Let's first talk about how to calculate the months. Add up the days of each month, add up the extra days, and complete a month. If it's not a full year, count it towards that year, and subtract when it's full. This is known as the leap month calculation; the remaining days are carried over to the following two years to complete a month again. For Venus and Mercury, add when they appear in the morning, and continue adding until evening; add when they appear at night, and continue adding until morning.
Next is to calculate the lunar cycle and the remaining lunar cycle. Add the lunar cycle and the remaining lunar cycle; if it exceeds a month, add an additional twenty-nine days (for a small remainder) or seven hundred seventy-three days (for a large remainder). When the small remainder is full, calculate according to the algorithm of the large remainder; the calculation method remains the same as previously described.
Then calculate the lunar day and the day remainder. Add the lunar day and the day remainder to complete a full day. If the previous lunar cycle is complete but there is still a remainder, subtract one day; if the small remainder exceeds seven hundred seventy-three, subtract twenty-nine days; if it's not enough, subtract thirty days. The remaining days are carried over to the next month as that month's lunar day.
Finally, calculate the degrees. Add up the degrees, and add up the extra degrees to complete a full day's worth of degrees, and you get a degree.
Next are the operational data for Jupiter, Mars, Saturn, Venus, and Mercury:
Jupiter: Hidden for 32 days, 3,484,646 minutes; visible for 366 days; hidden phase covering 5 degrees, 2,509,956 minutes; observed to operate for 40 degrees. (Retrograde for 12 degrees, actual operation for 28 degrees.)
Mars: Hidden for 143 days, 973,113 minutes; visible for 636 days; hidden phase covering 110 degrees, 478,998 minutes; observed to operate for 320 degrees. (Retrograde for 17 degrees, actual operation for 303 degrees.)
Saturn: Hidden for 33 days, 166,272 minutes; visible for 345 days; hidden phase covering 3 degrees, 1,733,148 minutes; observed to operate for 15 degrees. (Retrograde for 6 degrees, actual operation for 9 degrees.)
Venus: Hidden in the east in the morning for 82 days, 113,908 minutes; visible in the west for 246 days. (Retrograde for 6 degrees, actual operation for 240 degrees.) Hidden phase covering 100 degrees, 113,908 minutes; visible in the east. (The daily motion is similar to that of the west; hidden for 10 days, retrograde for 8 degrees.)
Mercury: Hidden in the east in the morning for 33 days, 612,505 minutes; visible in the west for 32 days. (Retrograde for 1 degree, actual operation for 31 degrees.) Hidden phase covering 65 degrees, 612,505 minutes; visible in the east. (The daily motion is similar to that of the west; hidden for 18 days, retrograde for 14 degrees.)
Goodness, this text looks overwhelming. Let's break it down sentence by sentence. First, we use a standard value (法) to subtract the degrees of the sun's daily movement (日度), resulting in a remainder. Then we add the degrees of the stars' daily movement (星合日度) to this remainder. If the remainder can be evenly divided by the standard value (余满日度法得一), then according to the previous calculation method, we can determine the angular difference between the positions of the stars and the sun.
Next, we multiply the denominator of the star's movement by the previously calculated angular difference (见度), and then divide the remainder by the standard value for the sun's degrees (日度). If it cannot be evenly divided, and the remainder exceeds half of the standard value, it is also considered divisible; then we add the value of the sun's daily movement to this result. If the total value equals or exceeds the denominator, we add one degree; note that the denominators used for direct and retrograde calculations differ, so we must multiply the current running denominator by the previous value and then divide by the original denominator to get the current running value.
The phrase 'if the planet is at rest (留)', we use the previous calculation result; if it is in retrograde (逆行), we subtract; if it cannot be evenly divided, we apply a specific method (经斗除分) to manage the fractions, using the running denominator as a standard. The values will have increases or decreases, and the results before and after will mutually influence each other.
Generally speaking, any terms that refer to 'full' or 'exact' require precise division; whereas 'to remove' and 'to divide' pertain to obtaining integer results from division.
Okay, the first paragraph is explained, let's look at the second paragraph. "Jupiter: In the morning, it conjuncts with the sun, moving forward. After sixteen days, Jupiter has traveled 1,742,323 minutes, and Jupiter has moved 2,323,467 minutes. At this time, Jupiter can be seen in the east in the morning, behind the sun." This paragraph talks about Jupiter; in the morning, Jupiter conjuncts with the sun, and it is moving forward. After 16 days, Jupiter has traveled 1,742,323 minutes, and Jupiter has moved 2,323,467 minutes, at which point Jupiter can be seen in the east, behind the sun.
"Moving forward quickly, Jupiter moves 11 minutes a day; in 58 days, it moves 11 degrees. Then moving forward slowly, Jupiter moves 9 minutes a day; in 58 days, it moves 9 degrees. It stops for 25 days and then moves backward, Jupiter moves 1/7 of a degree a day, retreating 12 degrees after 84 days. Then it stops for 25 days and then Jupiter resumes moving forward, traveling 9 minutes a day; in 58 days, it moves 9 degrees." This section describes the changes in Jupiter's speed: moving forward quickly, it travels 11 minutes a day, 11 degrees in 58 days; then moving forward slowly, it travels 9 minutes a day, 9 degrees in 58 days; then it stops for 25 days; then it moves backward, traveling 1/7 of a degree a day, retreating 12 degrees after 84 days; then it stops for another 25 days, and then Jupiter resumes moving forward, traveling 9 minutes a day, 9 degrees in 58 days.
"Moving forward quickly, Jupiter moves 11 minutes a day; in 58 days, it moves 11 degrees, in front of the sun, and it sets in the west in the evening. After sixteen days, Jupiter has traveled 1,742,323 minutes, and Jupiter has moved 2,323,467 minutes, aligning with the sun again. A complete cycle, lasting 398 days, during which Jupiter has traveled 3,484,646 minutes, and Jupiter has moved 43 degrees and 2,509,956 minutes." Finally, Jupiter moves forward quickly, traveling 11 minutes a day, 11 degrees in 58 days, at which point Jupiter is in front of the sun and sets in the west in the evening. After 16 days, Jupiter aligns with the sun again. One cycle ends, lasting a total of 398 days, during which Jupiter has traveled 3,484,646 minutes, and Jupiter has moved 43 degrees and 2,509,956 minutes. This entire second paragraph outlines the patterns and cycles of Jupiter's motion.
Sun: It appears in the morning alongside the sun, then it disappears. After that, it moves forward for a total of 71 days, traveling 1,489,868 minutes, which means the planet has moved 55 degrees 242,860.5 minutes. Then, it can be seen in the east in the morning, behind the sun. During its forward motion, it moves 14 minutes for every 23 minutes each day, covering 112 degrees in 184 days. Then it moves forward again and slows down, moving 12 minutes for every 23 minutes each day, covering 48 degrees in 92 days. After that, it stops for 11 days. Then it moves retrograde, covering 17 minutes for every 62 minutes each day, retreating 17 degrees in 62 days. It stops again for 11 days, then resumes its forward motion, moving 12 minutes each day, covering 48 degrees in 92 days. It moves forward again, speeding up, moving 14 minutes each day, covering 112 degrees in 184 days; at this time, it is in front of the sun and goes into hiding in the west at night. In 71 days, it traveled 1,489,868 minutes, and the planet will have moved 55 degrees 242,860.5 minutes, then it appears again with the sun. This entire cycle lasts a total of 779 days 973,113 minutes, and the planet will have moved 414 degrees 478,998 minutes.
Mars: It appears in the morning alongside the sun, then it disappears. After that, it similarly moves forward for a total of 16 days, traveling 1,122,426.5 minutes, which means the planet has moved 1 degree 1,995,864.5 minutes. Then, it can be seen in the east in the morning, behind the sun. During its forward motion, it moves 3 minutes for every 35 minutes each day, covering 7.5 degrees total in 87.5 days. Then it stops for 34 days. Then it moves retrograde, covering 1 minute for every 17 minutes each day, retreating 6 degrees in 102 days. After another 34 days, it resumes its forward motion, moving 3 minutes each day, covering 7.5 degrees total in 87 days; at this time, it is in front of the sun and goes into hiding in the west at night. In 16 days, it traveled 1,122,426.5 minutes, and the planet will have moved 1 degree 1,995,864.5 minutes, then it appears again with the sun. This entire cycle lasts a total of 378 days 166,272 minutes, and the planet will have moved 12 degrees 1,733,148 minutes.
Venus, when it appears in the morning with the sun, first moves in retrograde. Within five days, it will retrograde four degrees, then it will become visible in the eastern sky behind the sun. Continuing its retrograde motion, it will move three-fifths of a degree each day relative to the sun, and in ten days it will retrograde six degrees. Then it will stop moving for eight days. After that, it will begin to move forward, at a slower speed, completing one forty-six thousand nine hundred and fifty-fourth of a circle in forty-one days, with the planet also moving fifty-six thousand nine hundred and fifty-fourths of a circle. It will then meet the sun. A conjunction is five hundred eighty-four days and one hundred thirty-nine thousand eight hundred one-thousandths of a circle, with the planet moving the same distance.
When Venus appears in the evening with the sun, it will first move forward. It will complete one fifty-six thousand nine hundred and fifty-fourth of a circle in forty-one days, with the planet also moving fifty-six thousand nine hundred and fifty-fourths of a circle, and then it will be visible in the western sky in the evening in front of the sun. It will then continue to move forward, picking up speed, moving one twenty-second of a degree each day, and in ninety-one days it will move one hundred and thirteen degrees. The speed will then start to slow down, moving fifteen minutes of a degree each day, and in ninety-one days it will move one hundred and six degrees, at which point it will move forward. The speed will slow down, moving forty-six and one-third degrees each day, and in forty-six days it will move thirty-three degrees. Then it will stop for eight days. After that, it will start to retrograde, moving three-fifths of a degree each day, and in ten days it will retrograde six degrees, at which point it will appear in front of the sun, visible in the western sky in the evening. Continuing its retrograde motion, it will speed up, retrograding four degrees over five days, and then it will meet the sun. There will be two conjunctions, totaling five hundred eighty-four days and one hundred thirty-nine thousand eight hundred one-thousandths of a circle, with the planet moving the same distance.
When Mercury appears in the morning with the sun, it first moves in retrograde. Nine days later, it moves back seven degrees, then it moves behind the sun, and it can then be seen in the east in the morning. It continues to move in retrograde, speeds up, and retreats one degree per day. Then it stops for two days. After that, it begins to move forward at a slower speed, covering eight-ninths of a degree per day, which totals eight degrees in nine days. Then the speed increases, covering one and a quarter degrees per day, twenty-five degrees in twenty days, at which point it is behind the sun and appears in the east in the morning. Then it moves forward, covering one six-hundred forty-one million nine thousand sixty-seventh of a circle in sixteen days, and the planet also covers thirty-two degrees and one six-hundred forty-one million nine thousand sixty-seventh of a circle, and then it meets with the sun. One conjunction is fifty-seven days and one six-hundred forty-one million nine thousand sixty-seventh of a circle, with the planet moving the same distance.
Speaking of Mercury, it sets alongside the sun and then goes into hiding, following a direct trajectory. In sixteen days, it covers thirty-two degrees and one six-hundred forty-one million nine thousand sixty-sixth of a degree. At this point, it can be seen in the evening sky in the west, positioned ahead of the sun. When moving forward, it moves quite fast, covering one and a quarter degrees per day, twenty-five degrees in twenty days. If it moves slowly, covering only seven-eighths of a degree per day, it takes nine days to cover eight degrees. If it comes to a standstill, then it takes two days. If it retrogrades, moving backward and retreating one degree per day, it is in front of the sun, hiding in the west in the evening. When retrograding, it moves slowly, taking nine days to retreat seven degrees, and then eventually meets with the sun again.
From one conjunction to the next, it takes a total of one hundred fifteen days and five hundred fifty-fifths of a day, and this is how Mercury's movement repeats itself.
