Let’s start by discussing how to calculate the days. To find out which day of the lunar calendar corresponds to a given day, first divide the added days by 12 and see what the remainder is. If the remainder is 3, it indicates that this day falls slightly before a certain lunar calendar day; if the remainder is 0, it means this day is a little more than a certain day of the lunar calendar; if the remainder is 2, it means this day is much less than a certain day of the lunar calendar. The result obtained in this way is the angle of the moon's position relative to the ecliptic. To calculate the degree of the moon leaving the North Pole, for the solar calendar, add the number of days to the ecliptic angle, and for the lunar calendar, subtract the ecliptic angle. The excess is referred to as "strong," while the deficiency is termed "weak." When calculating, add when the signs match, and subtract when they differ. The same applies to subtraction; subtract if the signs are the same, add if the signs are different, and if there is no corresponding sign, ignore it, adding up two "strong" and then subtracting "weak."
From the Ji Chou year in the Shang Yuan era to the Bing Xu year in the Jian'an period, 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, and Bing Yin.
The five elements are: Wood (Jupiter), Fire (Mars), Earth (Saturn), Metal (Venus), and Water (Mercury). Each star has a cycle of operation and a degree for each day, which we respectively call the weekly rate and daily rate. Multiply the years by the weekly rate to obtain the monthly method; then multiply this by the number of days to find the monthly component; divide the monthly part by the monthly method to get the month number; finally, multiply the month number by the monthly method to obtain the daily method. The Dipper calculation is done by multiplying the Dipper by the weekly rate. (Because the daily method is calculated by multiplying the record method by the weekly rate, we also use the Dipper here to multiply.)
Next, let's calculate the major and minor residues for the five stars. Multiply the common method by the month number, then divide by the daily method to get the major residue, and subtract the major residue from 60 to find the corresponding value for the minor residue.
Then calculate the entry and residual values for the five stars within the month. Multiply the common method by the month residue, then multiply the combined month method by the minor residue of the month, add these two results, simplify the results, and finally divide by the daily method to obtain the final outcome.
Next, calculate the degrees and degree residues of the five stars. Subtract the excess, and the remaining value is the degree residue; then multiply by the weekly cycle to get the degree, divide by the daily method, and the remainder is the degree residue. If it exceeds the weekly cycle, subtract the weekly cycle, and then subtract the Dipper value.
Ji Yue is 7,285, Zhang Run is 7, Zhang Yue is 235, Sui Zhong is 12, Tong Fa is 43,226, Ri Fa is 1,457, Hui Shu is 47, Zhou Tian is 215,130, Dipper fraction is 145.
Jupiter: Orbital period is 6,722, Solar day method is 7,341, Total lunar months is 13, The lunar surplus is 64,810, Combined lunar method is 127,718, Solar day method is 3,959,258, Start of the month day is 15, Day surplus is 3,484,646, Shuo Xu Fen is 150, Dipper fraction is 974,690.
The total degree is 33, with a remainder of 2,956,000. Fire: Orbital period is 3,407, Solar day method is 7,271, Total lunar months is 26, The lunar surplus is 25,627, Combined lunar method is 64,733, Solar day method is 2,006,723, Shuo Da Yu is 47, Shuo Xiao Yu is 1,157, Start of the month day is 12, Day surplus is 973,013, Shuo Xu Fen is 300, Dipper fraction is 494,015.
Next is the data for another section. The total degree is 48, with a remainder of 19,001,706. Earth: Orbital period is 3,529, Solar day method is 3,653, Total lunar months is 12, The lunar surplus is 53,843, Combined lunar method is 67,051, Solar day method is 2,078,581, Shuo Da Yu is 54, Shuo Xiao Yu is 534, Start of the month day is 24, Day surplus is 166,272, Shuo Xu Fen is 923, Dipper fraction is 51,175.
Next, the degree is 12, with a remainder of 1,731,408. Metal: Orbital period is 9,022, Solar day method is 7,213, Total lunar months is 9, The lunar surplus is 152,293, Combined lunar method is 171,418, Solar day method is 5,313,958, Shuo Da Yu is 25, Shuo Xiao Yu is 1,129, Start of the month day is 27, Day surplus is 56,954, Shuo Xu Fen is 328, Dipper fraction is 1,308,190.
Finally, it is the water data. The degree is 292, with a remainder of degrees 56,954. Water: Zhou value is 11,561, daily calculation is 1,834, the total number of months is 1, with a month remainder of 211,331, total month calculation is 219,659, daily calculation is 6,809,429, major lunar remainder is 29, minor lunar remainder is 773, day of entering the month is 28, daily remainder is 641,967, and the fraction of the new moon is 684. These numbers look like some kind of astronomical calculation, right? Really complicated! First, we need to calculate some astronomical data, specifically, "Doufen" is 1,676,345, the degree is 57 with a remainder of 641,967. Then, we need to use the data from the first year for calculations. Multiply the Zhou value by this year, get an integer part, which we refer to as "Jihe," and the remaining value is "He Yu." Divide the Zhou value by He Yu, and you can get an integer, indicating how many years ago the stars aligned. If it does not divide evenly, it means the stars aligned in this year. Finally, subtract the Zhou value from He Yu to get the degree. The Jihe values for Venus and Mercury, odd numbers represent morning appearances, even numbers represent evening appearances. Next, we need to calculate the month. Multiply both the month number and the month remainder by Jihe. If the result is a multiple of the total month calculation, then it represents this month; the remaining remainder is the month remainder. Then subtract the Ji month from the Ji month, the remainder is the month of entering the Ji. Then use the leap month to multiply the month of entering the Ji; if the result is a multiple of the month calculation, it means there is a leap month. Then subtract this leap month from the month of entering the Ji, and then subtract the remaining from the years and months; this represents the total month outside of the Tianzheng calculations. If it falls during a leap month, adjust using the new moon. Then, multiply the common calculation by the month remainder, then multiply the total month calculation by the minor lunar remainder, add these two results together, then simplify using the total count. If the result is a multiple of the daily calculation, it means the stars were aligned on that day. If it is not a multiple, the remaining value is the daily remainder, which should be noted outside the new moon calculation. Next, multiply Zhou Tian by the degree; if the result is a multiple of the daily calculation, you get one degree, and the remaining value is represented using the first five of Niu Qian method. These are the methods for calculating star alignments.
Then, we add up the months and the remainders of the lunar months; if the result is a multiple of the lunar month, it indicates one month; if not, it indicates a period within the year. If it exceeds, subtract that amount, while considering leap months. The remainder indicates the next year, and if it exceeds further, it indicates the year after that. For Venus and Mercury, their appearances in the morning and evening balance each other out. When adding the new moon remainder to the lunar month remainder, if the result exceeds a month, add the remainder of 29 or 773, and subtract from the remainder of the lunar month if it exceeds the day method. Add the lunar day and the day remainder; if the result is a multiple of the day method, it represents one day. Subtract a day if the previous lunar month remainder exceeds the virtual minute, subtract 29 days if the remainder exceeds 773, otherwise subtract 30 days; the result is the subsequent lunar day. Finally, adding up the degrees and the degree remainder, if the result is a multiple of the day method, it represents one degree. Next are the specific data for Jupiter: Jupiter is invisible in the east for 32 days, 3484646 minutes; appears for 366 days; hides for 5 degrees, 2509956 minutes; appears for 40 degrees (retrograde 12 degrees, actual progress 28 degrees). Mars: hides for 143 days, 973113 minutes; appears for 636 days; hides for 110 degrees, 478998 minutes; appears for 320 degrees (retrograde 17 degrees, actual progress 303 degrees). Saturn: hides for 33 days, 166272 minutes; appears for 345 days. This passage outlines the methods used to calculate planetary motion in ancient astronomical calendars.
Next, this text starts explaining the calculation methods. It mentions that the specific position of a planet is calculated based on its orbital speed and time. The calculation process involves many complex steps, such as "adding the sun's position degrees and the remainder, along with the degrees when the planet is in conjunction with the sun, and applying a specific method to arrive at a result; from the total command as previously, obtain the planet's appearance time and position." This means that one must first calculate the days and degrees of the planet's retrograde motion, then add the degrees at which the planet is in conjunction with the sun, and finally use a specific algorithm to perform calculations to ultimately determine the time and position of the planet's appearance. This involves many technical terms, such as "solar degree method," "planetary motion denominator," and "division of factors," which require a certain level of expertise to understand.
The text also mentions the variations in planetary orbital speeds, such as sometimes being fast or slow, and even the phenomena of "stationary" and "retrograde." "Stationary" refers to the planet temporarily stopping its motion, while "retrograde" refers to the planet moving in reverse. These phenomena are caused by the relative motion between the planet and Earth. Calculating these variations requires taking into account several factors, such as the speed of the planet's motion, the speed of Earth's motion, and so on. The calculation methods are also quite complex, requiring multiple computations to ultimately arrive at accurate results.
Finally, it takes Jupiter as an example to detail its orbital situation over one year (398 days), including its retrograde time, position, orbital speed, and more, as well as the number and position of its conjunctions with the sun throughout the year. These data have undergone complex calculations, showcasing the sophisticated development of ancient astronomical calendars. In summary, this text describes an ancient and very complex method for calculating planetary motion, which is remarkable for its ability to accurately predict the trajectory and position of planets through a series of intricate calculations. Grasping and mastering this requires deep knowledge of mathematics and astronomy.
In the morning, the sun and Mars met, and Mars went into hiding. Then it started to move forward, moving for 71 days, a total of 1,489,868 minutes, which corresponds to 55 degrees and 242,860.5 arc minutes in its orbit. After that, people could see it rising in the east, positioned behind the sun. While moving forward, Mars traversed 14/23 of a minute each day, covering 112 degrees over 184 days. Then the speed of movement slowed down, traversing 12/23 of a minute each day, covering 48 degrees over 92 days. Then it went stationary for 11 days. Then it started moving backward, moving backward at a rate of 17/62 of a minute each day, moving back 6 degrees in 102 days. It stopped again for 11 days, then started moving forward again, traversing 12 minutes each day, covering 48 degrees over 92 days. Moving forward again, the speed increased, traversing 14 minutes each day, covering 112 degrees over 184 days. At this point, it moved ahead of the sun, and in the evening, it could be seen lurking in the west. After 71 days, it had traversed a total of 1,489,868 minutes, which corresponds to 55 degrees and 242,860.5 arc minutes in its orbit, and then it met the sun again. By this calculation, one complete cycle lasts 779 days and 97,313 minutes, traversing 414 degrees and 478,998 arc minutes in its orbit.
Next is Saturn. In the morning, the sun and Saturn met, and Saturn went into hiding. Then it started to move forward, moving for 16 days, a total of 1,122,426.5 minutes, which corresponds to 1 degree and 199,864.5 arc minutes in its orbit. After that, people could see it rising in the east, positioned behind the sun. While moving forward, Saturn traversed 3/35 of a minute each day, covering 7.5 degrees over 87.5 days. Then it stopped moving for 34 days. Then it started moving backward, moving backward at a rate of 1/17 of a minute each day, moving back 6 degrees in 102 days. After another 34 days, it started moving forward again, traversing 3 minutes each day, covering 7.5 degrees over 87 days. At this point, it moved ahead of the sun, and in the evening, it could be seen lurking in the west. After 16 days, it had traversed a total of 1,122,426.5 minutes, which corresponds to 1 degree and 199,864.5 arc minutes in its orbit, and then it met the sun again. By this calculation, one complete cycle lasts 378 days and 166,272 minutes, traversing 12 degrees and 173,148 arc minutes in its orbit.
Venus, when it meets the sun in the morning, first hides, then retrogrades, retreating four degrees over five days, and then you can see it in the east in the morning, behind the sun. During its retrograde phase, it moves three-fifths of a degree each day, retreating six degrees in ten days. Then it stops, motionless for eight days. It then begins to move forward, slowly, moving forty-six and a third degrees in a day, for forty-six days, then continues moving forward. Its speed increases, moving one degree and ninety-one-fifths in a day, and moving one hundred and sixty degrees in ninety-one days. Then the forward speed increases further, moving one degree and ninety-one twenty-seconds in a day, and moving one hundred and thirteen degrees in ninety-one days; at this point, it can be seen in the east, behind the sun in the morning. Continuing forward, it completes fifty-six thousand nine hundred and fifty-fourths of a circle in forty-one days, and the planet also completes fifty degrees fifty-six thousand nine hundred and fifty-fourths of a circle, and then it meets the sun again. One meeting totals two hundred and ninety-two days and fifty-six thousand nine hundred and fifty-fourths of a circle; the planet is the same.
When Venus meets the sun in the evening, it first hides, then moves forward, completing fifty-six thousand nine hundred and fifty-fourths of a circle in forty-one days, while the planet moves fifty degrees fifty-six thousand nine hundred and fifty-fourths of a circle, and then you can see it in the west in the evening, in front of the sun. The forward speed increases, moving one degree and ninety-one twenty-seconds in a day, and moving one hundred and thirteen degrees in ninety-one days. Then the forward speed slows down, moving one degree and fifteen minutes in a day, moving one hundred and sixty degrees in ninety-one days, and then continues moving forward. The speed slows down, moving forty-six and a third degrees in a day, moving thirty-three degrees in forty-six days. Then it stops, motionless for eight days. It then begins to move backward, moving three-fifths of a degree in a day, retreating six degrees in ten days; at this point, it appears in the west in the evening, in front of the sun. The retrograde motion speeds up, retreating four degrees in five days, and then it meets the sun again. Two meetings count as one cycle, totaling five hundred and eighty-four days and eleven thousand three hundred and ninety-eighths of a circle; the planet is the same.
Mercury, when it meets the sun in the morning, first hides, then it retrogrades, moving back seven degrees over nine days. Then it can be seen in the east in the morning, behind the sun. The retrograde speed increases, retreating one degree each day. Then it stops, not moving for two days. Then it begins to move forward slowly, moving eight-ninths of a degree each day, and eight degrees over nine days, and then it continues to move forward. The speed increases, moving one and a quarter degrees a day, and twenty-five degrees over twenty days. By this time, it is still behind the sun, appearing in the east each morning. It continues to move forward, traversing six hundred forty-one million nine thousand sixty-sevenths of a circle in sixteen days; the planet also traverses thirty-two degrees six hundred forty-one million nine thousand sixty-sevenths of a circle, and then meets the sun again. In one complete cycle, it takes a total of fifty-seven days and six hundred forty-one million nine thousand sixty-sevenths of a circle, and the planet is also like this.
The sun sets, meeting with Mercury. The pattern of Mercury's movement is as follows: when moving forward, it can cover thirty-two degrees in sixteen days plus six hundred forty-one million nine thousand six hundred sixty-sevenths of a degree. At this time, Mercury can be seen in the west in the evening, in front of the sun. When moving forward, the speed is fast, moving one and a quarter degrees a day, and twenty-five degrees over twenty days. When moving slowly, it only covers eight-sevenths of a degree each day, walking eight degrees in nine days. Sometimes Mercury stops, not moving for two days. Sometimes Mercury retrogrades, retreating one degree in a day; at this time, it can also be seen in the west in the evening, in front of the sun. When retrograding, the speed is slow, retreating seven degrees in nine days, and then meets the sun again.
From one meeting to the next, it takes a total of one hundred fifteen days plus six hundred one million two thousand five hundred fiveths of a day, and this is the pattern of Mercury's movement.
