Let's first talk about how to calculate the position of the sun at midnight each night. Start by multiplying the number of days by a fixed value (the calendar system), then subtract 360 degrees (the full circle). Divide the remaining number by that fixed value (the calendar system) to get the result in degrees. Starting from the fifth degree of the constellation of the Ox, subtract the degree of the constellation; if it's less than one full constellation, that indicates the position of the sun at midnight on the day of the new moon.
How do we calculate the position for the next day? Simply add one degree to the previous day's value, then divide by 360 degrees. If the remainder is insufficient, subtract one degree and add it to the fixed value (the calendar system).
Next, we consider the moon. Multiply the number of days in the month by a fixed value (the calendar system), subtract 360 degrees, and if the remaining number is divisible by the fixed value, that is the degree; the remainder represents the minutes. The method is the same as calculating the position of the sun, allowing us to determine the position of the moon at midnight on the day of the new moon.
How do we calculate it for the next month? For a short month, add 22 degrees and 258 minutes; for a long month, add 13 degrees and 217 minutes to the previous month's calculation. If it exceeds the fixed value, add one degree. For example, towards the end of winter, the moon is roughly near the Ox and Heart constellations.
Then, to calculate the positions of the sun and moon during the new moon (the first day of the lunar month), multiply the year by the remainder of the new moon day. The portion that is divisible by the number of new moons (the counting method) is the large minutes, and the remainder is the small minutes. Subtract the degree of the sun at midnight from the amount of the large minutes using the same method as before to find the positions of the sun and moon during the new moon.
How do we calculate the position for the next month? Add 29 degrees, 312 large minutes, and 25 small minutes. If the small minutes are sufficient for counting, subtract them from the amount of the large minutes. If the amount of the large minutes is sufficient for the calendar system, subtract it, then divide the remaining amount by 360 degrees.
Next, we calculate the position of the first quarter moon and full moon (the seventh, fifteenth, and twenty-third days of the lunar month). Add 7 degrees, 225 minutes, and 17.5 small minutes to the degree of the new moon, using the same method as before to find the position of the first quarter moon. By continuing this pattern, we can find the positions of the full moon, last quarter moon, and the new moon of the following month.
For the positions of the first quarter moon, full moon, and last quarter moon (the seventh, fifteenth, and twenty-third days of the lunar month), add 98 degrees, 480 large minutes, and 41 small minutes to the degree of the new moon, using the same method as before to find the position of the first quarter moon. By continuing this pattern, we can find the positions of the full moon, last quarter moon, and the new moon of the following month.
Finally, let's talk about how to calculate the degrees of solar and lunar eclipses (sunrise and sunset). Multiply the solar terms by the nighttime duration of the nearest solar term (calculated in time units), then divide by 200, and the result is the sunrise time. Subtract the number of days from the solar terms and the number of months from the number of weeks; the remaining time will give you the sunset time. Add these to the time at midnight to calculate the degrees.
Next, let's discuss how to calculate a solar eclipse. First, establish a starting year (Shangyuan year), then subtract the conjunction cycle (Hui Sui), multiply the remaining years by the conjunction cycle rate (Hui rate), and the result is the accumulated eclipse; if there is a remainder, add one. Then multiply the conjunction month by the remaining years; the result is the accumulated month, and the leftover is the month remainder. Multiply the remaining years by the leap year cycle; the part that can be divided by the chapter year is the accumulated leap year. Subtract this from the accumulated month, then subtract the number of years in the middle of the year; the remaining part starts counting from the Tianzheng calendar.
How do you calculate the next solar eclipse? Add five months; the month remainder is 1635. If it can be divided by the conjunction cycle rate, add one month, calculated based on the full moon.
Finally, based on the remainder of the winter solstice, multiply its small remainder, which indicates the day designated for earth-related activities. Then add the small remainder 175; if it can be divided by the value of the Qian hexagram, it is the day when the Zhongfu hexagram is used.
How do you calculate the next hexagram? Add the large remainder 6 and the small remainder 103. For the four positive hexagrams, use the value of one of the days, multiplying its small remainder.
First, we need to calculate the large and small remainders of the winter solstice day. The large remainder is 2356, and the small remainder is 927. Subtract 27 from the large remainder to get 2329, which is the day designated for earth-related activities. Subtract another 18 to get 2311, which is the day of wood use on Lichun. Subtract 73 more to get 2238, which is earth again. Since earth generates fire, there's no need to calculate metal and water; they are inherently linked to this method.
Next, multiply the small remainder by 12 to calculate a Chen (hour). Start counting from midnight; the small remainders of the days of the new moon, first quarter moon, and full moon need to be calculated separately. Multiply the small remainder by 100 to calculate a quarter of an hour; if it falls short of one-tenth, represent it as a fraction, then calculate from midnight based on the nearest solar term. If the water level is not full at night, use the nearest value.
During the calculation, there may be situations of advancement and retreat; add for advancement and subtract for retreat. The difference between advancement and retreat is calculated from two degrees, decreasing by four degrees each time. When it reaches half, multiply by three until the difference reaches three, then after five degrees, return to the initial state.
The speed of the moon's movement varies, but the cycle is constant. By combining various numbers from the heavens and the earth, using the product of the remainders divided by the number of days in a week, you can calculate the lunar calendar. The change in the speed of the moon's movement is regular. By using this rule to adjust the speed of the moon's movement, you can calculate the degrees and minutes of the moon's movement each day. The fluctuations in speed represent the profit and loss rate. Profit and loss affect each other, and the final result is the cumulative profit and loss. By multiplying half a small cycle by a standard method, then dividing by a standard number, and then subtracting the historical cycle, you can calculate the running fraction of the new moon.
Below are the specific calculation results presented in table form for easy reference:
Daily Rotation Degrees and Minutes | Column decline | Profit and Loss Rate | Cumulative Profit and Loss | Monthly Running Fraction
------- | -------- | -------- | -------- | --------
One day fourteen degrees and ten minutes | One retreat | Profit twenty-two | Initial profit | Two hundred and seventy-six
Two days fourteen degrees and nine minutes | Two retreats | Profit twenty-one | Profit twenty-two | Two hundred and seventy-five
Three days fourteen degrees and seven minutes | Three retreats | Profit nineteen | Profit forty-three | Two hundred and seventy-three
Four days fourteen degrees and four minutes | Four retreats | Profit sixteen | Profit sixty-two | Two hundred and seventy
Five days fourteen degrees | Four retreats | Profit twelve | Profit seventy-eight | Two hundred and sixty-six
Six days thirteen degrees and fifteen minutes | Four retreats | Profit eight | Profit ninety | Two hundred and sixty-two
Seven days thirteen degrees and eleven minutes | Four retreats | Profit four | Profit ninety-eight | Two hundred and fifty-eight
Eight days thirteen degrees and seven minutes | Four retreats | Loss | Profit one hundred and two | Two hundred and fifty-four
Nine days thirteen degrees and three minutes | Four retreats | Loss four | Profit one hundred and two | Two hundred and fifty
Ten days twelve degrees and eighteen minutes | Three retreats | Loss eight | Profit ninety-eight | Two hundred and forty-six
Eleven days twelve degrees and fifteen minutes | Four retreats | Loss eleven | Profit ninety | Two hundred and forty-three
This passage describes a complex ancient method of calendar calculation, which involves a lot of astronomical and mathematical knowledge. In short, it is to determine the daily movement of the moon through a series of complex calculations. On the twelfth, the moon reached the position of twelve degrees and eleven minutes. After subtracting three, then adding, and finally subtracting fifteen, the surplus is seventy-nine, bringing the total to two hundred and thirty-nine.
On the 13th, the moon's position is 12° 8′. Subtract 2, then add the previous value, then subtract 18, yielding a surplus of 64, which brings the total to 236.
On the 14th, the moon's position is 12° 6′. Subtract 1, then add the previous value, then subtract 20, yielding a surplus of 46, which brings the total to 234.
On the 15th, the moon's position is 12° 5′. Add 1, then subtract the previous value, then subtract 21, yielding a surplus of 26, which brings the total to 233.
On the 16th, the moon's position is 12° 6′. Add 2, then subtract the previous value, then subtract 20 (note: if the subtraction amount is insufficient, you can reverse the operation; for example, subtracting 5 means adding 5. If the surplus is 5, add 5, and the original subtraction of 20 is now reduced to 20). Yielding a surplus of 5, adjust the initial value, which brings the total to 234.
On the 17th, the moon's position is 12° 8′. Add 3, then subtract the previous value, yielding 18, reduce by 15, which brings the total to 236.
On the 18th, the moon's position is 12° 11′. Add 4, then subtract the previous value, yielding 15, reduce by 23, which brings the total to 239.
On the 19th, the moon's position is 12° 15′. Add 3, then subtract the previous value, yielding 11, reduce by 48, which brings the total to 243.
On the 20th, the moon's position is 12° 18′. Add 4, then subtract the previous value, yielding 8, reduce by 59, which brings the total to 246.
On the 21st, the moon's position is 13° 3′. Add 4, then subtract the previous value, yielding 4, reduce by 67, which brings the total to 250.
On the 22nd, the moon's position is 13° 7′. Add 4, then add the previous total, reduce by 71, which brings the total to 254.
On the 23rd, the moon's position is 13° 11′. Add 4, then add the previous total, reduce by 71, which brings the total to 258.
On the 24th, the moon's position is 13° 15′. Add 4, then add the previous total, reduce by 67, which brings the total to 262.
On the 25th, the moon's position is 14°. Add 4, then add the previous value again, reduce by 59, which brings the total to 266.
On the 26th, the moon's position is 14° 4′. Add 3, then add the previous value, reduce by 47, which brings the total to 270.
On the 27th, the moon's position is at fourteen degrees and seven minutes. This is a special day; add three units, then add three significant Sundays, subtract nineteen, and subtract thirty-one, resulting in a total of two hundred and seventy-three.
On Sunday, the moon's position is at fourteen degrees and nine minutes. Make a minor deduction, then add, and reduce by twelve, giving a total of two hundred and seventy-five.
Here are some astronomical data:
Sunday minutes count, three thousand three hundred and three.
Zhou Xu (a term for the remaining days of the week), two thousand six hundred and sixty-six.
Sunday law, five thousand nine hundred and sixty-nine.
Through Sunday, one hundred and eighty-five thousand thirty-nine.
Calendar Sunday, one hundred and sixty-four thousand four hundred and sixty-six.
A little big law, one thousand one hundred and one.
The first day of the lunar month, ten thousand eight hundred and one.
Small points, twenty-five.
Zhou Ban, one hundred and twenty-seven.
The last paragraph explains the calculation method: use the product of the lunar month and the new moon's position; if the small points reach thirty-one, subtract from the large points. If the large points reach one hundred and sixty-four thousand four hundred and sixty-six, subtract that as well. The remainder divided by the Sunday law (five thousand nine hundred and sixty-nine) gives the number of days; if the result is less than one day, it is considered the remainder. The remainder is calculated independently, and finally, the result of the new moon entering the calendar is obtained.
First paragraph:
Let's calculate the next month, add one more day, resulting in a total of 5832 days and an additional 25 small points.
Second paragraph:
Then calculate the dates of the crescent moon (the eighth and twenty-third of each lunar month); add seven days to each, resulting in a total of 2283 days and 29.5 small points. Convert the small points to days according to the rules; subtract when it reaches 27 days, and calculate the remaining days based on the number of weeks. If it is less than a week, subtract one day and add Zhou Xu (the remaining days of the week). Accumulate the gains and losses of the calendar, using the number of days in a week as the base. Then use the total number of days multiplied by the remainder of the days, multiplied by the gain and loss rate, to adjust the base; this is the gain and loss of time. Subtract the lunar month minutes from the calendar year, multiply by half the number of days in a week, to get the difference. Use it for division to obtain the remainder of the gain and loss, similar to how we calculate the gain and loss of days, adding time to the new moon (first day) in the preceding days. The crescent moon advances and retreats to determine the small remainder value.
Third paragraph:
Multiply the chapter year by the profit and loss of hours, divide by the method of differences, and the resulting full cycle number represents the magnitude of profit and loss. Adjust the position of the day and month using the profit and loss. If the profit and loss are insufficient, use the recording method to adjust the degrees and determine the positions of the day and month. Multiply half of the number of days in a week by the remainder of the new moon day, divide by the total number of days, and subtract from the remainder of the calendar day. If the subtraction is not sufficient, add the number of days in a week and subtract, then subtract one day. After subtracting, add the number of days in a week and its fraction to get the time of midnight entry into the calendar.
When calculating the second day, count back one day. If the remainder of the day reaches 27 days, subtract a fraction of the number of days in a week. If it is not a whole number of days in a week, make up the remainder with a virtual week, and the remaining is the remainder of the second day's entry into the calendar. Multiply the remainder of the day for the midnight entry into the calendar by the profit and loss rate. If it divides evenly by the number of days in a week, it yields an integer. The undivided portion represents the remainder, which is used to adjust the profit and loss accumulation. If the remainder is not enough to subtract, use the total number divided by the number of days in a week to subtract. This is the profit and loss at midnight. Full chapters are degrees, and insufficient ones are minutes. Multiply the total number of days by the fraction and remainder, handle the remainder as the number of days in a week, and handle the degrees as the recording method when the fraction is full. Add for profit and subtract for loss, adjust the degrees and remainder of the midnight, and determine the degrees.
Multiply the remainder of the entry into the calendar by the decay factor. If it divides evenly by the number of days in a week, the undivided portion represents the remainder, indicating the daily change and decay. Multiply the virtual week by the decay factor as a constant. When the calendar ends, add the change and decay, subtract when the column decay is full, and convert to the change and decay of the next calendar. Use the change and decay to adjust the conversion of calendar days to minutes, the profit and loss of fractions, which is the degree of entry and exit of chapters. Multiply the total number of days by the fraction and remainder, add the degrees determined at night, and it is the second day. If the calendar ends with a non-integer number of days in a week, subtract 138, then multiply it by the total number of days. If it is an integer number of days in a week, add the remainder 837, then add the fraction 899, add the change and decay of the next calendar, and calculate as before.
Either subtract from or add to the change and decay the profit and loss rate, obtaining the change in profit and loss rate, and use it to adjust the profit and loss at midnight. If the profit and loss are insufficient when the calendar ends, subtract the entry into the next calendar, and subtract the remainder as before.
Let's calculate the days. First, we need to look at the distance the moon travels each month, divided by the solar terms and nighttime hours, where every 200 units is considered a "ming fen." Then subtract the "ming fen" from the total distance the moon travels to get the "hun fen." "Fen" is like the four seasons of a year, with its own degrees. Multiply the total by "fen" and add that to the degree at midnight, and you can calculate the exact degree of sunrise and sunset. Any excess should be rounded up if more than half, and rounded down if less than half.
Second paragraph:
The moon has four reference points and three directions of entry and exit. These routes intersect in the sky, and dividing by the moon's speed allows us to calculate the number of days in the calendar. Multiply 360 degrees by the synodic month (the moon's full cycle), similar to the moon's monthly cycle of fullness and emptiness, which is the "shuo he fen." Multiply by the number of synodic months, divide the remainder by the number of synodic months, and you get the "tui fen." Calculate how many "fen" the moon advances each day. Calculate the difference every time there is a synodic month.
Third paragraph:
Next, we look at the rates of decline and increase in the lunar calendar, along with some related numbers:
Day 1, subtract one, increase by seventeen, initial value.
Day 2, subtract one, increase by sixteen, seventeen (limited to the remaining 1290, differential 457). This is the front limit.
Day 3, subtract three, increase by fifteen, total thirty-two.
Day 4, subtract four, increase by twelve, total forty-eight.
Day 5, subtract four, increase by eight, total sixty.
Day 6, subtract three, increase by four, total sixty-eight.
Day 7, subtract three (insufficient subtraction, which turns into addition, meaning it should increase by one, but instead it subtracts three, resulting in a deficit). Increase by one, total seventy-two.
Day 8, add four, subtract two, total seventy-three. If it exceeds the limit, subtraction is necessary, meaning when the moon reaches half a week, the degree exceeds the limit, so it should be subtracted.
Day 9, add four, subtract six, total seventy-one.
Day 10, add three, subtract ten, total sixty-five.
Day 11, add two, subtract thirteen, total fifty-five.
Day 12, add one, subtract fifteen, total forty-two.
Day 13, (limited to the remaining 3912, differential 1752). This represents the back limit.
Day 13, add one (the initial value of the calendar is larger, dividing by the day). Subtract sixteen, total twenty-seven.
For the division of days (5,203), after a few additions and subtractions, subtract 16, resulting in a total of 11. The law of few large numbers is 473.
Fourth paragraph: The calendar cycle lasts 107,565 days. The difference rate is 11,986. The conjunction value is 18,328. The micro-difference value is 914. The micro-difference calculation yields 2,209.
Subtract the lunar cycle (the cycle of the full and new moon) from the previous lunar month (a reference point in time), and multiply the remaining parts by the conjunction and micro-difference. If the micro-difference reaches its designated value, subtract it from the conjunction; if the conjunction reaches a full cycle, subtract it. If the remaining part is less than the calendar cycle, it is the solar calendar; if it exceeds the calendar cycle, subtract the cycle number, and the remaining part is the lunar calendar. Calculate the number of days based on the lunar cycle; this is an additional calculation. The conjunction of the moon with the calendar results in a remainder of time less than a day.
Fifth paragraph: After adding two days, the day remainder is 2,580, and the micro-difference is 914. Calculate the number of days using the specified method; if it reaches 13, subtract that amount and calculate the remaining days based on the division of days. Ultimately, the lunar and solar calendars intersect, with the lunar calendar entering before the limit and the solar calendar after, while the moon occupies a central position.
This passage describes more complex calculations involving the use of "universal methods" to calculate longer time periods, as well as how to deal with "surplus and deficit" and "yin-yang day remainders." The ultimate goal of these calculations is to determine the exact time of midnight. It also mentions how to calculate the time of solar terms and how to determine the moon's position based on these calculations, that is, the moon's distance from the ecliptic. The concept of "strong positive and weak negative" is used to indicate the positive or negative results of the calculations, with the ultimate goal of determining the specific position of the moon.
Finally, it covers the period from the Ji Chou year of the Shang Yuan era to the Bing Xu year of the Jian An era, totaling 7378 years. It then lists the stem-branch chronology during this period and mentions the five celestial bodies (Jupiter, Mars, Saturn, Venus, and Mercury), as well as how to use this information to calculate the weekly cycles and daily cycles, ultimately calculating the month and date. The passage also discusses "Doufen" and its use in calculating other astronomical data. Overall, this passage describes a rather complex method of calendar calculation involving a lot of astronomical and mathematical knowledge. "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 is the stem-branch chronology and does not need to be translated.
Wow, all these dense numbers are making my head spin! Let's take it step by step and explain slowly. First of all, this seems to be a record of ancient astronomical calculations, with various parameters and calculation methods that are dazzling. Starting with the beginning, "Five Stars New Moon Large Remainder, Small Remainder," means calculating the remainder of the five planets on the new moon day (first day of the lunar month), with large and small remainders being two different remainders. The calculation method is to multiply the month by the "universal method," then divide by the "daily method," the quotient is the large remainder, the remainder is the small remainder, and then subtract 60 from the large remainder. What are these universal and daily methods? We'll explain them later.
Next, "the five planets entering the month and the remaining days," this is to calculate the number of days and remainder when a planet enters a certain month. The calculation begins by multiplying the month's remainder using the standard method, then multiplying the synodic month remainder using the synodic month method, adding these two results together, simplifying, and finally dividing by the daily method to get the number of days the planet enters the month. The definitions of the month's remainder, synodic month method, and daily method will be explained later. "Five stars degrees, degree remainder" refers to the degrees and remainder of the planets; the calculation involves subtracting any excess degrees to determine the degree remainder, then multiplying by the week cycle to get the degree remainder, and finally dividing by the daily method to obtain the degrees and degree remainder. If it exceeds the week cycle, subtract the week cycle and convert to Dou fen. What are these week cycle and Dou fen units? This will be explained later.
Next, we have a series of numbers; these numbers represent various parameters: the recorded month is 7285, the leap month is 7, the chapter month is 235, the year is 12, the standard method is 43026, the daily method is 1457, the count is 47, the week cycle is 215130, and the Dou fen is 145. These numbers represent different astronomical cycles and calculation coefficients; the specific meanings need to be understood in conjunction with the knowledge of the calendar at that time. Next are the parameters for Jupiter, Mars, and Saturn, listing their synodic periods, daily rates, synodic months, month's remainder, synodic month method, daily method, as well as the calculated synodic excess, synodic deficiency, entry month days, days remaining, degrees, and degree remainder. These parameters and calculation results demonstrate the ancient astronomers' meticulous calculations of the planetary motion laws.
For example, for Jupiter, the synodic period is 6722, the daily rate is 7341, the synodic month is 13, the month's remainder is 64810, the synodic month method is 127718, the daily method is 3959258, the synodic excess is 23, the synodic deficiency is 1370, the entry month day is 15, the days remaining are 3484646, the degrees are 33, and the degree remainder is 2509956. The parameters for Mars and Saturn are similar, just with different values. Understanding the specific meanings and calculation processes of these numbers requires referencing ancient astronomical and calendrical texts. In conclusion, this text records the process and results of ancient astronomers' calculations of planetary motion, highlighting their sophisticated calculation skills and deep understanding of astronomical phenomena. Wow, all these dense numbers are quite overwhelming! Let's go through it sentence by sentence and try to translate this thing into plain language.
First of all, this initial series of numbers should be recording some astronomical observation data. For example, "the remaining degrees of the sun are one hundred sixty-six thousand two hundred seventy-two," meaning that the remaining degrees of the sun's movement are one hundred sixty-six thousand two hundred seventy-two; "the virtual division for the new moon is nine hundred twenty-three," referring to the virtual division for the new moon (the first day of the lunar month) being nine hundred twenty-three; "the degree of a certain constellation is five hundred eleven thousand seven hundred five," probably indicating the degree of a certain star; the terms "degrees" and "remaining degrees" that follow are similar units of measurement. In summary, these represent specific values used in ancient astronomical calculations, and there's no need to get too caught up in the exact meanings; just knowing they are astronomical data is sufficient.
Next, the characters for "gold" and "water" appear, which probably refer to Venus and Mercury. Then there's another set of numbers, similar to the previous ones, all recording various data about the movements of Venus and Mercury, such as the sidereal year, solar year, number of conjunctions, remaining months, etc. To be honest, I can't clarify the specific meanings of these numbers; the methods of ancient astronomical calculations are quite complex!
The final paragraph starts explaining the calculation methods. "Set the year you want to calculate, multiply it by the sidereal year; if it is an integer multiple of the solar year, it is called 'accumulated conjunction,' and if there is a remainder, it is called 'conjunction remainder,'" meaning that first, determine the year you want to calculate, then multiply the sidereal year by this year; if the result is exactly an integer multiple of the solar year, it is called "accumulated conjunction," and if there is a remainder, it is called "conjunction remainder." The following calculation methods are quite intricate and difficult to explain in modern terms; in short, they involve a series of complex multiplication and division operations, with the ultimate goal of calculating the positions of celestial bodies.
"Multiply the month number and remaining months by the accumulated conjunction; if the result is exactly an integer multiple of the conjunction month method, it represents the calculated number of months; if there is a remainder, it represents the new remaining month." The following phrases like "subtract the accumulated months from the recorded months; the remainder represents the entry month," "multiply by the leap month; if the leap month is complete, reduce the entry month, and the remainder is subtracted from the year, calculated by the day correctly, for the conjunction month," etc., are all technical terms used in ancient calendar calculations, and translating them into modern language would be quite challenging, so I'll just keep the original text.
The last few sentences also describe some calculation steps, such as "use the common method to multiply the months, multiply the months by the remainder of the synodic month, and round the result to obtain a full day; then the stars will align with the day of the month. If it’s not a full month, calculate the remainder based on the synodic month." These are specific steps in ancient astronomical calendar calculations, which are difficult to explain in modern language, so the original text should be retained directly.
In summary, this passage describes the method of calculating the positions of celestial bodies in ancient astronomical calendars, filled with complex numbers and professional terminology, even modern people would find it difficult to fully understand. However, we can roughly understand that ancient astronomers made great efforts to calculate the positions of celestial bodies and created very sophisticated calculation methods.
Let's calculate the days: first, add up the months and also add up the extra months. If the total is exactly one month, then there is no intercalary month in that year. If it is less than a year, then the extra part is carried over to the next year; if it exceeds a year, then carry it over to the following two years. For Venus and Mercury, if they appear in the morning, it is recorded as "morning"; if they appear in the evening, it is recorded as "evening." If they appear in the morning, add one day to switch to evening; if they appear in the evening, add one day to switch to morning.
Next, calculate the size of the new moon phase (first day of each lunar month). Add the remainder of the new moon's size and the remainder of the size of each month; if it exceeds a month, then add twenty-nine (large remainder) or seven hundred and seventy-three (small remainder). If the small remainder is full, subtract from the large remainder. The calculation method remains the same as previously described.
Then calculate the days in the month and the day remainder. Add the days in the month and the day remainder; if it exceeds one day, take one day. If the small remainder of the new moon is full, subtract one day; if the small remainder exceeds seven hundred and seventy-three, subtract twenty-nine days; if it does not exceed, subtract thirty days. The remainder will be carried over to the next month.
Finally, calculate the degrees: add the degrees and the remainder of the degrees; if it exceeds one day, take the equivalent of one day.
The following is the movement of Jupiter: Jupiter is invisible (meaning Jupiter is behind the sun and cannot be seen) for thirty-two days, three hundred and forty-eight thousand four hundred and sixty-four minutes of arc; visible (meaning Jupiter is in a position where it can be seen) for three hundred and sixty-six days; Jupiter is invisible for five degrees of movement, two hundred and fifty thousand nine hundred and fifty-six minutes of arc; visible movement for forty degrees (retrograde twelve degrees, actual movement twenty-eight degrees).
Mars: Hidden for one hundred forty-three days and ninety-seven thousand three hundred thirteen minutes; appeared for six hundred thirty-six days; hidden movement one hundred ten degrees, four hundred seventy-eight thousand nine hundred ninety-eight minutes; appeared for three hundred twenty degrees. (Retrograde by seventeen degrees, actual movement three hundred three degrees.)
Saturn: Hidden for thirty-three days and one hundred sixty-six thousand two hundred seventy-two minutes; appeared for three hundred forty-five days; hidden movement three degrees, one hundred seventy-three thousand one hundred forty-eight minutes; appeared for fifteen degrees. (Retrograde by six degrees, actual movement nine degrees.)
Venus: Hidden in the east during the morning for eighty-two days and eleven thousand three hundred ninety-eight minutes; appeared in the west for two hundred forty-six days. (Retrograde by six degrees, actual movement two hundred forty-six degrees.) In the morning, it moved one hundred degrees, eleven thousand three hundred ninety-eight minutes; appeared in the east. (Daily position as observed from the west; hidden for ten days, retrograde by eight degrees.)
Mercury: Hidden in the east during the morning for thirty-three days and 6,012,505 minutes; appeared in the west for thirty-two days. (Retrograde by one degree, actual movement thirty-two degrees.) Hidden movement sixty-five degrees, 6,012,505 minutes; appeared in the east. Daily position as observed from the west; hidden for eighteen days, retrograde by fourteen degrees.
First, let's calculate the positions of the sun and stars. Subtract the degrees the sun travels each day from the degrees the stars travel each day. If the remaining degrees equal the degrees the sun travels in a day, then the stars have appeared, just as we calculated earlier. Next, multiply the numerator of the degrees the stars travel each day by the difference between the degrees when the stars appear and the sun, then divide the remaining degrees by the degrees the sun travels in a day. If it doesn't divide evenly and exceeds half, treat it as if it divided evenly. Then, add the stars' daily travel degrees to the sun's daily travel degrees. If the total equals a full cycle, add one degree. The calculation methods for direct and retrograde motion differ; you need to multiply the current degrees the stars travel by the previous degrees, then divide by the previous numerator to get the current degrees the stars travel. The remaining degrees carry over the previous calculation result; subtract it if it's retrograde. If the degrees do not equal a full cycle, divide by the degrees of the Big Dipper, using the stars' travel degrees' numerator as a proportion, which will cause the degrees to increase or decrease, balancing each other out. In summary, terms like "surplus," "approximately," and "full" are used for precise division results, while terms like "remainder," "and," and "divide" aim for precise division results.
Next, let's talk about Jupiter. In the morning, Jupiter aligns with the Sun, and then it begins to move forward. After sixteen days, it travels 1,742,323 minutes, and Jupiter itself covers 2 degrees and 3,234,607 minutes, then it appears in the east, positioned behind the Sun. During its forward motion, it moves quickly, traveling 11/58 degrees each day, and 11 degrees in 58 days. When it continues to move forward, the speed slows down, covering 9 minutes of arc each day, and 9 degrees in 58 days. Then Jupiter halts for 25 days before it resumes motion. During its retrograde motion, it travels 1/7 degrees each day, retreating 12 degrees after 84 days. It then stops again and begins to move forward after 25 days, covering 9/58 degrees each day, and 9 degrees in 58 days. The forward speed is fast again, moving 11 minutes of arc each day, and 11 degrees in 58 days, at which point Jupiter is in front of the Sun, appearing in the west in the evening. After sixteen days, it travels 1,742,323 minutes, and Jupiter itself covers 2 degrees and 3,234,607 minutes, then it aligns with the Sun once more. A complete cycle takes a total of 398 days, covering 3,484,646 minutes, and Jupiter itself moves 43 degrees and 2,509,956 minutes.
The Sun: It rises in the morning alongside the Sun, and then it hides. The next phase is forward motion, which lasts for 71 days, with a total distance of 1,489,868 minutes, the planet travels 55 degrees and 242,860.5 minutes, then appears in the east, positioned behind the Sun in the morning. During its forward motion, it covers 14/23 degrees each day, which is about 0.61 degrees, and 112 degrees in 184 days. The forward speed increases slightly, then slows down, covering 12/23 degrees each day, and 48 degrees in 92 days. It then halts for 11 days. Next is the retrograde motion, traveling 17/62 degrees each day, retreating 17 degrees after 62 days. It stops moving again for 11 days and then resumes forward motion, covering 1/12 degrees each day, and 48 degrees in 92 days. It moves forward again, with an increased speed, covering 1/14 degrees each day, and 112 degrees in 184 days, at which point it is in front of the Sun, hiding in the west in the evening. After 71 days, it travels a total of 1,489,868 minutes, the planet moves 55 degrees and 242,860.5 minutes, and then appears again with the Sun. This completes one cycle, totaling 779 days and 973,113 minutes, with the planet moving 414 degrees and 478,998 minutes.
Mars: It appears in the morning together with the Sun, and then it hides. Next is direct motion, lasting 16 days, totaling 1,122,426.5 minutes, during which the planet moves 1 degree in 1,995,864.5 minutes. Then it is behind the Sun, visible in the east each morning. During direct motion, it moves 3/35 of a degree each day, totaling 7.5 degrees over 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 in 102 days. After another 34 days, it begins direct motion again, moving 1/3 of a degree each day, covering 7.5 degrees in 87 days, at which point it is in front of the Sun, lurking in the west in the evening. In total, during this period, it runs 1,122,426.5 minutes, with the planet moving 1 degree in 1,995,864.5 minutes, and then appears with the Sun again. This counts as one cycle, totaling 378 days and 166,272 minutes, with the planet moving 12 degrees in 1,733,148 minutes.
As for Venus, when it conjoins with the Sun in the morning, it first "lurks," which means it goes retrograde, retreating 4 degrees in 5 days, after which it becomes visible in the east behind the Sun. Continuing retrograde, it moves 3/5 of a degree each day, retreating 6 degrees in 10 days. Then it "stays," pausing for 8 days without moving. After that, it "turns," meaning it goes direct, moving relatively slowly at about 33 degrees and 46 minutes per day, covering 33 degrees in 46 days before it goes direct. Then the speed increases, moving 1 degree and 15 minutes per day, covering 160 degrees in 91 days. The speed continues to increase, moving 1 degree and 22 minutes per day, covering 113 degrees in 91 days, at which point it again goes behind the Sun, appearing in the east in the morning. Finally, during direct motion, it covers 1/56,954 of a circle in 41 days, with the planet also moving 50 degrees and 1/56,954 of a circle, before it conjoins with the Sun again. One conjoining cycle lasts 292 days and 1/56,954 of a circle, with the planet following the same cycle duration.
When Venus conjoins with the Sun in the evening, it first "hides" before moving forward, traveling one fifty-six-thousand nine-hundred fifty-fourth of a circle in forty-one days, covering fifty degrees and one fifty-six-thousand nine-hundred fifty-fourth of a circle. It can then be seen in front of the Sun in the west in the evening. Next, it moves forward, accelerating to travel twenty-two degrees and two ninety-firsts of a degree each day, covering one hundred thirteen degrees in ninety-one days. The speed then starts to decrease, moving one fifteen-th of a degree each day, covering one hundred six degrees in ninety-one days, and then it moves forward. The speed decreases again, moving three thirty-sixths of a degree each day, covering thirty-three degrees in forty-six days. Then it "pauses," stopping for eight days. Next, it "rotates," reversing direction and moving three-fifths of a degree each day, retreating six degrees in ten days, at which point it appears in front of the Sun in the west in the evening. "Hiding" and moving backward, it accelerates, retreating four degrees in five days, and then it again conjoins with the Sun. Two conjunctions complete one cycle, totaling five hundred eighty-four days and one hundred thirteen thousand nine hundred eight one-hundredth of a circle, with the planet following the same cycle.
As for Mercury, it conjoins with the Sun in the morning, first "hiding" before moving backward, retreating seven degrees in nine days, and then it can be seen in the east behind the Sun in the morning. Continuing to move backward, it accelerates, moving back one degree each day. It then "pauses," stopping for two days. After that, it "rotates," changing to forward motion, moving slowly at eight-ninths of a degree each day, covering eight degrees in nine days before moving forward. The speed increases, moving one quarter of a degree each day, covering twenty-five degrees in twenty days, at which point it moves behind the Sun again, appearing in the east in the morning. Finally, moving forward, it covers one six-hundred forty-one million nine hundred sixty-seven-thousandth of a circle in sixteen days, with the planet also moving thirty-two degrees and one six-hundred forty-one million nine hundred sixty-seven-thousandth of a circle before conjoining with the Sun again. The cycle for one conjunction is fifty-seven days and one six-hundred forty-one million nine hundred sixty-seven-thousandth of a circle, with the planet following the same cycle.
It is said that when Mercury sets alongside the sun, it is referred to as '伏' (occultation). The movement of Mercury follows this pattern: when it moves forward, it can travel 32 degrees and 641,961,967 parts of a degree in 16 days. During this time, you can see it in the western sky in the evening, positioned ahead of the sun. When it moves forward, it travels quickly, covering a quarter of a degree each day, and can cover 25 degrees in 20 days.
If it moves slowly, it only travels 7/8 of a degree in a day, taking 9 days to cover 8 degrees. If it "stays," it means it remains stationary for two days. If it retrogrades, it moves backwards by one degree each day, and you can see it in the west in the evening, positioned ahead of the sun. When retrograding, it is slow, taking 9 days to move back 7 degrees. From one conjunction with the sun to the next, it takes a total of 115 days and 601,255,505 parts of a day, and Mercury's orbital cycle follows the same pattern.
