First, we multiply a fraction by twelve, then divide the result by a certain fixed number (this number is referred to as "fa"), obtaining a remainder, which represents a "chen." The calculation starts from "zi," which is a term in the Chinese lunar calendar, and in addition, we need to determine this small remainder based on the new moon, first quarter, and full moon phases.
Next, we multiply one hundred by the small remainder obtained earlier, then divide by another fixed number (also "fa") to get a "ke." If it doesn't divide evenly, we need to find the decimal part. This decimal part should be determined based on the nearest solar term (jieqi), starting from midnight; if not completed by night, we use the nearest value as a substitute.
During the calculation, there will be instances of carrying and borrowing. For carrying, we add, and for borrowing, we subtract. The difference in carrying and borrowing calculations is determined starting from two degrees; for every four-degree increase, the difference is halved, and the pattern of reduction is tripled after three iterations, until the difference reaches three, after which it returns to the initial state after five degrees.
The moon's speed can vary, but its periodic changes remain constant. We use some numbers (heaven and earth common numbers) to calculate, multiplying the remainder by itself until the result equals a certain number (count number), resulting in a "guo zhou fen." Dividing this "guo zhou fen" by the moon's weekly orbit (which is the distance the moon travels in a week) allows us to calculate the days in the calendar. The change in the moon's speed follows a pattern, which is "shi." We use this pattern to correct the degrees the moon travels each day, obtaining the daily degrees and minutes of travel. This change pattern will be calculated on both sides, yielding a "shun yi rate." If it is "yi," it continues to accumulate; if it is "sun," it continues to decrease, ultimately leading to a "ying suo ji."
Multiplying half a small week (the moon's half cycle) by a fixed number (common method), then dividing by another number (common number), and finally subtracting the result from the weekly days will give us the running minutes of the new moon day.
Below are the specific numerical calculation results, presented in a clear table format:
Day, Degrees, and Minutes | Column Decline | Shun Yi Rate | Ying Suo Ji | Moon Travel Minutes
------- | -------- | -------- | -------- | --------
One day fourteen degrees ten minutes | One borrowing decrease | Increase twenty-two | Initial gain | Two hundred seventy-six
Two days fourteen degrees nine minutes | Two borrowing decreases | Increase twenty-one | Gain twenty-two | Two hundred seventy-five
Three days, fourteen degrees and seven minutes | Decrease by four | Gain nineteen | Net surplus forty-three | Two hundred seventy-three
Four days, fourteen degrees and four minutes | Decrease by four | Gain sixteen | Net surplus sixty-two | Two hundred seventy
Five days, fourteen degrees | Decrease by four | Gain twelve | Net surplus seventy-eight | Two hundred sixty-six
Six days, thirteen degrees and fifteen minutes | Decrease by four | Gain eight | Net surplus ninety | Two hundred sixty-two
Seven days, thirteen degrees and eleven minutes | Decrease by four | Gain four | Net surplus ninety-eight | Two hundred fifty-eight
Eight days, thirteen degrees and seven minutes | Decrease by four | Decrease | Net surplus one hundred twenty | Two hundred fifty-four
Nine days, thirteen degrees and three minutes | Increase by four | Decrease four | Net surplus one hundred two | Two hundred fifty
Ten days, twelve degrees and eighteen minutes | Increase by three | Decrease eight | Net surplus ninety-eight | Two hundred forty-six
Eleven days, twelve degrees and fifteen minutes | Increase by four | Decrease eleven | Net surplus ninety | Two hundred forty-three
Twelve days, twelve degrees and eleven minutes | Increase by three | Decrease fifteen | Net surplus seventy-nine | Two hundred thirty-nine
Thirteen days, twelve degrees and eight minutes | Increase by two | Decrease eighteen | Net surplus sixty-four | Two hundred thirty-six
Fourteen days, twelve degrees and six minutes | Increase by one | Decrease twenty | Net surplus forty-six | Two hundred thirty-four
This passage describes an ancient method for calculating astronomical calendars, which calculates the daily movement of the moon through a series of multiplication and division operations, as well as forward and backward operations. While it might seem complex, it showcases the deep understanding that ancient astronomers had of celestial motion.
On the fifteenth day, the moon moves to twelve degrees and five minutes. In the first calculation, decrease by twenty-one degrees, increase by twenty-six degrees; the total is two hundred thirty-three degrees.
On the sixteenth day, the moon moves to twelve degrees and six minutes. In the second calculation, decrease by twenty degrees (because it is insufficient; if there's a deficiency, we treat a decrease of five as a gain, which means decreasing by five degrees instead of twenty degrees, and the originally decreased twenty degrees is reduced to twenty degrees, so it is considered sufficient). Increase by five degrees, decrease the initial twenty degrees; the total is two hundred thirty-four degrees.
On the seventeenth day, the moon moves to twelve degrees and eight minutes. In the third calculation, increase by eighteen degrees, decrease by fifteen degrees; the total is two hundred thirty-six degrees.
On the eighteenth day, the moon moves to twelve degrees and eleven minutes. In the fourth calculation, increase by fifteen degrees, decrease by twenty-three degrees; the total is two hundred thirty-nine degrees.
On the 19th, the moon moved to a position of 12 degrees and 15 minutes. In the third calculation, it increased by 11 degrees and decreased by 48 degrees, for a total of 243 degrees.
On the 20th, the moon moved to a position of 12 degrees and 18 minutes. In the fourth calculation, it increased by 8 degrees and decreased by 59 degrees, for a total of 246 degrees.
On the 21st, the moon moved to a position of 13 degrees and 3 minutes. In the fourth calculation, it increased by 4 degrees and decreased by 67 degrees, for a total of 250 degrees.
On the 22nd, the moon moved to a position of 13 degrees and 7 minutes. In the fourth calculation, it increased by a loss of 8 degrees and decreased by 71 degrees, for a total of 254 degrees.
On the 23rd, the moon moved to a position of 13 degrees and 11 minutes. In the fourth calculation, it increased by a loss of 4 degrees and decreased by 71 degrees, for a total of 258 degrees.
On the 24th, the moon moved to a position of 13 degrees and 15 minutes. In the fourth calculation, it increased by a loss of 8 degrees and decreased by 67 degrees, for a total of 262 degrees.
On the 25th, the moon moved to a position of 14 degrees. In the fourth calculation, it increased by a loss of 12 degrees and decreased by 59 degrees, for a total of 266 degrees.
On the 26th, the moon moved to a position of 14 degrees and 4 minutes. In the third calculation, it increased by a loss of 16 degrees and decreased by 47 degrees, for a total of 270 degrees.
On the 27th, the moon moved to a position of 14 degrees and 7 minutes. This marks both the third occurrence and the first time we are adding three major Sundays, decreasing by 19 degrees and by 31 degrees, for a total of 273 degrees.
Sunday: 14 degrees and 9 minutes, decreased by an additional 21 degrees and decreased by 12 degrees, for a total of 275 degrees.
Total for weekdays: 3333.
Void Week: 2666.
Sunday calculation: 5969.
Total for the week: 18539.
Historical week: 16466.
Less big law: 1101.
Shuo Xing Da Fen: 1801.
Minor adjustments: 25.
Week half: 127.
The above are various parameters for calculating lunar motion. Using these parameters, combined with the lunar phases each month, one can calculate the date of the next new moon (the first day of the lunar calendar) using multiplication and addition or subtraction. The specific calculation method is as follows: multiply the accumulated number of months by the fractional value of the lunar phases; if the small fractional value reaches thirty-one degrees, subtract it from the large fractional value. If the large fractional value completes a lunar cycle, subtract the lunar cycle value. The remaining value is divided according to the weekly method; the quotient gives the number of days, while the remainder is noted separately. The day remainder should be noted separately, as it will be used when calculating the new moon date.
To calculate the new moon date for next month, add one day to today’s day count; the day remainder is 5,832, and the small fraction is 25. To calculate the full moon date (the fifteenth or sixteenth day of the lunar calendar), add seven days to today’s day count; the day remainder is 2,283, and the small fraction is 29 and a half. These fractional values must be converted into days using the method described above. If the number of days exceeds twenty-seven, subtract twenty-seven days, and process the remainder using the weekly method. If it is insufficient for division, subtract one day and then add the weekly fractional value.
Wow, this article looks pretty intense, so let’s take it slow and translate it into plain language, sentence by sentence.
First, the first paragraph discusses how to calculate the gains and losses in the calendar. It accumulates the excess or deficit days over the year and uses some complex calculation methods to determine how much time should be added or subtracted each day to adjust the calendar. "Subtract the lunar cycle from the monthly fraction, multiply by half the week for the difference method, and divide it," this is the key calculation formula. We don’t need to worry about the specific calculations; it's enough to know that it helps fine-tune the calendar's accuracy. The last sentence states that based on the calculation results, the new moon date (the first day of the lunar calendar) should be determined at what time.
Next, the second paragraph continues to explain the calculation method. It uses the previously calculated gain and loss values to adjust the positions of the sun and moon monthly, ensuring the accuracy of the calendar. "Multiply the small remainder of the new moon by half the week, as if it were a whole number, to subtract it from the calendar's day remainder," this is another calculation formula; we just need to know it's for fine-tuning the calendar's accuracy.
The third paragraph explains how to calculate "midnight's entry into the calendar," which corresponds to the calendar date at midnight. It uses a series of calculations based on the remainder calculated earlier to ultimately determine the accurate date and time corresponding to midnight. The phrase "then add the day of the week to the minutes, that is the midnight's entry into the calendar" is the calculation result.
The fourth paragraph discusses how to calculate the calendar date of the next day. It accumulates the surplus or deficit for each day, and if it exceeds one day, it subtracts one day, leaving the remainder as the surplus for the next day.
The fifth paragraph continues to calculate midnight's gains and losses, and uses these to adjust the date and time of midnight. The calculation method is complex, but the ultimate goal is to ensure the accuracy of the calendar. The phrase "a complete cycle of years is a degree, not all are minutes" indicates that the calculation results are in degrees and minutes.
The sixth paragraph explains how to calculate the daily "variations and declines," which should be a value related to time, used to further adjust the accuracy of the calendar. The phrase "multiply the remaining days by the column decline, as the weekly method yields one, not all are remaining" is another calculation formula that we do not need to delve into.
The seventh paragraph discusses how to update the value of "variations and declines" and use it to adjust the calendar. The phrase "multiply the weekly void by the column decline, as the weekly method serves as a constant, the calendar ends, then add the variations and declines" is also a calculation formula, and we only need to know that it continuously corrects the accuracy of the calendar. The last sentence states that if the calculation results are not accurate, further adjustments are needed.
The eighth paragraph continues to explain the calculation method, and points out that if the calculation results do not align directly with Sunday, additional adjustments are necessary. The phrase "the calendar does not align directly with Sunday, subtract one thousand thirty-eight, then multiply by the total" is another calculation formula that we do not need to delve into.
The ninth paragraph discusses how to calculate the adjusted gains and losses, and use it to adjust the calendar. The phrase "subtract the variations and declines and add the loss and profit rate, as the change in loss and profit rate, and then turn the loss and profit into midnight profit and shrinkage" is another calculation formula that we do not need to delve into.
The final paragraph explains how to calculate the "twilight and daylight divisions," which should refer to the length of day and night, and use them to further adjust the accuracy of the calendar. The phrase "multiply the historical month by the nearest solar term night leak, two hundred and one is the daylight division" is another calculation formula that we do not need to delve into.
In conclusion, this entire passage describes a very complex method of calendar calculation, and its aim is to accurately determine each day's date and time. Although we do not understand the specific calculation process, we understand its goal: to create a more accurate calendar.
First, let's explain what this article is about. In simple terms, this is an ancient algorithm for calendar calculations, involving many astronomical terms and complex calculation steps. Next, we will interpret it sentence by sentence, expressing it in contemporary spoken Chinese.
"The Four Tables of the Menstrual Cycle, entering and exiting three paths, intersecting and dividing the days, dividing by the lunar rate, to obtain the day of the calendar." This passage means: based on the data of the Four Tables of the Menstrual Cycle, calculate the number of days through three methods (entering and exiting three paths), intersect these days, and then divide by the lunar cycle (lunar rate) to obtain a day in the calendar.
"The week multiplied by the new moon and full moon combined refers to the cycle of the moon's phases; multiplying the number of weeks by the number of new moon and full moon cycles is like adding up the cycles of the moon's phases to get a score for the new moon and full moon."
"The total multiplied by the combined number, the remainder equals the number of lunar cycle phases, and the subtraction is also." Multiply the total number of days (total) by the number of new moon and full moon cycles (combined number); the remainder equals the number of lunar cycle phases, and this remainder is the subtraction.
"Based on the cycle of the moon's movement (lunar cycle), calculate the daily increase." Based on the cycle of the moon's movement (lunar cycle), calculate the daily increase.
"The number of cycles minus one gives the difference rate." Divide the number of cycles of the moon's phases (number of cycles) by 1 to get a difference rate.
Next is the content of the table, which lists the addition and subtraction operations that need to be performed each day, along with the corresponding values. These numbers represent the daily gains and losses in calendar calculations. Since these numbers are professional terms in ancient calendar calculations, we will retain the original text:
Day 1: Subtract 17, gain
Day 2: Subtract 16, gain of 17 (limit of 1290, minor difference of 457), this represents the previous limit
Day 3: Subtract 15, gain of 33
Day 4: Subtract 12, gain of 48
Day 5: Subtract 8, gain of 60
Day 6: Subtract 4, gain of 68
Day 7: Subtract 3 (insufficient to subtract, turn the loss into a gain, meaning gain one, should subtract three, due to insufficient amounts)
Gain of 1, total of 72
August 8: Add 4, subtract 2, resulting in 73
(When it surpasses the limit, it is said that the moon has completed half a cycle; if it has exceeded this limit, it should be subtracted.)
September 9: Add 4, subtract 6, resulting in 71
October 10: Add 3, subtract 10, resulting in 65
November 11: Add 2, subtract 13, resulting in 55
December 12: Add 1, subtract 15, resulting in 42
January 13 (remaining 3,912, minor division 1,752.)
This represents the later limit.
Add 1 (the beginning of the calendar, representing a fraction of a day); then subtract 16, resulting in 27.
Fraction of a day (5,203) minor addition and subtraction, subtract 16 major, resulting in 11.
Minor major law, 473.
"Calendar cycle: 107,565. Error rate: 19,986. New moon combined fraction: 18,328. Minor fraction: 914. Minor fraction law: 2,209." These numbers represent important parameters in calendar calculations; we retain the original text.
"Using the lunar cycle to subtract the upper yuan accumulated months, the remainder is multiplied by the new moon combined fraction and the minor fraction respectively; if the minor fraction meets its criteria, it is deducted from the combined fraction; if the combined fraction reaches the cycle of the heavens, it is subtracted from the cycle, and the remainder that does not fill the calendar cycle is considered the solar calendar; if it fills the cycle, the remainder is considered part of the lunar calendar. The remainder is calculated as one day for each lunar cycle; apart from these calculations, the requested lunar phases and the time difference from the calendar's start, where the remainder is less than one day."
"Add 2 days, the remaining days are 2,580, minor division 914; calculate the days according to the method; if it exceeds 13, subtract 13, and the remainder is the fraction. The solar and lunar calendars ultimately interweave; the calendar begins with the remainder before the upper limit, and after it starts, it reflects the lower limit remainder; this indicates that the moon has reached the midpoint of its cycle."
"Each parameter in the calendar is set to account for speed, gain and loss, size, and other factors, multiplying the number of lunar cycles by the small fractions to obtain differentials. Adjusting the remaining days of the lunar calendar by adding or subtracting gains and losses, we adjust the days if there is a surplus or deficit. Multiplying the determined remaining days by the gain/loss rate, if it results in one lunar month, we use the total gain/loss as the determined value for additional time.
In summary, this passage describes a very complex ancient calendar calculation method, involving a large number of specialized terms and calculation steps. It is difficult for modern people to fully understand its specific meaning without in-depth study of ancient astronomical calendars.
First, let’s calculate the 'new moon day,' which marks the beginning of the lunar month. Multiply the difference rate by the decimal portion of the remaining days until the new moon, similar to calculus, to calculate a number, then subtract it from the number of days in the calendar. If the result is insufficient, add the number of days in a month and subtract again, then subtract one more day. Add the remaining days to their fractional part, then simplify the differential fraction using multiplication to get the exact time of midnight on the new moon day.
Next, calculate the second day. Add one day; the remaining days are 31, and the small fraction is also 31. If the small fraction exceeds the multiplication number, subtract the number of days in a month. Add one more day; the calculation for the calendar is complete. If the remaining days exceed the fraction days, subtract the fraction days; this is the starting day of the calendar. If the remaining days do not exceed the fraction days, use them directly, then add 2720; the small fraction is 31, which is the starting day of the next calendar.
Then calculate the gain and loss of the moon each night in the calendar, as well as the remaining portions. If the remaining portion exceeds half a cycle, it will be treated as the small fraction. Add and subtract gains and losses to the remaining days; if the gain or loss is insufficient, use the lunar week to adjust the days. Multiply the determined remaining days by the gain/loss rate; if it equals the lunar week, use the total gain/loss to determine the midnight value.
Calculate the twilight time by multiplying the gain/loss rate by the night time of the nearest solar term, then divide by 200 to get the bright time. Subtract this value from the gain/loss rate to get the dark time. Using the midnight gain/loss value, we can determine the twilight time."
If you want to add time, use the specific values determined by dusk and dawn, divide it by 12 to get the degree. Multiply the remainder by one-third; if it is less than one, it is called "strong," and if it exceeds one, it is called "weak." Two "weak" equal one "strong." This calculates the angle of the moon's departure from the ecliptic. For the solar calendar, subtract the extreme position from the calendar of the day on the ecliptic; for the lunar calendar, add the extreme position to calculate the angle at which the moon departs from the extreme position. "Strong" represents a positive value, while "weak" indicates a negative value. Add values of the same type and subtract values of different types. When subtracting, values of the same type cancel out, while values of different types add together; there are no situations where values cancel each other out.
From the year of Ji Chou in the Shangyuan to the year of Bing Xu in Jian'an, a total of 7378 years have passed.
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
The five elements are: wood (Jupiter), fire (Mars), earth (Saturn), metal (Venus), water (Mercury). Combined with their final day and celestial degree, the week rate and day rate are obtained. Multiply the annual cycle by the week rate to get the lunar method; multiply the monthly cycle by the daily rate to get the month division; divide the month division by the lunar method to get the month number. Multiply the total by the lunar method to get the daily degree method. Multiply the Dipper star by the week rate to get the Dipper. (The daily degree method is multiplied by the discipline rate, so here it is also multiplied by the minute.)
Next, calculate the major and minor remainders of the five stars. (Multiply the general law by the month number separately, divide the daily law by the month number separately, get the major remainder, and the remainder is the minor remainder. Subtract 60 from the major remainder.)
Finally, calculate the number of days and the remaining days when the five stars enter the month. (Multiply the general law by the month remainder separately, multiply the combined month law by the minor remainder of the new moon, add them up, simplify with the meeting number, then divide by the daily degree method to get the final result.)
This passage outlines some data related to ancient astronomical calendars; let's break it down sentence by sentence. First, "degrees and remainders of the five stars. (Subtracting more for the remainder, multiplying by the week to get it, rounding it with the daily degree method, the result gives the degree, while any excess is the remainder, go beyond the week and the Dipper.)" This sentence explains the method of calculating the degrees and remainders of the movement of the five stars. In simple terms, calculate the degree first; the excess part is the remainder. If it exceeds one cycle (week), don’t get too caught up in the details; just know it is a calculation method.
Then, "Epoch Month: seven thousand two hundred eighty-five." This indicates that the total number of months in a certain era is seven thousand two hundred eighty-five months.
"Leap Months: seven." Seven leap months.
"Total Months: two hundred thirty-five." A total of two hundred thirty-five months.
"Months per Year: twelve." There are twelve months in a year, which everyone understands.
"Total Method: forty-three thousand twenty-six." This should be the total value of some calculation method, and the specific meaning needs to be understood in context.
"Day Count: one thousand four hundred fifty-seven." This probably refers to a standard value for some calendar calculation.
Next, "Total Count: forty-seven." "Week Days: two hundred eleven thousand five hundred thirty." "Dou Division: one hundred forty-five." These are some astronomical units or parameters, and their specific meanings require specialized knowledge to explain clearly. We ordinary people just need to know that these are numbers used in ancient astronomical calculations.
"Wood: Orbital Period: six thousand seven hundred twenty-two. Day Rate: seven thousand three hundred forty-one. Combined Month Count: thirteen. Month Remainder: sixty-four thousand eight hundred one. Combined Month Method: one hundred twenty-seven thousand seven hundred eighteen. Day Degree Method: three million nine hundred fifty-nine thousand two hundred fifty-eight. New Moon Large Remainder: twenty-three. New Moon Small Remainder: one thousand three hundred seven. Day of Entering Month: fifteen. Remaining Days: three million four hundred eighty-four thousand six hundred forty-six. New Moon Virtual Division: one hundred fifty. Dou Division: nine hundred seventy-four thousand six hundred ninety. Orbital Degrees: thirty-three. Degree Remainder: two million nine hundred fifty-nine thousand nine hundred fifty-six." This section is about various data regarding Jupiter's orbit, including orbital period, day rate, combined month count, etc., all results of observations and calculations by ancient astronomers. These numbers may seem abstract to us, and we don't need to fully understand them.
"Fire: Orbital Period: three thousand four hundred seven. Day Rate: seven thousand two hundred seventy-one. Combined Month Count: twenty-six. Month Remainder: twenty-five thousand six hundred twenty-seven. Combined Month Method: sixty-four thousand seven hundred thirty-three. Day Degree Method: two million six thousand seven hundred twenty-three. New Moon Large Remainder: forty-seven. New Moon Small Remainder: one thousand one hundred fifty-seven. Day of Entering Month: twelve. Remaining Days: ninety-seven thousand three hundred thirteen. New Moon Virtual Division: three hundred. Dou Division: four hundred ninety-four thousand one hundred fifteen. Orbital Degrees: forty-eight. Degree Remainder: one hundred ninety-nine thousand one hundred seventy-six." This section contains similar data about Mars.
Earth: Orbital period, 3529. Sun rate, 3653. Total number of lunar months, 12. Remaining days in the lunar month, 53843. Total lunar days calculated, 6751. Daily degree method, 278581. Major new moon excess, 54. Minor new moon excess, 534. Days in the lunar month, 24. Remaining days in the lunar month, 166272. New moon excess, 923. Dou excess value, 51175. Degree count, 12. Remaining degree count, 173148. This passage is about Saturn.
Venus: Orbital period, 9022. Sun rate, 7213. Total number of lunar months, 9. This section provides information about Venus, with only partial data included. Overall, this text records the observations and calculations of ancient astronomers on the movement of planets, containing a large amount of professional terminology and data. It is difficult for us to understand the calculation methods and astronomical significance behind it, but we can sense the exploration of the cosmos and the meticulous spirit of calculation exhibited by ancient astronomers.
Following this, one month has passed; the data is 152293. Using the total lunar days method, the result is 171418. Using the daily degree method, the result is 531958. Major new moon excess is 25. Minor new moon excess is 1129. Days in the lunar month is 27. Remaining days in the lunar month is 56954. New moon excess is 328. Dou excess value is 1308190. Degree count is 292. Remaining degree count is 56954.
Water: Orbital period is 11561. Sun rate is 1834. Total number of lunar months is 1. Next, one month has passed; the data is 211331. Using the total lunar days method, the result is 219659. Using the daily degree method, the result is 6809429. Major new moon excess is 29. Minor new moon excess is 773. Days in the lunar month is 28. Remaining days in the lunar month is 641967. New moon excess is 684. Dou excess value is 1676345. Degree count is 57.
The value is six million, four hundred nineteen thousand, nine hundred sixty-seven. First, substitute the previous year's data into the equation, multiply it by the lunar cycle; if the result divides evenly by the solar cycle to yield one, this is called the integrated sum, while the portion that cannot be divided is known as the sum remainder. Divide the integrated sum by the lunar cycle; if the result is one, it is the star integration in the previous year; if the result is two, it is the star integration in the previous two years; if it cannot be evenly divided, it is the integration in the current year. Subtract the lunar cycle from the sum remainder to get the degree. For the integration of metal and water, odd numbers represent morning while even numbers represent evening.
Multiply the number of months and the month remainder by the integrated sum separately; if the result can be evenly divided by the monthly integration method, the month is obtained, and the remainder that cannot be divided is the month remainder. Subtract the integrated month from the sum of months; the remainder is the entry month. Then multiply it by the chapter leap; if it divides evenly by the chapter month, it results in a leap, subtract it from the entry month, and subtract the remaining portion during the year; this calculation is outside the astronomical calendar and is referred to as month integration. If at the time of the leap month transition, use the new moon to control.
Multiply the common method by the month remainder, multiply the monthly integration method by the remainder of the new moon, then simplify it by the number of meetings; if the result can be evenly divided by the solar degree method to get one, it is the entry month day of the star integration; the remainder that cannot be divided is the day remainder, this part of the calculation is outside the calculation of the new moon.
Multiply the weekly cycle by the degree; if it can be evenly divided by the solar degree method to get one degree, the remainder that cannot be divided is the remainder; apply the first five days of the ox method to determine the degree. The above is the method of calculating the star integration.
Add the number of months and the month remainder; if it can be evenly divided by the monthly integration method to get a month, then place it in the current year; if it cannot be evenly divided, place it in the next year; if it can be evenly divided again, place it in the next two years. Gold and water added in the morning equals evening, added in the evening equals morning.
First, let's calculate the size remainder of the moon. Add up the size remainder of the new moon; if it exceeds one month, then add another 29 (large remainder) or 773 (small remainder). If the small remainder is full, use the algorithm for the large remainder; the method is the same as before.
Next, add up the entry day and the day remainder; if the remainder is enough for one day, add one day. If the small remainder perfectly fills the gap during the new moon, subtract one day; if the small remainder exceeds 773, subtract 29 days; if it does not exceed 773, subtract 30 days; the remaining days are determined based on the date of the new moon; this is the entry day of the month.
Next, sum the degrees and the degree remainders; if it is enough for one day of degree, add one degree.
Jupiter: It remains inactive for 32 days, (travels) 3,484,646 minutes; appears for 366 days; remains inactive and travels 5 degrees, (travels) 2,509,956 minutes; appears traveling 40 degrees. (Subtracting 12 degrees for retrograde motion, the actual travel is 28 degrees.)
Mars: It remains inactive for 143 days, (travels) 973,113 minutes; appears for 636 days; remains inactive and travels 110 degrees, (travels) 478,998 minutes; appears traveling 320 degrees. (Subtracting 17 degrees for retrograde motion, the actual travel is 303 degrees.)
Saturn: It remains inactive for 33 days, (travels) 166,272 minutes; appears for 345 days; remains inactive and travels 3 degrees, (travels) 1,733,148 minutes; appears traveling 15 degrees. (Subtracting 6 degrees for retrograde motion, the actual travel is 9 degrees.)
Venus: It remains hidden in the eastern morning sky for 82 days, (travels) 113,908 minutes; appears in the west for 246 days. (Subtracting 6 degrees for retrograde motion, the actual travel is 240 degrees.) It remains inactive and travels 100 degrees in the eastern morning sky, (travels) 113,908 minutes; appears in the east. (The daily rate is the same as in the west, remains inactive for 10 days, retrograde 8 degrees.)
Mercury: It remains hidden in the western morning sky for 33 days, (travels) 612,505 minutes; appears in the west for 32 days. (Subtracting 1 degree for retrograde motion, the actual travel is 31 degrees.) It remains inactive and travels 65 degrees, (travels) 612,505 minutes; appears in the east. (The daily rate is the same as in the west, remains inactive for 18 days, retrograde 14 degrees.)
First, let's calculate the positions of the Sun and the planets. Calculate the degrees the Sun travels each day, then add the degrees the planets travel each day. If the sum is a whole number, it indicates that the planet and the Sun are aligned. If it's not a whole number, continue calculating using the previous method until the day when the planet and the Sun meet is found. Then, multiply the degrees the planet travels each day by the degrees it travels when it meets the Sun. If the result, when divided by the degrees the Sun travels each day, has a remainder, and if the remainder is more than half, add one more; if the degrees the Sun travels each day can be divided evenly by the degrees the planet travels, it means the planet has traveled one degree. The calculation methods for direct and retrograde motion are different and need to be adjusted according to the direction of the planet's travel. During calculations, if there is a remainder, divide it by the planet's orbital period to find its specific position, and verify results both before and after the calculation. In summary, terms like "fullness" or "full phase" are actually used for precise calculations through division; and "divide, subtract, and divide again" refers to achieving the utmost precision in division.
Next, let's take a look at the situation of Jupiter. Jupiter conjuncts the sun in the morning, then Jupiter begins its direct motion. After 16 days, the sun has traveled 1,742,323 arcminutes, and Jupiter has traveled 2,323,467 arcminutes. At this time, Jupiter appears behind the sun, in the east. During the direct motion, Jupiter moves very fast, covering 58 arcminutes and 11 degrees each day; then, as it continues to move forward, its speed slows down, covering 9 arcminutes and 9 degrees each day for 58 days. Then Jupiter stops its direct motion and begins retrograde motion after 25 days. During retrograde motion, Jupiter moves 7 arcminutes and 1 degree each day, retreating 12 degrees after 84 days. Then Jupiter halts again and resumes direct motion after 25 days, covering 58 arcminutes and 9 degrees each day for 58 days. The forward speed increases again, covering 11 arcminutes and 11 degrees each day, and at this point, Jupiter appears in front of the sun, setting in the western sky during the evening. After 16 days, the sun has traveled 1,742,323 arcminutes, and Jupiter has traveled 2,323,467 arcminutes, and they meet again. The entire cycle is a total of 398 days, with the sun traveling 3,484,646 arcminutes and Jupiter traveling 43,259,956 arcminutes.
In the morning when the sun rises, Mars aligns with the sun, then Mars becomes obscured. It then begins its direct motion, continuing for 71 days, covering 1,489,868 arcminutes, equivalent to 55 degrees and 242,860.5 arcminutes. Then, we can see Mars in the east in the morning, behind the sun. During the direct motion, Mars moves 23/14 degrees each day, covering 112 degrees after 184 days. Then the forward speed slows down, moving 23/12 degrees each day, covering 48 degrees after 92 days. It then stops moving for 11 days. Then it starts retrograde motion, moving 62/17 degrees each day, retreating 17 degrees after 62 days. It stops again for 11 days, then starts moving forward again, covering 1/12 degrees each day, covering 48 degrees after 92 days. The forward speed then increases, covering 1/14 degrees each day, covering 112 degrees after 184 days, and at this point, it moves in front of the sun, visible in the western sky during the evening. After 71 days, it covers 1,489,868 arcminutes, equivalent to 55 degrees and 242,860.5 arcminutes, aligning with the sun again. In this way, one cycle is 779 days and 973,113 arcminutes, with Mars moving 414 degrees and 478,998 arcminutes.
Saturn, in the morning, conjuncts with the sun and then goes into conjunction. It then starts to move forward, continuing for 16 days, covering an arc of 1,122,426.5 minutes, which is equivalent to 1 degree 199,584.5 minutes. After that, we can see Saturn in the east in the morning, positioned behind the sun. During its forward movement, Saturn travels 11⅔ degrees each day, covering 7.5 degrees in 87.5 days. Then it stops moving for 34 days. It then starts moving backward, receding 6 degrees over 102 days, covering 17 degrees each day. After another 34 days, it resumes direct motion, covering 1/3 of a degree each day, moving 7.5 degrees in 87 days, at which point it moves in front of the sun and can be seen in the west at night. After 16 days, it travels an arc of 1,122,426.5 minutes, which is equivalent to 1 degree 199,584.5 minutes, aligning with the sun again. In this way, one cycle is 378 days and 166,272 minutes, with Saturn covering 12 degrees and 173,148 minutes.
As for Venus, when it aligns with the sun in the morning, it first enters retrograde motion, moving back 4 degrees in 5 days. Then it can be seen in the east in the morning, behind the sun. Continuing in retrograde, it moves ⅗ of a degree each day, moving back 6 degrees in 10 days. It then remains stationary for 8 days. It then starts moving forward, at a slow pace, covering 1⅓ degrees each day, moving 33 degrees in 46 days. The speed then increases, covering 1⅓ degrees each day, moving 160 degrees in 91 days. It then accelerates further, covering 1⅓ degrees each day, moving 113 degrees in 91 days, at which point it is behind the sun and appears in the east in the morning. Finally, it moves forward, covering 1/56 of a circle in 41 days, with the planet covering 50 degrees and 1/56 of a circle, before aligning with the sun again. The alignment cycle is 292 days and 1/56 of a circle, and the planet's movement follows this pattern.
When Venus meets the sun in the evening, it first "conjures," moving in direct motion, spending 41 days to travel 1/56,954 of a circle. The planet travels 50 degrees and 56,954/1 circles, and is visible in the western sky in the evening, positioned ahead of the sun. It then continues moving in direct motion, accelerating to travel 1 degree and 22/91 of a degree each day, completing 113 degrees in 91 days. The speed then decreases slightly, moving 1 degree and 15/1 of a degree each day, completing 160 degrees in 91 days, and then continues moving in direct motion. The speed slows down, traveling 46 degrees and 33/30 each day, completing 33 degrees in 46 days. Then it "pauses" for 8 days, remaining stationary. It then starts moving in the opposite direction, traveling 5 degrees and 3/1 each day, receding by 6 degrees over 10 days, appearing in the west in the evening, in front of the sun. It continues its retrograde motion, accelerating to retreat 4 degrees over 5 days, and then meets the sun again. Two conjunctions complete one cycle, totaling 584 days and 1/11,398 of a circle, and the planet's movement is like this.
Regarding Mercury, when it aligns with the sun in the morning, it first "conjures," moving retrograde, receding by 7 degrees over 9 days, then it can be seen in the east in the morning, behind the sun. It continues moving in the opposite direction, speeding up, receding by 1 degree each day. It then "pauses" for 2 days, remaining stationary. Then it starts moving in direct motion, at a slower speed, traveling 9 degrees and 8/1 each day, completing 8 degrees in 9 days. Afterward, the speed increases, traveling 1 degree and 4/1 each day, completing 25 degrees in 20 days, appearing in the east in the morning, behind the sun. It then moves in direct motion, covering 1/641,967 of a circle over 16 days, and then meets the sun again. One conjunction cycle lasts 57 days and 1/641,967 of a circle, and the planet's movement is like this.
Speaking of Mercury, it sets with the sun and then disappears, moving in the forward direction. Sixteen days later, it will move to the position of 32 degrees, 641,967 arcminutes in the ecliptic. At this time, it can be seen in the western sky in the evening, positioned ahead of the sun. When moving forward, it travels quite quickly, covering a quarter of a degree each day, so in twenty days it covers twenty-five degrees. If it slows down, it only travels seven-eighths of a degree per day, taking nine days to cover eight degrees. If it stops moving, it will remain stationary for two days.
If it starts moving backward, it moves in reverse, retreating one degree each day, still positioned ahead of the sun, lurking in the evening western sky. Moving backward is slow as well, taking nine days to move back seven degrees, and finally, it meets with the sun again. From one conjunction to the next, it takes a total of 115 days and 6,125,505 minutes for Mercury to complete its orbit, and that's the journey of Mercury.
