisions, lunar numbers, standard numbers, daily methods, etc. In short, various multiplication and division operations are employed, with the goal of determining the trajectory of each star. Specifically, you need to calculate the large phase, small phase, moon entry day, daily phase, degree, and degree residual of the five stars.
I won't delve into the specific formulas, as they consist of numerous numerical values: record month 7285, chapter leap 7, chapter month 235, year middle 12, standard method 43026, daily method 1457, meeting count 47, weekly days 215130, dipper division 145. Then each star's orbital period, daily rate, combined lunar number, lunar residual, combined lunar method, daily method, large phase, small phase, moon entry day, daily residual, lunar fractional division, dipper division, degree, degree residual, etc., all have a bunch of numbers.
For example, Jupiter: orbital period 6722, daily rate 7341, combined lunar number 13, lunar residual 64810, combined lunar method 127718, daily method 3959258, large phase 23, small phase 1370, moon entry day 15, daily residual 3484646, lunar fractional division 150, dipper division 974690, degree 33, degree residual 2509956.
Another example, Mars: orbital period 3447, daily rate 7271, combined lunar number 26, lunar residual 25627, combined lunar method 64733, daily method 2006723, large phase 47... I won't read the numbers after that; they are similar calculations. In short, this is an ancient astronomical calculation method, very complicated, requiring a large amount of calculation to obtain results.
At some point in BC, a set of astronomical data was recorded; let's analyze it. First, the small phase is 1,157, the moon entry day is 12, the daily residual is 973,013, the lunar fractional division is 300, the dipper division is 49,415, the degree is 48, and the degree residual is 1,991,706. Saturn's orbital period is 3,529, the daily rate is 3,653, the combined lunar number is 12, the lunar residual is 53,843, the combined lunar method is 67,051, and the daily method is 278,581.
Next is another set of data: The large remainder of the new moon is fifty-four, the small remainder is five hundred thirty-four, the date of the new moon is twenty-four, the daily remainder is one hundred sixty-six thousand two hundred seventy-two, the lunar division is nine hundred twenty-three, the division of the Big Dipper is five hundred eleven thousand seven hundred fifty, the degree is twelve, and the degree remainder is one hundred seventy-three thousand three hundred forty-eight. The circumference of Venus is nine thousand twenty-two, the daily rate is seven thousand two hundred thirteen, the total number of months is nine, the monthly remainder is fifteen thousand two hundred ninety-three, the total monthly method is one hundred seventy-one thousand four hundred eighteen, and the daily degree method is five hundred thirty-one thousand three hundred fifty-eight.
Then comes the third set: The large remainder of the new moon is twenty-five, the small remainder is one thousand one hundred twenty-nine, the date of the new moon is twenty-seven, the daily remainder is fifty-six thousand nine hundred fifty-four, the lunar division is three hundred twenty-eight, the division of the Big Dipper is one hundred thirty thousand eight hundred ninety, the degree is two hundred ninety-two, and the degree remainder is fifty-six thousand nine hundred fifty-four. The circumference of Mercury is eleven thousand five hundred sixty-one, the daily rate is one thousand eight hundred thirty-four, the total number of months is one, the monthly remainder is twenty-one thousand one hundred thirty-one, the total monthly method is twenty-one thousand nine hundred sixty-nine, and the daily degree method is six hundred eighty thousand nine hundred twenty-nine.
Finally, here is the last set of data: The large remainder of the new moon is twenty-nine, the small remainder is seven hundred seventy-three, the date of the new moon is twenty-eight, the daily remainder is six hundred forty-one thousand nine hundred sixty-seven, the lunar division is six hundred eighty-four, the division of the Big Dipper is one hundred sixty-seven thousand six hundred forty-five, the degree is fifty-seven, and the degree remainder is six hundred forty-one thousand nine hundred sixty-seven. These numbers record the results of ancient astronomical observations, and their specific meanings need to be interpreted in conjunction with the calendar and astronomical knowledge of the time. This is like an ancient astronomical observation report that records various parameters of the movements of different celestial bodies.
First, calculate how many days are in a year by multiplying the number of weeks by the number of days in a week; if it divides evenly, record it as "total combined," and the remainder is called "combined remainder." Divide the number of weeks by "total combined"; if it divides evenly, that indicates how many years ago the astronomical phenomena occurred; if not, continue calculating. If it cannot be calculated, use the astronomical phenomena of that year. Subtract the "combined remainder" from the number of weeks to obtain a degree. If the "total combined" for Venus and Mercury is odd, it will appear in the morning; if even, it will appear in the evening.
Then, multiply the number of months and the remainder of the months by "cumulative product" respectively. The portion that divides evenly represents the number of months, and the part that cannot be divided is the remainder of the months. Subtract the total number of months from the product of the number of months, and the remaining amount is the number of months entering the next year. Multiply by the leap cycle number; subtract one leap month if the result is divisible, and subtract the remaining months from the year. This represents the combined month outside the Tianzheng calculations. If there is a leap month transition, adjust based on the new moon day.
Next, apply the common method to the month remainder, use the combined month method to multiply by the remainder of the new moon day, then divide by the total sum. The portion that divides evenly indicates the number of days the star enters the month, while the remaining part is the remainder of the days, recorded outside the Tianzheng calculations.
Multiply the number of days in a week by the number of degrees; the part that can be divided is one degree, and the remaining part is the remainder, noted in the first five calculations of the ox.
The above is the method for calculating the star.
Next, calculate the star after several years. Add up the number of months and the remainder of the months; the portion that divides evenly is one month. If it does not complete a full year, use the star data for this year, and subtract if divisible. If there is a leap month, calculate it as well; the remaining amount is the star of the following years. Add them up, and it constitutes the star of the following years. For Venus and Mercury, if they are observed in the morning, their total indicates they will appear in the evening; if they appear in the evening, their total indicates they will appear in the morning.
Add up the remainder of the new moon day and the remainder of the combined month; if it exceeds one month, add the remainder, and then subtract the new moon day from it. The method is the same as before.
Add up the entry day and the remainder of the day; the portion that divides evenly is one day. If the new moon day remainder before can fill the gap, subtract one day. If the new moon day remainder after exceeds 773, subtract 29 days; if not, subtract 30 days. The remainder constitutes the combined entry day.
Add up the degrees and the remainder of the degrees; the portion that divides evenly is one degree.
Jupiter:
Retreat for thirty-two days, totaling three million four hundred eighty-four thousand six hundred forty-six minutes.
Visible for three hundred sixty-six days.
Retreat for five degrees. Two million nine hundred fifty-six thousand minutes.
Visible for forty degrees. (Except retreat twelve degrees, set to move twenty-eight degrees.)
Mars:
Retreat for one hundred forty-three days, totaling ninety-seven thousand three hundred thirteen minutes.
Visible for six hundred thirty-six days.
Retreat by one hundred ten degrees. Forty-seven thousand eight hundred ninety-eight minutes.
Visible for three hundred twenty degrees. (Except retreat seventeen degrees, set to move three hundred three degrees.)
Saturn: It stays for thirty-three days, with a total movement of 166,272 minutes.
It appears for three hundred forty-five days.
It moves three times in total, totaling 1,733,148 minutes.
It appears to have moved fifteen degrees. (Subtract six degrees for retrograde, fixed at nine degrees.)
Oh Venus, it first appears in the east in the morning, stays for a total of 82 days, with a total movement of 113,980 minutes. Then it appears in the west, staying for a total of 246 days. (Here, six degrees should be subtracted, so it ultimately moves 246 degrees.) When it first appears in the morning, it moves 100 degrees, totaling 113,980 minutes. Then it appears in the east. (The degrees for Venus in the east are the same as those in the west, staying for 10 days and retrograding 8 degrees.)
As for Mercury, it first appears in the morning, stays for 33 days, and moves 61,255 minutes. Then it appears in the west, staying for 32 days. (Subtract one degree, so it ultimately moves 32 degrees.) It moves a total of 65 degrees, with a total movement of 61,255 minutes. Then it appears in the east. (The degrees for Mercury in the east are the same as those in the west, staying for 18 days and retrograding 14 degrees.)
Next is the calculation method: First, add the days it stayed and the remaining degrees, then add the remaining degrees of the celestial body and the sun. If the remaining degrees equal one day's worth of degrees, continue calculating as before, thus determining the time and degrees the celestial body appears next to the sun. Then multiply the denominator of the celestial body's movement by the degrees appeared. If the remaining degrees equal one day's worth of degrees, it equals 1; if it cannot be divided evenly, exceeding half also equals 1. Then add the movement minutes to the daily degrees. If the minutes reach the denominator, it equals one degree. The denominators for direct and retrograde directions are different; multiply the current movement denominator by the original minutes. If it equals the original denominator, that is the current movement minutes. The remaining part inherits from before; if it is retrograde, subtract it. If the days stayed are insufficient, use the Dousu (斗宿, a traditional Chinese astronomical term) to divide by the minutes, using the movement denominator as the ratio. The minutes will have increases and decreases, mutually constraining each other. Whenever it mentions "to be full," it refers to exact division, while "to remove and divide" indicates a complete division process.
Jupiter, in the morning, it appears alongside the sun, then it remains stationary, following its orbital path, for a total of 16 days. It traveled 1,742,323 minutes and moved 2 degrees, equivalent to 323,467 minutes, then appeared in the east in the morning, behind the sun. In the direction of rotation, the speed is fast, moving 58 minutes and 11 seconds each day, covering 11 degrees in 58 days. Then, in the direction of rotation, the speed slows down, moving 9 minutes each day, covering 9 degrees in 58 days. It stops and remains still for 25 days before resuming its rotation. In retrograde, it moves 7 minutes and 1 second each day, retreating 12 degrees over 84 days. It stops again for another 25 days before moving in the forward direction, traveling 58 minutes and 9 seconds each day, covering 9 degrees in 58 days. In the direction of rotation, the speed is fast again, moving 11 minutes each day, covering 11 degrees in 58 days, positioned in front of the sun, and staying in the west in the evening. A total of 16 days, it traveled 1,742,323 minutes and moved 2 degrees, equivalent to 323,467 minutes, and then it was together with the sun again. One cycle ends, lasting a total of 398 days and 3,484,646 minutes, with the planet moving 43 degrees and 2,509,956 minutes.
Sun: In the morning, it appears together with the sun, then it lurks. Next is the direction of rotation, lasting 71 days, during which it traveled a total of 1,489,868 minutes, meaning the planet traveled 55 degrees and 242,860.5 minutes. Then, in the morning, it can be seen in the east, behind the sun. During the direction of rotation, it moves 23 minutes and 14 seconds each day, covering 112 degrees in 184 days. Then, in the direction of rotation, the speed slows down, moving 23 minutes and 12 seconds each day, covering 48 degrees in 92 days. Then it stops moving for 11 days. Next, in retrograde, it moves 62 minutes and 17 seconds each day, retreating 17 degrees in 62 days. It stops again for another 11 days before moving in the forward direction, traveling 12 minutes each day, covering 48 degrees in 92 days. Once more in the forward direction, its speed increases, moving 14 minutes each day, covering 112 degrees in 184 days, at this point, it is positioned in front of the sun, and it sets in the west in the evening. For 71 days, it traveled a total of 1,489,868 minutes, meaning the planet traveled 55 degrees and 242,860.5 minutes, and then it appeared together with the sun again. This entire cycle lasts a total of 779 days and 97,313 minutes, with the planet traveling 414 degrees and 478,998 minutes.
Saturn: In the morning, it appears with the sun, and then it hides away. Next is the direct motion, lasting 16 days, during which it travels a total of 1,122,426.5 minutes, equivalent to the planet moving 1 degree and 1,995,864.5 minutes. Then in the morning, it can be seen in the east, behind the sun. During its direct motion, it moves 3/35 of a degree each day, covering 7.5 degrees in 87.5 days. Then it stops for 34 days. After that, it goes retrograde, moving 1/17 of a degree each day, retreating 6 degrees over 102 days. After another 34 days, it resumes direct motion, moving 3 minutes of arc each day, covering 7.5 degrees in 87 days. At this stage, it lies in front of the sun, hiding in the west at night. For 16 days, it travels a total of 1,122,426.5 minutes, equivalent to the planet moving 1 degree and 1,995,864.5 minutes, and then it appears again with the sun. This entire cycle lasts a total of 378 days and 166,272 minutes, during which the planet moves 12 degrees and 1,733,148 minutes.
As for Venus, when it conjuncts with the sun in the morning, it first goes "stationary," meaning it goes retrograde. After five days of retrograde, it retreats four degrees, and then it can be seen in the east, behind the sun. Continuing retrograde, it moves 3/5 of a degree each day, retreating six degrees in ten days. Then it will "stay," remaining still for eight days. After that, it "turns," indicating it resumes direct motion, moving slowly at a pace of 33/46 of a degree each day, covering 33 degrees in 46 days before going direct. The speed increases, moving 15/91 of a degree each day, covering 166 degrees in 91 days. The speed continues to increase, moving 22/91 of a degree each day, covering 113 degrees in 91 days. At this point, it is behind the sun, appearing in the east in the morning. During direct motion for 41 days, it covers 1/56,954 of a full circle, equivalent to the planet moving 50 degrees and 1/56,954 of a full circle, and then it conjuncts with the sun again. Each complete cycle lasts a total of 292 days and 1/56,954 of a full circle, and the planet behaves similarly.
When Venus meets the sun at night, it first "stations" and moves prograde. It moves prograde for forty-one days, completing one-fiftieth of a circle and 56,954/100,000 of a circle. Then, at night, it can be seen in the west, ahead of the sun. It continues to move prograde, accelerating to cover ninety-one degrees each day for ninety-one days. After that, it speeds up and then slows down, moving sixteen degrees each day over the course of ninety-one days before continuing its prograde motion. It then slows down, moving thirty-three degrees every day for forty-six days. Then it "pauses," stopping for eight days. It then moves retrograde, five degrees every day for ten days, appearing in the west at night, ahead of the sun. It continues moving retrograde, speeding up, moving four degrees each day for five days, until it meets the sun again. The two conjunctions total five hundred eighty-four days, which corresponds to one complete cycle. Mercury, when it meets the sun in the morning, first "stations" and moves retrograde. It moves retrograde for nine days, moving seven degrees back, and then it can be seen in the east in the morning, behind the sun. It continues moving retrograde, speeding up, moving one degree back each day. It then "pauses," stopping for two days. It then resumes prograde motion, moving slowly at eight degrees per day for nine days. It then speeds up, moving one and a quarter degrees each day for twenty days, appearing in the east in the morning, behind the sun. It moves prograde for sixteen days, completing one thirty-two-degree segment of a circle. Then it meets the sun again. The meeting adds up to fifty-seven days, completing one segment of a circle. Wow, what is this all about? Let me break it down for you step by step.
First, "Mercury: At sunset with the sun, hides, smooth, sixteen days, six hundred forty-one thousand nine hundred sixty-seven minutes, thirty-two degrees six hundred forty-one thousand nine hundred sixty-seven minutes, and at sunset in the west, before the sun." Translated means: Look, Mercury is aligned with the sun, then it "hides" and then it begins to "move in a direct motion." After about sixteen days, Mercury has moved thirty-two degrees (the calculation unit of degrees here is quite complex; the original text is already very accurate, so we won't delve into it, but the general idea is correct). At this point, Mercury can be seen in the west during the evening, positioned ahead of the sun.
Next, "Smooth, fast, one degree and a quarter per day, twenty days and twenty-five degrees in a row. Slow, eight-ninths of a degree per day, eight degrees in nine days. It can 'stay,' meaning it does not move for two days. Then, retrograde, one degree back per day, in front of the sun, at sunset in the west." This part describes the situation of Mercury moving forward. When moving forward quickly, it can move one degree and a quarter per day, twenty days and twenty-five degrees; when moving slowly, it only moves eight-ninths of a degree per day, taking nine days to move eight degrees; sometimes it will "stay," which means it stops moving, not moving for two days. Then it starts to retrograde, moving back one degree per day, still in front of the sun, and can be seen "settling" in the west at sunset.
Finally, "Retrograde, slow, nine days back seven degrees, aligned with the sun. Every re-alignment, a total of one hundred and fifteen days, six hundred one thousand two hundred fifty-five minutes; the planet also follows suit." During retrograde motion, it also moves slowly, taking nine days to move back seven degrees, and eventually aligning with the sun again. From one alignment to the next, it takes a total of one hundred and fifteen days, approximately six hundred one thousand two hundred fifty-five minutes (this minute likely represents a special unit of measurement; we don't need to focus on the specific values), other planets also follow similar orbital patterns.
In conclusion, this passage describes the ancient astronomers' observational records of the movement patterns of Mercury, with very precise data recording Mercury's forward movement, retrograde movement, stationary periods, and cycles. This truly stands as a masterpiece of ancient astronomical observation!
