First, we use one hundred multiplied by a decimal; according to the regulations, we can get a quarter, but it does not reach one-tenth, so it needs to be further subdivided. The calculation method is to start counting from the night according to the solar terms at that time and continue counting until the end of the night; if the water level at night has not been counted, use a nearby value instead.
Next, there will be situations of carry-over and borrowed value in the calculation process. Add when carrying over, and subtract the obtained value when borrowing. The difference between carry-over and borrowed value starts at two degrees, decreases by four degrees each time, and halves each time after three decreases. If the difference reaches three times, stop, then after five degrees, return to the initial state.
The moon's speed varies, but the overall movement pattern is constant. The calculation method combines astronomical numbers with actual numbers, multiplies by the remainder, then divides by the total to get the portion that exceeds one complete cycle. Then subtract the number of days in one cycle from this portion to get the actual number of days. The change in the moon's speed is determined by its orbital pattern. According to the speed change, adding the rate of the moon's movement can calculate the degrees and fractions of movement each day. The total of the speed changes represents the profit and loss rate. The profit and loss rates will affect each other, eventually leading to the accumulation of profits or losses. Multiply half of a small cycle by the common calculation method, divide by the total, then subtract the number of days in one cycle to get the degree of the moon's movement on the first day.
Below are the specific calculation results; the table lists the degrees and fractions of movement, the depreciation value, the profit and loss rate, the accumulated profit and loss, and the degrees of the moon's movement for each day:
Day 1: 14 degrees and 10 minutes, 1 decrease, profit 22, initial profit, 276
Day 2: 14 degrees and 9 minutes, 2 decreases, profit 21, profit 22, 275
Day 3: 14 degrees and 7 minutes, 3 decreases, profit 19, profit 43, 273
Day 4: 14 degrees and 4 minutes, 4 decreases, profit 16, profit 62, 270
Day 5: 14 degrees, 4 decreases, profit 12, profit 78, 266
Day 6: 13 degrees and 15 minutes, 4 decreases, profit 8, profit 90, 262
Day 7: 13 degrees and 11 minutes, 4 decreases, profit 4, profit 98, 258
Day 8: 13 degrees and 7 minutes, 4 decreases, loss, resulting in a total profit of 102, 254
On the 9th, twelve degrees and three minutes, subtract four, then add four, lose four, gain one hundred and two, resulting in two hundred and fifty.
On the 10th, twelve degrees and eighteen minutes, subtract three, then add four, lose eight, gain ninety-eight, resulting in two hundred and forty-six.
On the 11th, twelve degrees and fifteen minutes, subtract four, then add four, lose eleven, gain ninety, resulting in two hundred and forty-three.
On the 12th, twelve degrees and eleven minutes, subtract three, then add four, lose fifteen, gain seventy-nine, resulting in two hundred and thirty-nine.
On the 13th, twelve degrees and eight minutes, subtract two, then add four, lose eighteen, gain sixty-four, resulting in two hundred and thirty-six.
On the 14th, twelve degrees and six minutes, subtract one, then add four, lose twenty, gain forty-six, resulting in two hundred and thirty-four.
On the 15th, twelve degrees and five minutes, subtract one, then add one, lose twenty-one, gain twenty-six, resulting in two hundred and thirty-three.
This passage describes an ancient method of astronomical calculation, using a series of addition, subtraction, multiplication, and division operations to calculate the trajectory and time of the moon. While the specific algorithms may seem complex to modern readers, it shows the exquisite skills of ancient people in astronomical observation and calculation.
On the 16th, twelve degrees and six minutes, subtract twenty, then add five (because it is insufficient). The total comes to two hundred and thirty-four.
On the 17th, twelve degrees and eight minutes, subtract eighteen, then add five, finally subtract fifteen. The total comes to two hundred and thirty-six.
On the 18th, twelve degrees and eleven minutes, subtract fifteen, then add five, finally subtract twenty-three. The total comes to two hundred and thirty-nine.
On the 19th, twelve degrees and fifteen minutes, subtract eleven, then add five, finally subtract forty-eight. The total comes to two hundred and forty-three.
On the 20th, twelve degrees and eighteen minutes, subtract eight, then add five, finally subtract fifty-nine. The total comes to two hundred and forty-six.
On the 21st, thirteen degrees and three minutes, subtract four, then add five, finally subtract sixty-seven. The total comes to two hundred and fifty.
On the 22nd, thirteen degrees and seven minutes, add five, then subtract seventy-one. The total comes to two hundred and fifty-four.
On the 23rd, thirteen degrees and eleven minutes, add five, then subtract four, finally subtract seventy-one. The total comes to two hundred and fifty-eight.
On the 24th, thirteen degrees and fifteen minutes, add five, then subtract eight, finally subtract sixty-seven. The total comes to two hundred and sixty-two.
On the 25th, fourteen degrees, add five, then subtract twelve, finally subtract fifty-nine. The total comes to two hundred and sixty-six.
On the 26th, fourteen degrees and four minutes, add five, then subtract sixteen, finally subtract forty-seven. The total comes to two hundred and seventy.
On the 27th, at fourteen degrees and seven minutes, this is a special day; add a large weekly value, then subtract nineteen, and finally subtract thirty-one. The total is two hundred seventy-three.
On Sunday, it's fourteen degrees and nine minutes. This day is also special; add a value, then subtract twenty-one, and finally subtract twelve. The total is two hundred seventy-five.
Sunday points total three thousand three hundred and three.
Zhou Xu (a concept in lunar calculations), two thousand six hundred sixty-six.
Total Zhou is one hundred eighty-five thousand thirty-nine.
Historical Zhou totals one hundred sixty-four thousand four hundred sixty-six.
A little big method, one thousand one hundred one.
Shuo Xing Da Fen, ten thousand one thousand eight hundred one.
Small points total twenty-five.
Zhou half, one hundred twenty-seven.
These numbers above are used to calculate the movement of the moon. How to calculate it specifically? First, multiply the value of each month by the value of the moon's movement. If the small points reach thirty-one, subtract from the large points; if the large points reach one hundred sixty-four thousand four hundred sixty-six, subtract it. Then divide the remaining by five thousand nine hundred sixty-nine. The quotient is the number of days, and the remainder is the remaining days. The calculated number of days is the day of the conjunction and history.
To calculate the next month, add one day to this basis; now the remainder is five thousand eight hundred thirty-two, and the small points are twenty-five.
To calculate the first quarter moon (lunar calendar on the 7th, 15th, and 23rd), add seven days to this basis; now the remainder is two thousand two hundred eighty-three, and the small points are twenty-nine point five. Calculate the number of days according to the method above; if it exceeds twenty-seven days, subtract twenty-seven. The remaining will be calculated according to the week points. If it is not enough to divide, subtract one day and add Zhou Xu.
Wow, this ancient text is really confusing! Let's break it down piece by piece and explain it in simple terms.
First, this discusses how to calculate the calendar, using accumulated gains and losses to adjust the calendar. "Set the accumulated gains and losses of the calendar to adjust, using the constant (known as Zhou Cheng) as the base value." This means multiplying the previously accumulated calendar gains and losses by a constant (Zhou Cheng) to establish a base value. "Multiply the total number by the daily remainder, then by the adjustment ratio (gain/loss rate) to adjust the base value; this is the method for adjusting the calendar time." Then, use another constant (total number) to multiply by the daily remaining fraction, and then multiply by an adjustment ratio (gain/loss rate) to adjust the base value; this is the calculation method for adjusting the time. "Subtract the monthly fraction from the annual total, multiply by half of the cycle to get the difference, and divide it; the resulting gains and losses will help determine the larger and smaller remainders. If the solar calculation is insufficient, adjust the timing on the first and last days of the month. The large remainder of the waxing and waning will determine the small remainder." Subtract the lunar running fraction from the annual total, multiply by half of the cycle to obtain a difference, and divide it by the previous value to get the remainder of gains and losses. Based on the gains and losses, adjust the time of the new moon (the first day of the lunar calendar) and the waxing and waning (the eighth and twenty-third days of the lunar calendar).
Next, continue calculating more precise dates. "Multiply the annual total by the adjusted time, divide by the difference to obtain the complete cycle count as the large and small fractions of gains and losses, and adjust the current day's lunar position based on the gains and losses; if insufficient, adjust the degree according to the established calendar method to determine the lunar position." Multiply the annual total by the previously calculated adjusted time, then divide by that difference to obtain a full cycle number (which roughly refers to a cycle); this number represents the magnitude of gains and losses. Adjust the position of the sun and moon for that day based on the gains and losses, ultimately determining their positions. "Multiply half of the cycle by the small remainder from the new moon calculation; if equal to the total number, reduce it from the remaining days of the calendar. If there are not enough remaining days, add a cycle, then subtract, and finally retreat one day. Add the weekly day and its fraction to obtain the time of midnight entry into the calendar." Multiply half of the cycle by the remaining value of the new moon, then divide it by the total number, and subtract from the remaining days of entry into the calendar. If there are not enough remaining days, add a cycle and then subtract, then retreat one day, and add the weekly day and its fraction to get the time of midnight entry into the calendar.
This section is about how to calculate the next day. "To calculate the next day, shift forward by one day, due to having 27 days remaining; divide the surplus by the number of days in a week, rather than treating it as a full week. If the surplus is insufficient, add the remaining part of the week, which will then be counted as the next day's historical days." To calculate the next day, begin with the remaining days from the previous day and continue counting up to 27 days. If the remaining days exceed a full cycle, subtract a full cycle. If it is less than a full cycle, add the remaining portion of that cycle, and the result will be the remaining days for the next day's historical record. "Entering the historical days at midnight, multiply by the rate of gain and loss; if the weekly method results in one, it does not all count as surplus. With the accumulation of gains and losses, the surplus remains unaffected, which breaks all conventional loss laws, thus representing the midnight gain and loss. A complete year is considered a degree, while anything less is treated as a fraction. Multiply the common values by fractions and remainders; the remainder is calculated using the weekly method based on the fraction, while the fraction is derived from the record method based on the degree, adding and subtracting the remaining half degree and the remainder to establish the final degree." Multiply the remaining days entered into history at midnight by the rate of gain and loss; if the result is divisible by the cycle, use this value to adjust the accumulated gains and losses. If it is not divisible, use the quotient to make adjustments to the accumulated gains and losses, ultimately determining the gain and loss values at midnight.
Finally, this section calculates the decay of the calendar and the final date. "Using the remaining days of the lunar calendar multiplied by the decay value (decay series), if it can be evenly divided by the cycle, the daily decay value is obtained. 'Using the weekly cycle multiplied by the decay series, if the weekly method is a constant, when the calendar concludes, the decay value is adjusted by adding the variable decay, and the full decay is removed, turning into the next calendar variable decay.' The remaining portion of the cycle is multiplied by the decay value to derive a constant, which is used to adjust the decay value after the calendar ends. 'Using the variable decay to add or subtract the number of days in the lunar calendar, if the division is either complete or insufficient, it corresponds to the annual cycle. Multiply the total number by the division and the remainder, and the day is then added to determine the following day. If the calendar does not align precisely with the weekly cycle, subtract 1,038, then multiply by the total; if it equals the whole weekly day, add 837, then adjust using 899, and proceed with the calculations.' Adjust the remaining fraction of each day based on the decay value; if there is a surplus or deficit in the remaining fraction, adjust the annual chapter. Finally, calculate the next day. If the calendar cycle does not equal an integer weekly day, subtract 1,338, multiply by the total; if it equals an integer weekly day, add 837, then adjust using 899, and continue calculating. 'Using the variable decay to add the profit and loss rate, it becomes the variable profit and loss rate, and the profit and loss at midnight is adjusted for surplus and deficit. If the calendar ends with insufficient profit and loss, reverse the adjustment, using the same method as above.' Adjust the profit and loss rate based on the decay value, then adjust the midnight profit and loss. If there is insufficient profit and loss at the end of the calendar, reverse the adjustment using the same method as above. 'By multiplying the lunar month's fractional movement by the nighttime duration of the nearest solar term, 201 corresponds to the bright fraction. Subtract the monthly fractional movement to get the dim fraction. The fraction is like the annual chapter for degrees; multiply the total number by the fraction, and add the midnight adjustment to determine the dim-bright degree. If the remaining fraction exceeds half, it is retained; if not, it is discarded.' In summary, this text outlines a complex method for calculating the calendar, involving many constants and ratios, aimed at accurately calculating the date and time each day. This is essentially the work of an ancient 'algorithm engineer'!
This text describes an ancient calendar calculation method, which is quite complex. Let's break it down sentence by sentence and try to explain it in plain language as much as possible.
First, "The lunar cycle is divided into four phases, entering and exiting along three pathways, intersecting and dividing the sky, and then dividing it by the lunar rate to determine the day in the calendar." This sentence means that based on the pattern of the moon's movement (the four phases and three pathways refer to certain calculation parameters), the year is divided into several days, which is the "day" in the calendar. This is similar to how we use the Gregorian calendar now, where we first determine how many days are in a year.
"Multiplying the weeks of the year by the conjunction of the new moon and the full moon, this will align with the lunar cycle, and the conjunction and division calculations will follow. Multiplying the general number by the conjunction number, the remainder will correspond to the conjunction number, and the subtraction will follow. By following the moon weekly, the daily progression will be determined. The conjunction number will be the difference rate." This part is even more difficult to understand. "Weeks of the year" refer to a year, "conjunction of the new moon and the full moon" refers to the phases of the moon, and "monthly conjunction" refers to a cycle. "Conjunction and division," "difference rate," and so on are some intermediate results in the calculations, which can only be accurately expressed using modern mathematical formulas. In simple terms, it is calculating important values based on the moon's movement pattern for subsequent calendar calculations.
Next is a table of data showing the daily changes in the "gain-loss rate," which is the essence of the calendar calculations, adjusting the deviation between the calendar and the actual lunar movement through daily increments and decrements. "The day count is reduced by seventeen on the first day" means reducing the day count by seventeen on the first day and adding seventeen of what unit (specific unit not mentioned). The subsequent entries for "second day," "third day," and so on outline the daily adjustment values. The content in parentheses is additional explanation, such as "upper limit of one thousand two hundred and ninety, differential four hundred and fifty-seven" referring to some intermediate values in the calculation process.
"On the thirteenth day (upper limit of three thousand nine hundred and twelve, differential one thousand seven hundred and fifty-two.) This is the later limit." This part is similar to the previous one, also data in the calculation process.
"On the divided day (five thousand two hundred and three) a decrease of sixteen and an increase of eleven." This describes a special calculation method, which is more complex to explain in modern mathematical language. In simple terms, it adjusts the calendar using different calculation methods based on different situations.
"Total calendar days: one hundred seventy-five thousand six hundred and five. Difference rate: eleven thousand nine hundred and eighty-six. Conjunction and division: eighteen thousand three hundred and twenty-eight. Differential: nine hundred and fourteen. Differential calculation method: two thousand two hundred and nine." These numbers are all calculation results, representing key parameters in the calendar.
"To calculate the new moon, the accumulated months are added to the months that have passed since the last new moon, and then multiplied by the difference between the synodic month and the sidereal month. The product is divided by the difference to yield the remainder, which is then subtracted from the difference to find the date in the solar calendar. If the remainder is insufficient to complete a week, it is classified as a solar calendar date; otherwise, it is a lunar calendar date. The calculation is adjusted by adding one day each month to ensure accuracy in determining the calendar date.
After adding two days to the remainder of 2580 and multiplying the difference of 914 by 2, the result is divided by 13, with the remainder being considered the date. The solar and lunar calendars are interconnected, with the solar calendar preceding the lunar calendar, and the lunar calendar following the solar calendar.
To adjust the calendar, the surplus and deficit are calculated and adjusted based on their magnitude. The surplus or deficit determines the day, and the adjustment rate is determined based on the lunar week. This summary explains how the calendar is adjusted to align with the lunar cycle.
In conclusion, this text outlines a complex ancient method for calculating calendars that involves mathematics and astronomy. It showcases the wisdom of ancient people in astronomy and calendar calculations, but understanding the specific process requires modern mathematical tools. First, let's discuss how to calculate the new moon. By multiplying the difference rate by the new moon's remainder, akin to calculus, a result is obtained, which is then subtracted from the day's remainder in the calendar. If the subtraction is insufficient, a month's worth of weeks is added before performing the subtraction again, followed by subtracting one day. By adding the obtained day to its fraction and simplifying the difference, the calendar date of the new moon is determined."
The next step is to calculate the next day. Adding one day, the remainder of the day is 31, and the small remainder is also 31. If the small remainder exceeds the count, subtract one month’s worth of weeks. Then add one more day, calculate the calendar until the end; if the remainder of the day exceeds the small remainder count, subtract the small remainder count; this marks the beginning of the calendar. If the remainder of the day is either in surplus or insufficient, keep it, add 2720, and the small remainder remains 31; this gives the date of the next calendar.
Next is to calculate the length of day and night. Multiply the total by the delay and acceleration of the night and the remainder of the entered calendar; if the remainder exceeds half a week, it becomes the small remainder. Add the surplus to the reduction, then subtract the yin and yang remainder of the day. If the remainder of the day is surplus or insufficient, adjust the date with the number of weeks in the month. Multiply the determined remainder of the day by the profit and loss rate; if it equals the number of weeks in the month, use the total profit and loss as the constant of the night.
Calculate the twilight moments. Multiply the profit and loss rate by the night leakage time of the nearest solar terms, divide by 200 to get the bright moments, then subtract this result from the profit and loss rate to get the dark moments. Use the nocturnal half of the profit and loss as the constant of the twilight moments.
Finally, calculate how far the moon is from the ecliptic. If the added time equals the twilight constant, divide by 12 to get the degree; multiply the remainder by one-third; less than one is weak, more than one is strong, and two weak counts as one strong. This determines the degree of the moon’s departure from the ecliptic. For the solar calendar, subtract the calendar from the extremes of the day; for the lunar calendar, add the extremes of the day to get the degree of the moon leaving the extremes. Strong is positive, weak is negative; same names are added, different names are subtracted. When subtracting, same names cancel out, different names add up; there are no opposite cases, and two strong plus one weak.
Starting from the year of Ji Chou in the first year of the Yuan dynasty, to the year of Bing Xu in the eleventh year of Jian'an, it is a total of 7378 years.
Ji Chou, Wu Yin, Ding Mao, Bing Chen, Yi Si, Jia Wu, Gui Wei
Ren Shen, Xin You, Geng Xu, Ji Hai, Wu Zi, Ding Chou, Bing Yin
Five Elements: Wood (Jupiter), Fire (Mars), Earth (Saturn), Metal (Venus), Water (Mercury). Calculate the weekly and daily rates using the number of days and degrees they traverse. Multiply the chapter by the weekly rate to obtain the monthly portion; multiply the chapter by the monthly rate to obtain the monthly part; divide the monthly part by the monthly method to get the month. Multiply the total by the monthly method to get the daily degree method. Multiply the Dipper's value by the weekly rate to calculate the Dipper. (The daily degree method is multiplied by the weekly rate using the record method, so we also multiply using minutes here.)
Finally, calculate the large remainder and small remainder of the new moon day for the five stars. (Multiply by the number of months respectively, divide by the number of days respectively, get the large remainder, and the remaining part after division is the small remainder. Subtract the small remainder from 60.)
There are also the entry month day and day remainder for the five stars. (Multiply by the number of months respectively, multiply by the small remainder of the new moon day using the combined month method, add them together, simplify with the total number, then divide by the result using the day method, and you will get the final result.)
This is a bunch of numbers written above, looking like astronomical calculation records. First of all, it explains how to calculate the degrees and the degree remainder for the five stars. The specific method is: subtract the integer part, the remaining part is the degree remainder, then multiply by the number of days in a lunar month, then divide by the day method, and the result obtained is the degree, the remaining part after division is the new remainder; if it exceeds the number of days in a lunar month, subtract the number of days in a lunar month and then calculate the Dipper division. Anyway, it's a bunch of complex calculation steps, which makes my head hurt just looking at it.
Next, a series of numbers is recorded, like some kind of calendar or astronomical observation data. For example, the recorded numbers include: the lunar month is 7285, the leap month is 7, the chapter moon is 235, the year is 12, the general method is 43026, the daily method is 1457, the total number is 47, the number of days in a lunar month is 215130, and the Dipper division is 145. I completely don't understand what these numbers specifically mean; only professionals can interpret them.
Then, it starts to list some data for Jupiter, Mars, Saturn, and Venus separately. For example, Jupiter: the weekly rate is 6722, the daily rate is 7341, the combined month count is 13, the month remainder is 64810, the combined month method is 127718, the day method is 3959258, the new moon large remainder is 23, the new moon small remainder is 1370, the entry month day is 15, the day remainder is 3484646, the new moon virtual division is 150, the Dipper division is 974690, the degree is 33, the degree remainder is 2509956. The data for Mars, Saturn, and Venus are also similar, all a long string of numbers, making it dizzying to look at.
These numbers are probably used by ancient astronomers to calculate the trajectories of planetary movements and predict celestial phenomena. As someone from the modern era, I completely do not understand what these numbers represent. However, I can feel how much effort ancient astronomers put into calculating this data; what calculating skills they must have had! Finally, the combined month count for Venus is 9, and there doesn't appear to be anything after that; maybe the record is incomplete.
One month has passed, and the result is 152,293 using the lunar month method. According to the combined month method, the result is 171,418. Using the day method, the result is 5,319,358.
Shuo Dayu is twenty-five.
Shuo Xiaoyu is one thousand one hundred twenty-nine.
The date of entry into the month is twenty-seven.
Dayyu is fifty-six thousand nine hundred fifty-four.
Shuo virtual points are three hundred twenty-eight.
Dou points are one million three hundred eight thousand one hundred ninety.
Degrees are two hundred ninety-two.
Degreeyu is fifty-six thousand nine hundred fifty-four.
Water's circumference is eleven thousand five hundred sixty-one.
The circumference of the sun is one thousand eight hundred thirty-four.
The total number of months is one.
In the next month, the remaining month is twenty-one thousand one hundred thirty-one.
The result calculated using the total month method is two hundred nineteen thousand six hundred fifty-nine.
The result calculated by the daily degree method is six hundred eighty-nine thousand four hundred twenty-nine.
Shuo Dayu is twenty-nine.
Shuo Xiaoyu is seven hundred seventy-three.
The date of entry into the month is twenty-eight.
Dayyu is six hundred forty-one million nine hundred sixty-seven.
Shuo virtual points are six hundred eighty-four.
Dou points are one million six hundred seventy-six thousand three hundred forty-five.
Degrees are fifty-seven.
Degreeyu is six hundred forty-one million nine hundred sixty-seven.
First, multiply the values from the previous year by the circumference. If it can be evenly divided by the daily rate to yield 1, it is called the product, and the part that cannot be divided is the remainder. Divide the remainder by the circumference; if it can be divided by 1, it is the star combination in the previous year; if it can be divided by 2, it is the star combination in the previous two years; if it cannot be divided, it is the combination in the current year. Subtract the circumference from the remainder to get the degree. When gold and water combine, odd numbers represent the morning, while even numbers represent the evening.
Then, multiply the month number and month remainder by the product. If the result can be divided by the total month method, the month is obtained, and the part that cannot be divided is the month remainder. Subtract the accumulated month from the integrated month, and the remaining part is the entry month. Then multiply it by the chapter leap; if it can be divided by the chapter month to yield 1, subtract a leap month. The remaining part is subtracted in the middle of the year; this part is outside the calculation of the astronomical calendar and is called the total month. If it is at the time of the leap month transition, use the new moon to adjust.
Next, multiply the month remainder by the common method and the moon remainder by the total month method, and then simplify the resulting numbers. If the result can be evenly divided by the daily degree method to yield 1, it indicates the entry month and day of the star combination. The part that cannot be divided is the remainder, which is outside the calculation of the new moon.
Then use the circumference to multiply the degree. If it can be divided by the daily degree method to get one degree, the part that cannot be divided is the remainder, and use the method of the first five cows to determine the degree.
The above outlines the method for finding the star combination.
Finally, first, add the total months to the remaining months. If you can use the lunar calendar to divide the total months evenly to get a full month, then it will be in the current year; if it cannot be divided evenly, then it will be in the middle of the year, taking into account any leap month, and the remaining months will be in the following year; if it can be divided evenly again, then it will be in the following two years. Adding metal and water results in morning, and adding morning results in evening.
First, let's calculate the remainder of the lunar cycle. Add up the remainder of the new moon day; if it exceeds one month, then add another twenty-nine days (large remainder) and seven hundred seventy-three minutes (small remainder). If the small remainder is filled, then follow the algorithm for the large remainder, using the same method as before.
Next, calculate the days from the new moon to the day the moon enters and the remaining days. Add the entry day and the remaining days together; if the remaining days are enough for a full day, then count it as a full day. If the small remainder was exactly filled at the new moon, then subtract one day; if the small remainder exceeds seven hundred seventy-three minutes, then subtract twenty-nine days; if it doesn't exceed, then subtract thirty days, and the remaining days will be determined by the date of the next new moon.
Finally, calculate the total degrees and the remaining parts. If it reaches a full degree, count it as one degree.
Here are the orbital data for Jupiter, Mars, Saturn, Venus, and Mercury:
Jupiter: Hidden for 32 days, 3,484,646 minutes; visible for 366 days; invisible movement of five degrees, 2,956,000 minutes; visible movement of forty degrees. (Retrograde subtracts twelve degrees, resulting in a final movement of twenty-eight degrees.)
Mars: Hidden for 143 days, 973,013 minutes; visible for 636 days; invisible movement of one hundred ten degrees, 47,898 minutes; visible movement of three hundred twenty degrees. (Retrograde subtracts seventeen degrees, resulting in a final movement of three hundred thirty degrees.)
Saturn: Hidden for 33 days, 166,272 minutes; visible for 345 days; invisible movement of three degrees, 173,148 minutes; visible movement of fifteen degrees. (Retrograde subtracts six degrees, resulting in a final movement of nine degrees.)
Venus: At dawn, Venus lurks in the east for 82 days and 113,908 minutes; then appears in the west for 246 days. (Subtract six degrees for retrograde motion, ultimately covering 246 degrees.) At dawn, it lurks in the east, covering 100 degrees in 113,908 minutes; it appears in the east. (The solar movement is the same as in the west, lurking for ten days and retrograding eight degrees.)
Mercury: At dawn, it lurks in the west for 33 days and 6,012,555 minutes; then appears in the west for 32 days. (Subtract one degree for retrograde motion, ultimately covering 32 degrees.) It lurks and moves for 65 degrees, taking 6,012,555 minutes; it appears in the east. (The solar movement is the same as in the west, lurking for eighteen days and retrograding fourteen degrees.)
First, let's calculate the movement of the sun and planets. First, calculate the degrees the sun travels each day, then add the degrees the planet moves each day. If the total exceeds the degrees covered by the sun each day, divide the excess by the degrees covered by the sun each day to obtain a quotient. Then, using the previous calculation method, we can determine when the planet will be visible and its degrees of movement. Next, divide the degrees of the planet's movement by the appropriate denominator. If the remainder divided by the degrees the sun travels each day has a remainder, discard it if it's less than half, and add one if it's more than half; then, add this quotient to the planet's movement degrees. If the degrees exceed the denominator, add one degree. The denominators for direct and retrograde motion are different and should be calculated according to the actual situation using the appropriate denominator. For the calculated degrees, if they have been calculated before, use the previous result; if the motion is retrograde, subtract the value. If the calculation result is less than one degree, use the Dou method for calculation, using the planet's movement denominator as a proportion, adjusting the degrees according to the situation, ensuring all calculations are coordinated. In summary, anything that requires "precise division" demands accurate division; while "to remove and divide" and "to take everything and divide" require complete division.
The situation of Jupiter is as follows: it appears in the morning with the sun, then moves behind the sun; this is known as direct motion. 16 days later, the sun has moved 1742323 minutes, while Jupiter has moved 2323467 minutes. At this time, Jupiter appears behind the sun, in the east. During direct motion, when moving quickly, it travels 58/11 degrees each day; when moving slowly, it travels 1/9 degrees each day. When Jupiter comes to a stop, it will remain still for 25 days, then begin retrograde motion, moving 1/7 degrees each day, retreating 12 degrees in 84 days. It will then stop for another 25 days, then begin direct motion, moving 58/9 degrees each day, completing 9 degrees in 58 days. During fast direct motion, it moves 11/1 degrees each day, completing 11 degrees in 58 days. At this time, Jupiter appears in front of the sun, and in the evening it sets in the west. 16 days later, the sun has moved 1742323 minutes, while Jupiter has moved 2323467 minutes, and they are together again. One complete cycle lasts a total of 398 days, with the sun moving 3484646 minutes and Jupiter moving 43 degrees 2509956 minutes.
As for the sun, when it appears together with the sun, it becomes obscured. Then, it moves forward for a total of 71 days, covering 1489868 minutes, shifting 55 degrees 242860.5 minutes in the sky. Then, in the morning, it can be seen in the east, behind the sun. During direct motion, it travels 23/14 degrees each day, covering 112 degrees in 184 days. Moving forward again, the speed slows down, traveling 23/12 degrees each day, covering 48 degrees in 92 days. Then it stops, remaining still for eleven days. It then moves in reverse, traveling 62/17 degrees each day, retreating 17 degrees in 62 days. It then stops for another eleven days, then resumes direct motion, traveling 12 degrees each day, covering 48 degrees in 92 days. Moving forward again, the speed increases, traveling 14 degrees each day, covering 112 degrees in 184 days. At this point, it is in front of the sun, setting in the west at night. After 71 days, covering 1489868 minutes, shifting 55 degrees 242860.5 minutes in the sky, it appears together with the sun again. This full cycle lasts a total of 779 days and 97313 minutes, shifting 414 degrees 478998 minutes in the sky.
Mars, when it appears with the sun, also becomes latent. Then it moves forward for a total of 16 days, traversing the sky 1,122,426.5 minutes and moving 1 degree and 1,995,864.5 minutes in the sky, becoming visible in the east behind the sun in the morning. During the forward movement, it moves 3 minutes and 35 seconds each day, covering 7.5 degrees in 87.5 days. After that, it remains stationary for 34 days. Then, it moves in reverse, moving back 1 minute and 17 seconds each day, retreating 6 degrees in 102 days. After another 34 days, it moves forward again, moving 3 minutes each day, covering 7.5 degrees in 87 days; at this point, it appears in front of the sun, disappearing in the west at night. After 16 days, covering 1,122,426.5 minutes and moving 1 degree and 1,995,864.5 minutes in the sky, it appears with the sun again. In this cycle, it totals 378 days and 166,272 minutes, moving 12 degrees and 1,733,148 minutes in the sky.
Venus, when it appears with the sun in the morning, first "hides," which means moving backward, retreating 4 degrees in five days, and then it can be seen in the east behind the sun in the morning. Continuing to move backward, it moves 5/3 degrees each day, moving back 6 degrees in ten days. Then it stops for eight days. Then it starts moving forward, slowly, moving 46/33 degrees each day, covering 33 degrees in 46 days. Then the speed increases, moving 91/15 degrees each day, covering 160 degrees in 91 days. Then it speeds up, moving 91/22 degrees each day, covering 113 degrees in 91 days; at this point, it is behind the sun, appearing in the east in the morning. Finally, it moves forward for 41 days, covering one-fiftieth of a full circle, which is 50 degrees, and then it appears with the sun again. From one conjunction to the next, it totals 292 days and one-fiftieth of a full circle; Venus also travels this distance.
When Venus goes into conjunction with the sun in the evening, it first "hides," which means it moves in the same direction, moving one fifty-six-thousand-nine-hundred-fifty-fourth of a circle over forty-one days, which is fifty degrees and fifty-six-thousand-nine-hundred-fifty-fourth of a circle. Then in the evening, it can be seen in the west, in front of the sun. It continues to move in the same direction, speeding up, moving one degree and ninety-two twenty-second of a degree each day, traversing one hundred sixty degrees over ninety-one days. Then the speed slows down, moving one degree and fifteen fifteenths of a degree each day, traversing one hundred six degrees over ninety-one days, and then moving in the same direction again. The speed decreases, moving three degrees and forty-six minutes each day, traversing thirty-three degrees in forty-six days. Then it remains stationary for eight days. After that, it starts moving in the opposite direction, moving three-fifths of a degree each day, retreating six degrees over ten days; at this point, it is in front of the sun, appearing in the west in the evening. Continuing to move in the opposite direction, the speed increases, moving back four degrees over five days, and then it meets the sun again. The total duration from one conjunction to the next is five hundred eighty-four days and one hundred thirty-nine thousand eight hundredths of a circle, and Venus also traverses this much.
When Mercury goes into conjunction with the sun in the morning, it first "hides," which means it moves in retrograde, retreating seven degrees over nine days, making it visible in the east behind the sun in the morning. Continuing to move in the opposite direction, it speeds up, retreating one degree each day. Then it stops, remaining stationary for two days. After that, it starts moving in the same direction at a slow speed, moving eight-ninths of a degree each day, traversing eight degrees over nine days. Then the speed increases, moving one and a quarter degrees each day, traversing twenty-five degrees over twenty days; at this point, it is behind the sun, appearing in the east in the morning. Finally, it moves in the same direction for sixteen days, traversing one six hundred forty-one million nine thousand sixty-seventh of a circle, which is thirty-two degrees and six hundred forty-one million nine thousand sixty-seventh of a circle, and then it meets the sun again. The total duration from one conjunction to the next is fifty-seven days and six hundred forty-one million nine thousand sixty-sevenths of a circle, and Mercury also traverses this much.
Speaking of Mercury, it sets together with the Sun and then disappears, moving in a direct path. Sixteen days later, Mercury reaches the position of 32 degrees and 641,967 minutes of ecliptic longitude. At this time, it can be seen in the west during the evening, positioned ahead of the Sun. When in direct motion, it travels quickly, covering a total of twenty-five degrees in twenty days. If it slows down, it can only move seven-eighths of a degree each day, taking nine days to move back eight degrees. If it comes to a standstill, it will remain stationary for two days. If it moves in the opposite direction, known as retrograde motion, it will regress by one degree each day, at which point it is positioned ahead of the Sun, disappearing in the west at dusk. When retrograding, its speed is similarly slow, taking nine days to move back seven degrees, eventually aligning again with the Sun. From one conjunction to the next, it takes a total of 115 days and 601,255 minutes, and Mercury's motion follows this repetitive cycle.
