Let's first talk about calculating the days. Add the number of days from the new moon plus the remaining days. If the total is exactly a full month (a month's worth of days), then it's correct. If when the new moon happens, the remaining days are not enough, then subtract one day; if the remaining days exceed 773, subtract 29 days; if it’s not enough, subtract 30 days, and the remaining days will indicate how many days until the next new moon.
Next, calculate the degrees. Add the degrees plus the remainder; if the total is exactly a full day's worth of degrees (a full cycle of degrees), then it's correct.
Jupiter: Is in conjunction for 32 days, 3484646 minutes; appears visible for 366 days; is in conjunction and travels 5 degrees, 2509956 minutes; appears and travels 40 degrees (retrograde 12 degrees, actual movement 28 degrees).
Mars: Is in conjunction for 143 days, 973113 minutes; appears visible for 636 days; is in conjunction and travels 110 degrees, 478998 minutes; appears and travels 320 degrees (retrograde 17 degrees, actual movement 303 degrees).
Saturn: Is in conjunction for 33 days, 166272 minutes; appears visible for 345 days; is in conjunction and travels 3 degrees, 1733148 minutes; appears and travels 15 degrees (retrograde 6 degrees, actual movement 9 degrees).
Venus: Is in conjunction in the east in the morning for 82 days, 113908 minutes; appears in the west for 246 days (retrograde 6 degrees, actual movement 240 degrees); is in conjunction and travels 100 degrees in the east in the morning, 113908 minutes; appears in the east (the daytime movement is similar to that in the west, is in conjunction for 10 days, retrograde 8 degrees).
Mercury: Is in conjunction in the west in the morning for 33 days, 612505 minutes; appears in the west for 32 days (retrograde 1 degree, actual movement 31 degrees); is in conjunction and travels 65 degrees, 612505 minutes; appears in the east (the daytime movement is similar to that in the west, is in conjunction for 18 days, retrograde 14 degrees).
Finally, we calculate the days and degrees of the appearance of the constellations. Add the days and remainder of the hidden constellations together. If the sum totals a whole number based on the method of days, that’s correct. If it doesn’t divide evenly but exceeds half, it’s also considered full. Then multiply the denominator of the constellation's movement by the number of days it appears, and the remaining remainder. If it is a full number according to the method of days, then it is correct. Then add the fraction of movement to the days, and if the fraction is full, add one degree to the denominator. The denominator is different for direct and retrograde movements, so multiply the actual movement denominator by the original fraction. If the result equals the original denominator, then it is the actual movement fraction. For retrograde movements, carry over the remainder from the previous step and subtract. If the hidden days are not enough for a full degree, divide using the standard of the Dipper's fraction, with fractions increasing or decreasing, and they should complement each other. Whenever it says "approximately full," it refers to real number division; "subtract and divide," refers to the Dipper division method.
In the morning, when the Sun and Jupiter appear simultaneously, Jupiter hides. The trajectory of Jupiter's movement is as follows: it moves forward for 16 days, traveling 1,742,323.2 minutes, then moves 2 degrees and 323,467 minutes, and then appears in the east in the morning behind the Sun. It then speeds up, moving at a rate of 58/11 minutes per day, covering 11 degrees in 58 days. Then the speed slows down, moving 9 minutes per day, covering 9 degrees in 58 days. It then stops for 25 days, and then begins to move backward at a rate of 1/7 minutes per day, covering 12 degrees in 84 days. It stops again for 25 days, then starts moving forward, covering 9 degrees in 58 days at a speed of 58/9 minutes per day. The speed increases again, moving 11 minutes per day, covering 11 degrees in 58 days, appearing in the west in the evening ahead of the Sun. After another 16 days, it travels 1,742,323.2 minutes again, moving 2 degrees and 323,467 minutes, and appears simultaneously with the Sun once more. This completes one cycle, which takes a total of 398 days, during which it travels 3,484,646 minutes and covers 43 degrees and 25,099.56 minutes.
In the morning, when the sun and Mars appear at the same time, Mars disappears. The trajectory of Mars is as follows: it first moves forward, covering an angular distance of 1,489,868 minutes over 71 days, then moves an additional 55 degrees, covering 1,242,860.5 minutes in angular measurement, appearing in the east in the morning behind the sun. Next, its speed increases, traversing 23 minutes and 14 seconds of angular distance each day, covering 112 degrees in 184 days. Then, its speed decreases, covering 23 minutes and 12 seconds of angular distance each day, 48 degrees in 92 days. It then comes to a standstill for 11 days, then starts moving backward, retreating by 62 minutes and 17 seconds of angular distance each day, moving back 17 degrees in 62 days. It stops again for 11 days, then resumes its forward motion, covering 12 minutes of angular distance each day, 48 degrees in 92 days. The speed increases again, covering 14 minutes of angular distance each day, 112 degrees in 184 days, appearing in the west in the evening, ahead of the sun. After 71 days, it covers an angular distance of 1,489,868 minutes, moves 55 degrees and 1,242,860.5 minutes in angular measurement, and appears simultaneously with the sun again. One cycle is completed, taking a total of 779 days, covering an angular distance of 973,130 minutes, and moving 414 degrees and 47,998 minutes.
In the morning, when Saturn and the sun appear at the same time, Saturn disappears and then begins its movement. According to its movement pattern, it covers an angular distance of 1,122,426.5 minutes in 16 days, moving 1 degree and 1,995,864.5 minutes in angular measurement, appearing in the east behind the sun. When moving forward, it covers 3 minutes and 35 seconds of angular distance each day, covering 7.5 degrees in 87.5 days. It remains stationary for 34 days. Then it moves backward, retreating by 1 minute and 17 seconds of angular distance each day, moving back 6 degrees in 102 days. After another 34 days, it resumes its forward motion, covering 1 minute of angular distance each day, 7.5 degrees in 87 days, appearing in the west in the evening, ahead of the sun. It covers an angular distance of 1,122,426.5 minutes in 16 days, moves 1 degree and 1,995,864.5 minutes in angular measurement, then appears simultaneously with the sun again. This marks the completion of one cycle, taking a total of 378 days, covering an angular distance of 166,272 minutes, and moving 12 degrees and 173,148 minutes in angular measurement.
In the morning, Venus and the sun appear at the same time; Venus then conceals itself, retrogrades, moves back three-fifths of a degree each day, and after ten days, it moves back six degrees. It stays still for eight days. Then it starts moving forward slowly, covering forty-six and one-third minutes each day for forty-six days, before moving forward. The speed increases, covering one degree and fifteen minutes each day, totaling one hundred and sixty degrees after ninety-one days. The forward speed further accelerates, with it covering one degree and twenty-two minutes each day, totaling one hundred and thirteen degrees after ninety-one days. At this point, it's positioned behind the sun and hides in the east in the morning. It moves forward for forty-one days, covering fifty degrees and fifty-six minutes, and then appears simultaneously with the sun again. Overall, the conjunction happens after a total of two hundred ninety-two days and fifty-six thousand nine hundred fifty-four minutes, with the planet covering the same angle during its orbit.
In the evening, Venus and the sun appear simultaneously; Venus then conceals itself and moves forward for forty-one days, covering fifty-six thousand nine hundred fifty-four minutes during its orbit, and then appears in the west before the sun. Moving forward at a fast pace, it covers one hundred and thirteen degrees after ninety-one days, then slows down to cover one degree and fifteen minutes each day, totaling one hundred and sixty degrees after ninety-one days. The speed decreases further, covering thirty-three and one-third minutes each day for forty-six days. It stays still for eight days, then it retrogrades, moving back three-fifths of a degree each day, and after ten days, it moves back six degrees. At this point, it is in front of the sun and hides in the west in the evening during retrograde, moving back four degrees after five days, and then appears simultaneously with the sun again. Two conjunctions make up one cycle, totaling five hundred eighty-four days and one hundred thirty-nine thousand eight minutes, with the planet covering the same angle during its orbit.
Mercury, when it meets the sun in the morning, first it hides, then it moves backward, retreating seven degrees over nine days, and can be seen in the east, positioned behind the sun in the morning. Then it moves backward again, quickly, retreating one degree in a day. After that, it stops and remains still for two days. Then it starts to move forward again, slowly, covering eight-ninths of the sun's apparent motion in a day, and eight degrees over nine days. Once it speeds up, it moves one degree and a quarter in a day, covering twenty-five degrees in twenty days, and runs behind the sun. In the morning, it appears in the east, then moves forward, covering thirty-two degrees and six hundred forty-one million nine thousand sixty-seven minutes in sixteen days and six hundred forty-one million nine thousand sixty-seven minutes, before meeting the sun again. The total duration of this conjunction is fifty-seven days and six hundred forty-one million nine thousand sixty-seven minutes, with Mercury covering the same degrees.
When Mercury meets the sun at night, it first it hides, then it moves forward, covering thirty-two degrees and six hundred forty-one million nine thousand sixty-seven minutes in sixteen days and six hundred forty-one million nine thousand sixty-seven minutes, and can be seen in the west, positioned in front of the sun at night. Then it moves forward quickly, covering one degree and a quarter in a day, and twenty-five degrees in twenty days. After slowing down, it covers eight-ninths of the sun's apparent motion in a day, retreating eight degrees over nine days. Then it stops and remains still for two days. After that, it starts to move backward, retreating one degree in a day, moving in front of the sun, and hiding in the west at night. Then it moves backward slowly, retreating seven degrees over nine days, and finally meets the sun again. The total duration of these two conjunctions is one hundred fifteen days and six million two thousand five hundred five minutes, with Mercury covering the same degrees.
