The speed of the moon's movement is sometimes fast and at other times slow, but overall it is relatively stable. The calculation method is as follows: first, based on various data between heaven and earth, calculate the number of lunar cycles and the remainder of the moon's movement, then square the remainder until the result equals the number of lunar cycles, to obtain a "lunar excess fraction." Then divide this "lunar excess fraction" by the number of days in a lunar cycle to obtain the number of degrees the moon moves each day. The variation in the moon's speed is consistent, and this trend can be described as a "decline." By subtracting this "decline" from the average speed, we can determine the specific degrees the moon travels each day. The value of "decline" will vary, contributing to the "profit and loss rate." "Profit" will accumulate, while "loss" will decrease; this represents the accumulation of gains and losses. Finally, multiply the value of half a small cycle by a constant, then divide by the total, and subtract the result from the total number of lunar cycles to calculate the degrees the moon moves on the new moon day (lunar New Year). Next, let's look at the specific calculation table. The table shows the degrees the moon moves each day, the decline value, the profit and loss rate, the accumulated profit and loss value, and the total number of degrees the moon moves. For example, on the first day, the moon moves 14 degrees and 10 minutes, the decline value is 1 degree, the profit and loss rate increases by 22 degrees, the accumulated profit and loss totals 276 degrees, and the total number of degrees is 276. On the second day, the moon moves 14 degrees and 9 minutes, the decline value is 2 degrees, the profit and loss rate increases by 21 degrees, the accumulated profit and loss totals 275 degrees, and so on.
From the third day to the sixteenth day, the data is as follows: on the third day, 14 degrees 7 minutes, decrease 3, gain 19, net 43, 273; on the fourth day, 14 degrees 4 minutes, decrease 4, gain 16, net 62, 270; on the fifth day, 14 degrees, decrease 4, gain 12, net 78, 266; on the sixth day, 13 degrees 15 minutes, decrease 4, gain 8, net 90, 262; on the seventh day, 13 degrees 11 minutes, decrease 4, gain 4, net 98, 258; on the eighth day, 13 degrees 7 minutes, decrease 4, loss, net 102, 254; on the ninth day, 13 degrees 3 minutes, decrease 4, loss 4, net 102, 250; on the tenth day, 12 degrees 18 minutes, decrease 3, loss 8, net 98, 246; on the eleventh day, 12 degrees 15 minutes, decrease 4, loss 11, net 90, 243; on the twelfth day, 12 degrees 11 minutes, decrease 3, loss 15, net 79, 239; on the thirteenth day, 12 degrees 8 minutes, decrease 2, loss 18, net 64, 236; on the fourteenth day, 12 degrees 6 minutes, decrease 1, loss 20, net 46, 234; on the fifteenth day, 12 degrees 5 minutes, increase 1, loss 21, net 26, 233.
The sixteenth day's situation is unique. The moon moves 12 degrees 6 minutes, which should have resulted in an increase of 2 and a decrease of 20, but due to the calculation method of losses and gains, the result was insufficient, so adjustments had to be made. The final result is a net of 5, an initial reduction of 20, with a total of 234 degrees.
Starting from the seventeenth day, the data continues: on the seventeenth day, 12 degrees 8 minutes, increase 3, no change, gain 18, reduction 15, 236; on the eighteenth day, 12 degrees 11 minutes, increase 4, no change, gain 15, reduction 23, 239.
At 12:15 PM on the 19th day, increase 3, decrease 1, add 11, reduce 48, the total is 243.
At 12:18 PM on the 20th day, increase 4, decrease 1, add 8, reduce 59, the total is 246.
At 1:03 PM on the 21st day, increase 4, decrease 1, add 4, reduce 67, the total is 250.
At 1:07 PM on the 22nd day, increase 4, loss added, reduce 71, the total is 254.
At 1:11 PM on the 23rd day, increase 4, loss 4 added, reduce 71, the total is 258.
At 1:15 PM on the 24th day, increase 4, loss 8 added, reduce 67, the total is 262.
At 2 PM on the 25th day, increase 4, loss 12 added, reduce 59, the total is 266.
On the 26th, at 2:04 in the afternoon, add 3, subtract 16, and reduce by 47; the total is 270. On the 27th, at 2:07 in the afternoon, add 3 for the initial advance, add 3 for the major Sunday, subtract 19, reduce by 31; the total is 273. On Sunday, at 2:09 in the afternoon, advance and subtract 21, reduce by 12; the total is 275. Sunday, 3333. Week Void, 2666. Sunday calculation, 5969. Through the week, 18539. Historical week, 16466. Lesser method, 1101. New moon phase, 11801. Minor fraction, 25. Half of the week, 127.
The above are the methods for calculating the product of the lunar month and the minor fraction. If the minor fraction reaches 31, it should be subtracted from the major fraction. If the major fraction reaches 16466, it is subtracted. If the remainder can be divided by 5969, it is the number of days, and the remainder is set aside. The calculated number of days is the date of the new moon.
To calculate the next month, add one day to this number of days; the remainder is 5832 and the minor fraction is 25. To calculate the first quarter, add seven days to this number of days; the remainder is 2283 and the minor fraction is 29.5. Then convert these fractions into days according to the method. If the number of days exceeds 27 days, subtract 27 days; the remaining portion represents the week fraction. If it is not enough, subtract one day and add 2666 (Week Void).
Calculate the surplus and deficit by multiplying 18539 (through the week) by it as the base number. Then multiply the total by the daily remainder fraction, and then multiply by the profit and loss rate to adjust the base number; this is the added time surplus and deficit. Subtract the month from the year and multiply by 127 (half of the week) to calculate the difference method. Use it for division to obtain the surplus and deficit of the major and minor parts. If the daily method does not yield sufficient results, the new moon date may need to be adjusted by one day. The first quarter advances and retreats with the major remainder, which is used to determine the minor remainder.
First paragraph: First calculate the number of surplus and loss days in a year, using a special method to calculate the size of the surplus and loss. Then add this surplus and loss to the current day's position of the sun and moon. If it is not enough or exceeds, adjust it using another method to finally determine the position of the sun and moon.
Second paragraph: Multiply half of the week by the remainder of the new moon, then divide by the number of weeks and subtract the remainder of the calendar days. If the remainder is insufficient, add the number of weeks and then subtract, which is equivalent to going back one day. Finally, add the number of weeks and its fraction to obtain the time at which midnight transitions into the calendar.
Third paragraph: When calculating the second day, start from the remainder of the previous day and accumulate up to twenty-seven days. If the remainder is exactly a multiple of the number of weeks, subtract it; if it is not a multiple, add the insufficient portion of the week, and the remaining amount is the entry into the calendar for the second day.
Fourth paragraph: Multiply the remainder that enters the calendar at midnight by the gain-loss rate, divide by the number of weeks; if it cannot be divided evenly, the remainder is what is left. Use this remainder to adjust the accumulation of gains and losses; if the remainder cannot be adjusted, use the number of weeks to adjust, obtaining the gain-loss value at midnight. A year serves as the unit, with anything less considered a fraction. Multiply the number of weeks by the fraction and remainder, divide the remainder by the number of weeks, and when the fraction is full, use the calendar method to adjust the degrees, adding the gains and subtracting the losses to get the final degree.
Fifth paragraph: Multiply the remainder that enters the calendar by the decay number, divide by the number of weeks; the part that cannot be divided evenly is the remainder, allowing you to understand the daily decay situation.
Sixth paragraph: Multiply the insufficient part of the week by the decay number, divide by the number of weeks to derive a constant. At the end of the calendar, add this constant to adjust the decay value; if it exceeds the decay number, subtract it, and then convert it to the decay value of the next calendar.
Seventh paragraph: Use the decay value to adjust the fraction of calendar days; if the fraction is either insufficient or excessive, adjust the degrees of the chapter years accordingly. Multiply the number of weeks by the fraction and remainder, then add the degree determined at midnight to get the degree for the second day. If the total does not equal the number of weeks at the end of the calendar, subtract 1338. Then, multiply the result by the number of weeks; if it equals the number of weeks, add the remainder of 837, then add the small fraction of 899, and finally add the decay value of the next calendar to continue the calculation.
Eighth paragraph: Subtract or add the decay value to the gain-loss rate to get a new gain-loss rate, then use it to adjust the gains and losses at midnight. If at the end of the calendar the gain-loss is insufficient, reverse the adjustment, enter the next calendar, and calculate the remaining part according to the method above.
Section Nine: Multiply the monthly running score by the number of nighttime hourglasses of the latest solar term, then divide by 200 to obtain the score for the bright portion. Subtract this score from the monthly score to get the score for the dark part. Calculate the degree based on the number of days in a chapter, multiply by the number of weeks, and add the degree determined at midnight to get the degree of darkness and light determined. If the remainder is more than half, round up; if less, discard.
Section Ten: The lunar calendar consists of four tables, three routes for entry and exit, with days arranged in a crisscross manner. Divide the monthly rate by these days to obtain the days in the calendar. Multiply the number of weeks by the conjunction number, then divide by the number of meetings to obtain the conjunction score. Multiply the number of weeks by the number, divide the remainder by the number of meetings to get the deduction. Then, based on the month and week, calculate the daily progress. Divide by the number of meetings to get the difference rate.
Yin and Yang calendar, decline, profit and loss rate, multiple
One day, subtract one, add seventeen, start
On the first day, the limit is 1290, the decimal part is 457. This is the starting limit.
Subtract 16 from the first day, then add 17; the result is 16.
On the second day, subtract 3, then add 15; the result is 33.
On the third day, subtract 4, then add 12; the result is 48.
On the fourth day, subtract 4, then add 8; the result is 60.
On the fifth day, subtract 3, then add 4; the result is 68.
On the sixth day, subtract 3 (indicating that 3 should be subtracted, but since there isn’t enough to do so, add 1 instead), then add 1; the result is 72.
On the seventh day, add 4, subtract 2; the result is 73. (If the limit is exceeded, subtract, as the moon has already passed halfway through its cycle.)
On the eighth day, add 4, subtract 6; the result is 71.
On the ninth day, add 3, subtract 10; the result is 65.
On the tenth day, add 2, subtract 13; the result is 55.
On the eleventh day, add 1, subtract 15; the result is 42.
On the twelfth day, the limit for the twelfth day is 3912, the decimal part is 1752. This is the ending limit.
On the thirteenth day, add 1 (just starting the calendar, calculating the date), subtract 16; the result is 27.
For daily distribution (5203), subtract 16 from those with lesser additions; the result is 11.
Minor Adjustment Method, 473.
Historical week, 107565.
Difference rate, 11986.
Conjunction score, 18328.
Differential: 914.
Differential Method: 2290.
Subtract the accumulated months from the conjunction month, then multiply the synodic and differential conjunctions by the remainder separately. Subtract from the synodic conjunction when the differential conjunction is full, and subtract when the synodic conjunction is full. What remains that does not complete the complete week is the solar calendar; subtract when it is full, and what is left is the lunar calendar. The remaining calculations are similar to those of month weeks, yielding one day. Beyond the calculation, what is sought is the integration of the month and the synodic conjunction into the calendar; not all are leftover days. Add two days; the leftover days are 2580, and the differential is 914. Calculate the date according to the established method: subtract when it reaches 13, and calculate according to the day when it is not full. The lunar and solar calendars ultimately converge at their respective endpoints, with the entry date before the limit and after the limit, and the moon runs halfway. Set the rate of calendar entry as either slow or fast; multiply the determined number by the small fraction to get the differential. Subtract the surplus and add the lunar and solar leftover days; if the surplus is not enough, adjust the day accordingly. Multiply the determined leftover days by the profit and loss rate, like the month weeks to get one, and use the profit and loss as the added time constant. Multiply the difference rate by the synodic small remainder, just like the differential method to get one. Subtract it from the historical day leftover; if not enough, add the month week and subtract again, and subtract one more day. Then add the day to its fraction; use the number to approximate the differential as the small fraction, which is the conjunction of the synodic day and night into the calendar. Speaking of this calculation of the calendar, you must first calculate the days. Add one day on the second day, for a total of thirty-one days; the small fraction is also thirty-one. If the sum of the small fraction and remaining days completes a month, subtract a month and add one day. Keep calculating like this until the end of the calendar; if the remaining days are exactly a month, subtract a month and enter the beginning of the new calendar. If the remaining days are not a full month, keep them, add 2720; the small fraction is still 31, and enter the next calendar cycle. Next, multiply the total number of days by the variable rate of the calendar and the remaining days. If the remaining days complete half a cycle, count it as a small fraction. Subtract the surplus from the deficit, and adjust the remaining days of the lunar and solar calendars. Adjust the number of days based on the lunar cycle according to the surplus or deficit of the remaining days. Next, multiply the determined remaining days by the profit and loss rate. If a month equals 1, use the profit and loss value to find the value at midnight. Then multiply the profit and loss rate by the nighttime duration of the most recent solar term; one two-hundredth represents brightness. Subtract the profit and loss rate to get dark, then use the profit and loss value at midnight to calculate the determined values of both dark and bright.
If the sum of the time equals a certain value, divide by 12 to get the degree; one-third of the remainder indicates weakness, while less than one degree indicates strength, and two degrees of weakness indicate weakness. This represents the angle at which the moon departs from the ecliptic. For the solar calendar, subtract the extreme from the added days in the ecliptic calendar, while for the lunar calendar, add it. Strong is positive, weak is negative; add the strong and weak, like terms cancel each other out, while unlike terms are added together. When subtracting, the same cancels out, the different adds up; there’s no overlap in cancellation, with two strong adding one weak and subtracting one weak.
From the year of Ji Chou in the Shangyuan period to the year of Bing Xu in the Jian'an period, 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 (year star), fire (Venus), earth (filling star), metal (Venus), water (Chen star). Each uses the whole day and the celestial degree to obtain the weekly rate and daily rate. Multiply the annual chapter by the weekly rate to get the lunar method; multiply the chapter month by the daily rate to get the month; divide the month by the lunar method to get the month. Multiply the total number of days by the lunar method to obtain the daily degree method. Multiply the Dipper measure by the weekly rate to calculate the Dipper value. (The daily degree method is multiplied by the law by the weekly rate, so we also use minutes here.)
The larger and smaller remainders of the five stars. (Multiply by the common calculation method to get the month, divide by the daily method; the remainder is the smaller remainder. Subtract the larger remainder from 60.)
The five stars enter the month and the day. (Multiply by the common calculation method to get the month, multiply by the conjunction method to get the smaller remainder, add, reduce, divide by the daily degree method, and you will obtain the results.)
The degree and degree remainders of the five stars. (Subtract the excess to get the degree remainder, multiply by the weekly day by the degree remainder, divide by the daily degree method; the remainder is the degree remainder. If it exceeds the weekly day, subtract the weekly day and the Dipper measure.)
Record month: 7285. Leap month: 7. Chapter month: 235. There are twelve months in a year. The common calculation method is forty-three thousand twenty-six. The daily value is 1457. The count is 47. The weekly day number is two hundred and fifteen thousand one hundred and thirty. The Dipper measure is one hundred and forty-five.
Jupiter: The circumference is six thousand seven hundred twenty-two, the daily value is seven thousand three hundred forty-one, which totals thirteen months, with a remainder of sixty-four thousand eight hundred one. The combined lunar method is one hundred twenty-seven thousand seven hundred eighteen, and the solar method is three hundred ninety-five thousand nine hundred fifty-eight. The major surplus is twenty-three, and the minor surplus is one thousand three hundred seven. The entry month day is fifteen, and the solar surplus is three hundred forty-eight thousand four hundred sixty-four. The virtual surplus is one hundred fifty, the division is ninety-seven thousand four hundred sixty-nine, the degree is thirty-three, and the remainder is two hundred fifty-nine thousand nine hundred fifty-six.
Mars: The circumference is three thousand four hundred seven, the daily value is seven thousand two hundred seventy-one, which totals twenty-six months, with a remainder of twenty-five thousand six hundred twenty-seven. The combined lunar method is sixty-four thousand seven hundred thirty-three, and the solar method is two million six hundred seventy-two thousand three hundred. The major surplus is forty-seven, and the minor surplus is one thousand one hundred fifty-seven. The entry month day is twelve, and the solar surplus is ninety-seven thousand three hundred thirteen. The virtual surplus is three hundred, the division is forty-nine thousand four hundred fifteen, the degree is forty-eight, and the remainder is one hundred ninety-nine thousand one hundred seventy-six.
Saturn: The circumference is three thousand five hundred twenty-nine, the daily value is three thousand six hundred fifty-three, which totals twelve months, with a remainder of fifty-three thousand eight hundred forty-three. The combined lunar method is sixty-seven thousand fifty-one, and the solar method is two hundred seventy-eight thousand five hundred eighty-one. The major surplus is fifty-four, and the minor surplus is five hundred thirty-four. The entry month day is twenty-four, and the solar surplus is sixteen thousand six hundred seventy-two. The virtual surplus is nine hundred twenty-three, the division is fifty-one thousand seven hundred five, the degree is twelve, and the remainder is one hundred seventy-three thousand one hundred forty-eight.
Venus: The circumference is nine thousand twenty-two, the daily value is seven thousand two hundred thirteen, which totals nine months, with a remainder of fifteen thousand two hundred ninety-three. The combined lunar method is one hundred seventy-one thousand four hundred eighteen, and the solar method is five hundred thirty-one thousand three hundred fifty-eight. The major surplus is twenty-five, and the minor surplus is one thousand one hundred twenty-nine.
This text records a series of astronomical data, which may be related to ancient calendar calculations. The specific meanings need to be accurately understood in conjunction with the literary context of that era. Terms such as "Tongfa," "Zhou rate," "He Yue method," and "Ridu method" need to be interpreted by professionals. These numbers represent the laws and cycles of different celestial bodies, reflecting the observations and calculations of ancient astronomers on the laws of cosmic motion.
The first day is the twenty-seventh day of the lunar calendar.
After fifty-six thousand nine hundred and fifty-four days have passed.
The virtual division is three hundred and twenty-eight.
The Dou division is one million three hundred and eighty-one thousand nine hundred.
The degree is two hundred ninety-two.
The remainder of the degrees is fifty-six thousand nine hundred and fifty-four.
The Zhou rate of Mercury is eleven thousand five hundred and sixty-one.
The daily rate is one thousand eight hundred and thirty-four.
The He Yue value is one.
The lunar remainder is two hundred and eleven thousand three hundred and thirty-one.
The He Yue method is two hundred nineteen thousand six hundred fifty-nine.
The Ridu method is six hundred eighty-nine thousand four hundred twenty-nine.
The Shuo Da remainder is twenty-nine.
The Shuo Xiao remainder is seven hundred and seventy-three.
The second day is the twenty-eighth day of the lunar calendar.
After sixty-four million nine hundred and sixty-seven days have passed.
The virtual division is six hundred and eighty-four.
The Dou division is one million six hundred and seventy-six thousand three hundred and forty-five.
The degree is fifty-seven.
The remainder of the degrees is sixty-four million nine hundred and sixty-seven.
Next, multiply the Shangyuan (referring to the Shangyuan calendar in this context) of the year you want to calculate by the Zhou rate. If the result can be divided by the daily rate, the quotient is the product, and the remainder is the He Yu remainder. Divide the He Yu remainder by the Zhou rate; if the quotient is one, it means the stars were in conjunction in the previous year; if the quotient is two, it means the stars were in conjunction in the previous two years; if it cannot be divided evenly, then it is in conjunction in that year. Subtract the He Yu remainder from the Zhou rate to get the degree. The conjunction product of Venus and Mercury, odd numbers are in the morning, and even numbers are in the evening.
Then, multiply the month number and month remainder by the product; if the result can be divided by the He Yue method, the quotient is the month, and the remainder is the month remainder. Subtract the Ji Yue from the product month, and the remainder is the Ruji month. Then, multiply the Zhangrun by the Ruji month; if it can be divided by the Zhangyue, the quotient is the number of leap months. Subtract this leap month number from the Ruji month; deduct this remaining part in the year; this part is calculated outside the astronomical framework, it is the He Yue. If during the transition of the leap month, use the new moon for regulation.
Using the standard method, multiply by the number of months, then multiply by the number of lunar months using the lunar month method, and then divide by the number of conjunctions. If the result can be divided using the daily calculation method, the quotient is the date of the lunar conjunction; the remainder is the leftover days, which is calculated separately from the lunar month. Multiply by the number of degrees in a week; if the result can be divided using the daily calculation method, the quotient is one degree. The remainder is the leftover, which is then used to calculate the five planets (a method of astronomical calculation). This is the method for finding the day of the conjunction.
Add the number of months and the month leftover; if the result can be divided by the lunar month method, the quotient is one month. If it is within a year, remove the whole parts, consider the intercalary month, and the remainder indicates the following year. Add the lunar months and the month leftover; if the result can form a month, then add 29 and 773 to the leftover. If the leftover can be divided using the daily calculation method, subtract from the greater part. The method is the same as above.
First, let's talk about how to calculate the days. Add the date of the lunar conjunction and the remaining days; if the total equals the number of days in a month, then it is correct. If there are remaining days when the conjunction occurs, subtract one day. If the remaining days exceed 773, subtract 29 days; if less than 773, subtract 30 days, and the result is the date of the following month. Next, we calculate the degrees. Add the degrees and the remainder of the degrees; if it equals a daily degree, then it is correct.
For Jupiter:
- Jupiter is in hiding for 32 days, 3484646 minutes; appears for 366 days.
- In hiding, it moves 5 degrees, 2509956 minutes; appears and moves 40 degrees (retrograde 12 degrees, actual movement 28 degrees).
For Mars:
- Mars is in hiding for 143 days, 973113 minutes; appears for 636 days.
- In hiding, it moves 110 degrees, 478998 minutes; appears and moves 320 degrees (retrograde 17 degrees, actual movement 303 degrees).
For Saturn:
- Saturn is in hiding for 33 days, 166272 minutes; appears for 345 days.
- In hiding, it moves 3 degrees, 1733148 minutes; appears and moves 15 degrees (retrograde 6 degrees, actual movement 9 degrees).
For Venus:
- Venus is in hiding in the east for 82 days, 113908 minutes; appears in the west for 246 days (retrograde 6 degrees, actual movement 240 degrees).
- In hiding, it moves 100 degrees in the east, 113908 minutes; appears in the east (the daily degree is calculated the same way as in the west, hides for 10 days, retrograde 8 degrees).
Part of Mercury: Mercury hides in the east for 33 days and 612,505 minutes; it appears in the west for 32 days (with a retrograde of 1 degree, making the actual movement 31 degrees); hidden movement is 65 degrees, 612,505 minutes; it appears in the east (the solar degree is the same as in the west, hidden for 18 days, retrograde 14 degrees).
Finally, let's talk about how to calculate the days and degrees of the star's appearance. Use the method to calculate the days and remainder of the planet's hiding, adding the remainder of the celestial body’s conjunction with the solar degree. If the remainder equals one solar degree, then it’s correct; according to the previous method, you can calculate the days and degrees of the star's appearance. Multiply the denominator of the celestial body's movement by the degrees of its appearance, and calculate the remainder according to the solar degree method; if it cannot be divided evenly and exceeds half, consider it as one. Then add the running fraction; if the fraction is full, the denominator equals one degree. The denominators for retrograde and direct motion are different; multiply the actual running denominator by the original fraction, and the result divided by the original denominator gives the actual running fraction. The remaining part carries over from the previous calculation, and for retrograde, you should subtract it. If the hiding days are not enough for one degree, use doufen (a traditional Chinese unit of measurement) to divide the fraction, using the running denominator as a proportion; the fraction will increase or decrease, affecting each other. Whenever it refers to "like fullness nearing completion," it seeks precise division; "to take and divide" is to take complete division.
In the morning, the sun and Jupiter appear simultaneously, and Jupiter hides away. Jupiter's direction of movement is direct, and its cycle is 16 days; in these 16 days, it traveled a total of 1,742,323 minutes, and the planet moved 2°32'34.67". Then, in the morning, Jupiter will be visible in the east, behind the sun. After that, Jupiter’s speed increases, moving 11/58 degrees each day, traveling 11 degrees in 58 days. Then the speed slows down again, moving 9/58 degrees each day, traveling 9 degrees in 58 days. Then Jupiter stops moving, and after 25 days, it begins to move in reverse. During retrograde, it moves backward 1/7 degree each day, retreating 12 degrees in 84 days. Then it stops again, and after 25 days, it resumes direct motion, moving 9/58 degrees each day, traveling 9 degrees in 58 days. After that, the speed increases again, moving 11/58 degrees each day, traveling 11 degrees in 58 days; at this time, Jupiter is in front of the sun, and it will be visible hiding in the west in the evening. After 16 days, it has traveled a total of 1,742,323 minutes, and the planet has moved 2°32'34.67", then appears simultaneously with the sun. One cycle concludes, lasting a total of 398 days, traveling 3,484,646 minutes, and the planet has traveled a total of 43°25'09.956".
In the morning, the sun and Mars appeared simultaneously, and Mars hid away. Mars moves in a prograde motion with a cycle of 71 days, during which it traveled 1,489,868 minutes, while the planet covered an angle of 55° 1,242,860.5 minutes. Then, in the morning, Mars could be seen in the east, positioned behind the sun. After that, Mars traveled 14/23 of a degree each day, covering 112 degrees in 184 days. It then slowed down, traveling 12/23 of a degree each day, covering 48 degrees in 92 days. Afterward, Mars stopped and began retrograde motion after 11 days. During the retrograde motion, it receded 17/62 degrees each day, retreating 17 degrees after 62 days. It then stopped again and resumed prograde motion after 11 days, traveling 12/23 of a degree each day, covering 48 degrees in 92 days. After that, it sped up, traveling 14/23 of a degree each day, covering 112 degrees in 184 days, at which point Mars was in front of the sun and could be seen setting in the west in the evening. After 71 days, it traveled another 1,489,868 minutes, and the planet moved 55° 1,242,860.5 minutes, and then reappeared alongside the sun. One cycle ended, totaling 779 days, with a movement of 973,113 minutes, and the planet covered 414° 478,998 minutes.
In the morning, the sun and Saturn appeared simultaneously. Saturn initially moved in a prograde direction, then in sixteen days, Saturn traveled 1,122,426.5 minutes (referring to ancient units of measurement, the same hereafter), and the planet covered an angle of 1 degree 1,995,864.5 minutes. At this point, Saturn was visible in the east, positioned behind the sun. During the forward motion, Saturn traveled 3/35 of a degree each day, covering 7.5 degrees in 87.5 days. Then Saturn stopped moving for 34 days. Next, it moved in reverse, traveling 1/17 of a degree each day, retreating 6 degrees after 102 days. After another 34 days, it resumed prograde motion, traveling 1/3 of a degree each day, covering 7.5 degrees in 87 days, at which point Saturn was in front of the sun and could be seen in the west in the evening. Over the course of the sixteen days, Saturn traveled 1,122,426.5 minutes, while the planet covered an angle of 1 degree 1,995,864.5 minutes, and then reappeared alongside the sun. One cycle ended, totaling 378 days and 166,272 minutes, with the planet covering 12 degrees and 1,733,148 minutes.
In the morning, the sun and Venus rise together. Venus initially follows a retrograde trajectory, retreating four degrees after five days. Then, in the morning, Venus can be seen in the east, behind the sun. During its retrograde motion, Venus moves three-fifths of a degree each day, retreating six degrees after ten days. Then, Venus pauses its movement for eight days. It then moves forward slowly, moving thirty-three-fifths of a degree each day, totaling thirty-three degrees after forty-six days. The speed then increases, with Venus moving one degree and ninety-one minutes each day, reaching one hundred and sixty degrees after ninety-one days. The speed continues to increase, with Venus moving one degree and ninety-one minutes each day, reaching one hundred and thirteen degrees after ninety-one days. At this point, Venus is behind the sun and visible in the eastern sky in the morning. After moving forward for forty-one days and fifty-six thousand nine hundred and fifty-four minutes, Venus covers fifty degrees. Venus and the sun then appear simultaneously again. This conjunction occurs every two hundred and ninety-two days, with Venus traveling the same distance.
In the evening, the sun and Venus also rise together. Venus's trajectory is direct, covering fifty degrees after forty-one days and fifty-six thousand nine hundred and fifty-four minutes. In the evening, Venus can be seen in the west, in front of the sun. Moving quickly, Venus covers one hundred and thirteen degrees in ninety-one days, then the speed decreases to one degree and fifteen minutes daily, totaling one hundred and sixty degrees after ninety-one days. Venus moves thirty-three-fifths of a degree each day, totaling thirty-three degrees after forty-six days. Then, Venus halts for eight days. It then goes retrograde, moving three-fifths of a degree each day, retreating six degrees after ten days. At this point, Venus is in front of the sun and can be seen in the west in the evening. Moving retrograde quickly, Venus retreats four degrees after five days, then appears simultaneously with the sun again. The cycle concludes after two conjunctions, totaling five hundred and eighty-four days and eleven thousand three hundred and ninety-eight minutes, with Venus traveling the same distance.
It is said that Mercury, in the morning, when it meets the Sun, it first hides, then moves in the opposite direction, moves back seven degrees in nine days, and then in the morning it can be seen in the east, behind the Sun. Then it moves in the opposite direction, faster, moving back one degree in a day. Then it stops, not moving for two days. Then it turns, moves in the correct direction, slower, travels eight-ninths of the Sun's path in a day, moving back eight degrees in nine days. When the speed picks up, it moves one degree and a quarter in a day, twenty-five degrees in twenty days, still behind the Sun. In the morning it appears in the east, then moves straight; after sixteen days, it travels 32 degrees plus six hundred forty-one million nine thousand sixty-seven millionths of a degree, and meets the Sun again, a total of fifty-seven days plus six hundred forty-one million nine thousand sixty-seven millionths of a degree; the distance Mercury travels is the same.
Next, let's talk about the situation when Mercury meets the Sun in the evening. When it meets the Sun in the evening, it first hides, then moves straight; after sixteen days, it travels 32 degrees plus six hundred forty-one million nine thousand sixty-seven millionths of a degree, then in the evening it can be seen in the west, ahead of the Sun. Then it moves straight, faster, moving one degree and a quarter in a day, twenty-five degrees in twenty days, still moving straight. When the speed slows down, it travels eight-ninths of the Sun's path in a day, moving back eight degrees in nine days. Then it stops, not moving for two days. Then it turns, moves in the opposite direction, moving back one degree in a day; in front of the Sun, it hides in the west in the evening. Moving in the opposite direction, slower, it moves back seven degrees in nine days, and meets the Sun again. Adding up the two meetings, it is a total of one hundred fifteen days plus six hundred one million two thousand five hundred five millionths of a degree; the distance Mercury travels is also the same.
