First, add the sizes of the new moon's remainders; if the total exceeds one month, then add either twenty-nine (for a large remainder (大余)) or seven hundred seventy-three (for a small remainder (小余)). When the small remainder is full, calculate it according to the large remainder method, using the same method as before.
Next, calculate the lunar date and solar remainder. Add the lunar date and the solar remainder together; if the remainder is full, count it as one day. If the small remainder perfectly fills the fractional part during the new moon calculation, then subtract one day; if the small remainder exceeds seven hundred seventy-three, subtract twenty-nine days; if it does not exceed that, subtract thirty days. The remaining value will then be incorporated into the lunar date calculation according to the subsequent method.
Finally, calculate the degrees. Add the degrees together, and also add the remainder of the degrees; if it is full, count it as one degree.
Here are the operational data for Jupiter, Mars, Saturn, Venus, and Mercury:
Jupiter: Lurking (伏) for 32 days, 3,484,646 minutes; Appeared for 366 days; Lurking for 5 degrees, 2,509,956 minutes; Appeared running 40 degrees. (After retrograding 12 degrees, the actual position is 28 degrees.)
Mars: Lurking for 143 days, 973,113 minutes; Appeared for 636 days; Lurking for 110 degrees, 478,998 minutes; Appeared running 320 degrees. (After retrograding 17 degrees, the actual position is 303 degrees.)
Saturn: Lurking for 33 days, 166,272 minutes; Appeared for 345 days; Lurking for 3 degrees, 1,733,148 minutes; Appeared running 15 degrees. (After retrograding 6 degrees, the actual position is 9 degrees.)
Venus: Lurking in the eastern sky in the morning for 82 days, 113,908 minutes; Appeared in the west for 246 days. (After retrograding 6 degrees, the actual position is 240 degrees.) Lurking in the morning for 100 degrees, 113,908 minutes; Appeared in the east. (The solar position matches that in the west, lurking for 10 days, retrograding 8 degrees.)
Mercury: Lurking in the morning for 33 days, 612,505 minutes; Appeared in the west for 32 days. (After retrograding 1 degree, the actual position is 31 degrees.) Lurking for 65 degrees, 612,505 minutes; Appeared in the east. (The solar position matches that in the west, lurking for 18 days, retrograding 14 degrees.)
First, let's calculate the position of this star. First, subtract the daily movement of the star from the daily movement of the sun. If the remaining number of degrees is enough for one solar day (the number of degrees the sun moves in one day), this indicates that the star has completed one solar day. Next, as before, determine when the star can be seen. Divide the number of degrees the star moves by the total degrees in a solar day. If it does not divide evenly, count it as one solar day if the remainder exceeds half of the solar day's degrees. Then add the star's daily movement to its total movement thus far. If the sum is enough for one solar day, count an additional solar day. The methods for calculating direct and retrograde motion differ based on the star's current degree position. If the star is in direct motion, use the previous result; if it is in retrograde motion, subtract from it. If the calculated number of degrees is not enough for one solar day, use the division method, treating the star's movement as the base value, and adjust for any increases or decreases during the calculation process. In summary, terms such as "full," "approximate," and "exact" pertain to the outcomes of precise division, while terms like "remainder," "and," and "divide" refer to the results of exhaustive division.
Next, let's take a look at the situation of the planet Jupiter. Jupiter appears in the morning with the sun, then disappears and continues its motion. It travels for sixteen days, covering a total of one million seven hundred forty-two thousand three hundred twenty-three minutes, and the planet moves two degrees and three hundred twenty-three thousand four hundred sixty-seven minutes. Then it appears in the east, lagging behind the sun. During its forward movement, it moves quickly, moving eleven-fifty-eighths of a degree per day, and covering eleven degrees in fifty-eight days; then, during its slower forward motion, it moves nine minutes in a day, covering nine degrees in fifty-eight days. It remains stationary for twenty-five days, then starts moving backward. During its retrograde phase, it moves one-seventh of a degree per day, retreating a total of twelve degrees over eighty-four days. Then it stops moving for another twenty-five days, after which it resumes its forward motion, moving nine-fifty-eighths of a degree per day, and covering nine degrees in fifty-eight days. During its forward movement, it moves quickly, moving eleven minutes in a day, and covering eleven degrees in fifty-eight days; at this point, it is in front of the sun, setting in the west in the evening. It travels for sixteen days, covering a total of one million seven hundred forty-two thousand three hundred twenty-three minutes, and the planet moves two degrees and three hundred twenty-three thousand four hundred sixty-seven minutes, then it appears again with the sun. One complete cycle lasts three hundred ninety-eight days, covering a total of three hundred forty-eight thousand four hundred sixty-six minutes, and the planet travels forty-three degrees and two hundred fifty-nine thousand nine hundred fifty-six minutes.
In the morning when the sun rises, Mars is together with the sun and then it starts to "station," beginning its prograde motion. It moves prograde for 71 days, traveling a total of 1,489,868 minutes, which is equivalent to 55 degrees and 242,860.5 arcminutes. After that, it becomes visible in the morning sky to the east, positioned behind the sun. During its prograde motion, it travels approximately 36 seconds each day, covering a total of 112 degrees over the course of 184 days. Then it slows down, moving approximately 31.3 seconds each day, covering a total of 48 degrees over 92 days. Next, it remains stationary for 11 days. Then it begins to retrograde, moving 17/62 of a minute each day, retreating 17 degrees in 62 days. It remains stationary again for 11 days, then resumes prograde motion, traveling 12 minutes each day and covering a total of 48 degrees over 92 days. After that, it speeds up, moving 14 minutes each day, covering a total of 112 degrees over 184 days, at which point it moves ahead of the sun, becoming visible as it sets in the west in the evening. After 71 days of motion, it has traveled a total of 1,489,868 minutes, which is equivalent to 55 degrees and 242,860.5 arcminutes, and is again together with the sun. Over the entire cycle, it totals 779 days and 973,113 minutes, covering 414 degrees and 478,998 arcminutes.
As for Saturn, it also starts "resting" in the morning when it is together with the sun, then moves prograde. It moves prograde for 16 days, traveling a total of 1,122,426.5 minutes, which is equivalent to 1 degree and 1,995,864.5 arcminutes. Then it becomes visible in the morning sky to the east, positioned behind the sun. During its prograde motion, it travels approximately 8.6 seconds each day, covering a total of 7.5 degrees over 87.5 days. Then it remains stationary for 34 days. Next, it retrogrades, moving 1/17 of a minute each day, retreating 6 degrees in 102 days. After another 34 days of being stationary, it resumes prograde motion, traveling 3 minutes each day and covering a total of 7.5 degrees over 87 days, at which point it moves ahead of the sun, becoming visible as it sets in the west in the evening. After 16 days of motion, it has traveled a total of 1,122,426.5 minutes, which is equivalent to 1 degree and 1,995,864.5 arcminutes, and is again together with the sun. Over the entire cycle, it totals 378 days and 166,272 minutes, covering a total of 12 degrees and 1,733,148 arcminutes.
Venus, when it aligns with the Sun in the morning, lurks, moving retrograde, retreating four degrees in five days, and then it appears in the east behind the Sun. During retrograde motion, it moves three-fifths of a degree each day, retreating six degrees over ten days. Then it remains stationary for eight days. After that, it turns and begins prograde motion, moving slowly, covering thirty-three degrees over forty-six days, and then it is in prograde motion. The speed increases, moving one degree and fifteen minutes daily, covering one hundred sixty degrees over ninety-one days. Then the prograde motion speed increases further, moving one degree and twenty-two minutes each day, covering one hundred thirteen degrees over ninety-one days, at which point it is positioned behind the Sun, appearing in the east in the morning. In prograde motion, it covers one five hundred sixty-four-thousand-nine-hundred-fifty-fourth of a full orbit over forty-one days, and the planet also covers fifty degrees of that fraction before aligning with the Sun again. One alignment totals two hundred ninety-two days and one five hundred sixty-four-thousand-nine-hundred-fifty-fourth of a full orbit, and the planet behaves similarly.
As for Venus in the evening, when it aligns with the Sun, it lurks, moving in prograde motion, covering one five hundred sixty-four-thousand-nine-hundred-fifty-fourth of a full orbit over forty-one days, and the planet also covers fifty degrees of that fraction, then it can be seen in the west in front of the Sun in the evening. In prograde motion, the speed increases, moving one degree and twenty-two minutes each day, covering one hundred thirteen degrees over ninety-one days. Then its prograde motion slows, moving one degree and fifteen minutes per day, covering one hundred sixty degrees over ninety-one days, thus remaining in prograde motion. The speed slows down, moving thirty-three degrees over forty-six days. Then it remains stationary for eight days. After that, it turns and goes into retrograde motion, moving three-fifths of a degree each day, retreating six degrees over ten days, at which point it is in front of the Sun, appearing in the west in the evening, moving in retrograde, speeding up, retreating four degrees over five days, and then aligning with the Sun again. Two alignments together count as one complete cycle, totaling five hundred eighty-four days and one hundred thirteen thousand nine hundred eighty-eighth of a full orbit, and the planet behaves similarly.
Mercury, when it aligns with the sun in the morning, remains hidden, retrograding seven degrees over nine days, and then can be seen in the eastern sky in the morning, behind the sun. Then its retrograde speed increases, retreating one degree per day. After two days of coming to a standstill, it then turns and starts moving forward, slowly, moving eight-ninths of a degree per day. After retreating eight degrees over nine days, it then moves forward. As it speeds up, it moves one and a quarter degrees per day, traversing twenty-five degrees over twenty days. At this point, it is behind the sun, appearing in the eastern sky in the morning. In direct motion, it traverses six hundred forty-one million nine thousand sixty-seven parts of a circle over sixteen days; the planet also traverses thirty-two degrees and six hundred forty-one million nine thousand sixty-seven parts of a circle, before aligning with the sun once more. In total, it takes fifty-seven days to traverse six hundred forty-one million nine thousand sixty-seven parts of a circle; the planet follows this pattern as well.
Speaking of Mercury, it sets with the sun, and then it appears to hide; sometimes its orbit is direct, sometimes it is retrograde. Specifically, every sixteen days, Mercury will reach the position of thirty-two degrees six hundred forty-one million nine thousand six hundred sixty-seven minutes in longitude. At this point, it can be seen in the western sky in the evening, and it is always in front of the sun.
If Mercury is in direct motion, it moves quickly, moving one and a quarter degrees per day, allowing it to traverse twenty-five degrees in twenty days. But if it moves slowly, it only traverses eight-sevenths of a degree per day, taking nine days to traverse eight degrees. If it comes to a standstill, you must wait two days for it to move again. If Mercury is retrograde, it moves in reverse, retreating one degree each day; at this point, it remains in front of the sun, appearing in the western sky in the evening. During retrograde motion, it also moves slowly, retreating seven degrees over nine days, before finally aligning with the sun once more.
In general, it takes Mercury one hundred fifteen days and six hundred twelve thousand five hundred five minutes from one conjunction with the Sun to the next, and other planets have similar orbital patterns. "At sunset, during the conjunction with the Sun, Mercury is stationary and direct for sixteen days, six hundred forty-one thousand nine hundred sixty-seven minutes, at thirty-two degrees, as the sunset occurs in the west, just ahead of the Sun." This describes the specific details of Mercury's orbit.
