First, let's calculate the moon. Multiply the number of months by the month remainder, and add the results together. If the result meets the standard for a month, calculate by month; otherwise, record the remainder as the month remainder. Then subtract the accumulated number of months from the recorded month; the remainder is the number entered in the recorded month. Next, multiply it by the chapter leap; if it meets the standard for a chapter month, subtract a leap month. The remaining number is subtracted from the year, and the remainder is noted outside the calculations of the astronomical calendar. This is the method of conjunction. If at the juncture of the leap month, use the new moon to adjust.
Next, multiply the common method by the month remainder and the conjunction method by the remainder of the new moon calculation, then add them together. Divide by the total; if the result meets the standard for a day, then the day when the stars align with the moon has arrived. If not enough, the remainder is the day remainder, recorded outside the new moon calculation.
Then, multiply the week by the degrees; if it meets the standard for a day, record one degree; if not enough, record it as a remainder. Use the previous method to record the degrees. The above is the method of seeking the conjunction.
Next, calculate the year. Add the number of months and the month remainder; if the total meets the standard for a month, record it as such. If not enough, calculate within the year, subtract if sufficient, and consider the leap month. The remainder is carried over to the next year; if sufficient for a month, record it for the next two years. Venus and Mercury, add morning to evening, add evening to morning. (This sentence is the original text, no need to translate).
Then, add the remainder from the new moon calculation and the remainder from the conjunction method; if it exceeds a month, add a large remainder of twenty-nine and a small remainder of seven hundred and seventy-three. If the small remainder is enough for a standard day, subtract from the large remainder; the method is the same as before.
Add the days of the month and the day remainder, then add to the conjunction days and remainder. If the remainder meets the standard for a day, count it as a day; if the small remainder from the previous conjunction is sufficient for division, subtract a day. If the small remainder exceeds seven hundred and seventy-three, subtract twenty-nine days; if not enough, subtract thirty days. The remainder is noted for the next calculation; this concludes the calculation for the day of the month.
Finally, add up the degrees and the degree remainder; if it meets the standard for a day, record one degree.
The following is the movement of Jupiter, Mars, Saturn, and Venus:
Jupiter: Hidden for thirty-two days, three million four hundred eighty-four thousand six hundred forty-six minutes; appears for three hundred sixty-six days; hidden travel of five degrees, totaling two million five hundred ninety-nine thousand five hundred sixty minutes; apparent motion of forty degrees. (Except for twelve degrees retrograde, fixed travel for twenty-eight degrees.)
Mars: Hidden for one hundred forty-three days, nine hundred seventy-three thousand thirteen minutes; appears for six hundred thirty-six days; hidden travel of one hundred ten degrees, forty-seven thousand eight hundred ninety-eight minutes; apparent motion of three hundred twenty degrees. (Except for seventeen degrees retrograde, fixed travel for three hundred three degrees.)
Saturn: Hidden for thirty-three days, one hundred sixty-six thousand two hundred seventy-two minutes; appears for three hundred forty-five days; hidden travel of three degrees, one hundred seventy-three thousand one hundred forty-eight minutes; apparent motion of fifteen degrees. (Except for six degrees retrograde, fixed travel for nine degrees.)
Venus: In the morning, it was obscured in the east for eighty-two days, eleven thousand three hundred ninety-eight minutes; visible in the west for two hundred forty-six days. (Except for six degrees retrograde, fixed travel for two hundred forty-six degrees.) Morning hidden travel for one hundred degrees, eleven thousand three hundred ninety-eight minutes; visible in the east. (Daily degrees as in the west. Hidden for ten days, retrograde for eight degrees.)
Mercury, it appears in the morning for a total of thirty-three days, with a total distance of six million two hundred fifty-five thousand minutes. It then becomes visible in the western sky for thirty-two days. (Subtracting one degree retrograde, the total travel is thirty-two degrees.) Then it moves forward by sixty-five degrees, with a total distance of six million two hundred fifty-five thousand minutes. Then it appears in the east. Its speed in the east matches that in the west, lasting for eighteen days, with retrograde motion of fourteen degrees.
Calculate the degrees and remaining degrees that Mercury runs each day, plus the remaining degrees at the time of its conjunction with the sun. If the remaining degrees reach a full day's degrees, continue the calculations using the same method to determine the time and degrees of Mercury's appearance. Multiply the denominator of Mercury's movement by its degrees of appearance. If the remaining degrees are evenly divisible by the daily degree, you get a whole number; if not, round up if it exceeds half. Add the obtained whole number to its movement fraction. If the fraction reaches a full degree, add one degree, using different methods for retrograde and direct motion. Multiply its current movement denominator by the previous fraction. If the result equals the previous denominator, you have found the current movement fraction. Retain means to carry over the previous calculation results, while retrograde subtracts. If the movement degrees are not enough, divide by the fraction, treating the movement denominator as a ratio. The fraction will increase or decrease and should be coordinated before and after. Any mention of "如盈约满" refers to precise division; "去及除之，取尽之除也" means complete division.
As for Jupiter, it appears with the sun in the morning, then disappears and begins its direct motion for 16 days, covering a total distance of 1,742,323 minutes. The planet moves 2 degrees and 3,234,670 minutes, then reappears in the east, behind the sun. It moves quickly at 58 minutes and 11 seconds per day, covering 11 degrees in 58 days. The speed slows down to 9 minutes per day, covering 9 degrees in 58 days. It stops for 25 days, then starts moving again. During retrograde, it moves 7 minutes per day, retreating 12 degrees in 84 days. It stops for another 25 days, then moves direct at 58 minutes and 9 seconds per day, covering 9 degrees in 58 days. Moving quickly at 11 minutes per day, it covers 11 degrees in 58 days, appearing ahead of the sun and setting in the west in the evening. Over 16 days, it covers a total distance of 1,742,323 minutes, moving 2 degrees and 3,234,670 minutes in total, then meets the sun again. A complete cycle takes 398 days, covering a total distance of 3,484,646 minutes, with the planet moving 43 degrees and 2,509,956 minutes.
The Sun appears together with the Sun in the morning, and then remains hidden. Next comes the direct motion, lasting 71 days, covering a total of 1,489,868 minutes, which means the planet moved 55 degrees and 242,860.5 minutes. Then, in the morning, it can be seen in the east, behind the Sun. During direct motion, it moves 14 minutes and 23 seconds each day, covering 112 degrees in 184 days. Then, during direct motion, its speed slows down, moving 12 minutes each day, covering 48 degrees in 92 days. Then it stops for eleven days. After that, it retrogrades, moving 17 minutes and 62 seconds each day, moving back 17 degrees in 62 days. After stopping again for eleven days, it resumes direct motion, moving 3 minutes each day, covering 7.5 degrees in 87 days. Once again, during direct motion, its speed increases, moving 14 minutes each day, covering 112 degrees in 184 days; at this point, it moves ahead of the Sun, lurking in the west at night. After 71 days and 1,489,868 minutes, the planet has also moved 55 degrees and 242,860.5 minutes, then it meets with the Sun again. This way, one cycle is calculated to be a total of 779 days and 973,113 minutes, with the planet moving 414 degrees and 478,998 minutes.
As for Saturn, it also appears with the Sun in the morning, then remains hidden. Next comes the direct motion, lasting 16 days, covering a total of 1,122,426.5 minutes, with the planet moving 1 degree and 1,992,864.5 minutes. Then, in the morning, it can be seen in the east, behind the Sun. During direct motion, it moves 3 minutes and 35 seconds each day, covering 7.5 degrees in 87.5 days. Then it stops for 34 days. After that, it retrogrades, moving 1 minute and 17 seconds each day, moving back 6 degrees in 102 days. After another 34 days, it resumes direct motion, moving 3 minutes each day, covering 7.5 degrees in 87 days; at this point, it moves ahead of the Sun, lurking in the west at night. After 16 days and 1,122,426.5 minutes, the planet has also moved 1 degree and 1,992,864.5 minutes, then it meets with the Sun again. This way, one cycle is calculated to be a total of 378 days and 166,272 minutes, with the planet moving 12 degrees and 1,733,148 minutes.
Venus, when it meets the sun in the morning, first "retrogrades." It retrogrades four degrees within five days, then in the morning it can be seen in the east, positioned behind the sun. It continues to retrograde, moving back three-fifths of a degree each day, for a total of six degrees in ten days. Then it "pauses," remaining stationary for eight days. Next, it starts to "rotate," or move forward, at a slower pace, covering three degrees and thirty-six minutes each day, for a total of thirty-three degrees over forty-six days. After that, its speed increases, moving one degree and fifteen minutes each day, for a total of one hundred and sixty degrees in ninety-one days. The speed continues to increase, moving one degree and twenty-two minutes each day, for a total of one hundred and thirteen degrees in ninety-one days, at which point it is once again positioned behind the sun and can be seen in the east in the morning. Finally, it moves forward, covering fifty-six thousand nine hundred and fifty-four minutes in forty-one days, with a total of fifty degrees during that period, ultimately meeting the sun. From one conjunction to the next, a total of two hundred and ninety-two days and fifty-six thousand nine hundred and fifty-four minutes pass, with the same total movement for the planet.
When Venus meets the sun in the evening, it first "retrogrades," this time moving forward. It covers fifty-six thousand nine hundred and fifty-four minutes in forty-one days, with a total of fifty degrees during that period, and can be seen in the west in the evening, ahead of the sun. Then it accelerates its forward movement, covering one hundred and thirteen degrees in ninety-one days, moving one degree and twenty-two minutes each day. The speed then begins to slow down, moving one degree and fifteen minutes each day, for a total of one hundred and sixty degrees in ninety-one days, before starting to move forward again. The speed slows down further, moving three degrees and thirty-six minutes each day, for a total of thirty-three degrees over forty-six days. Then it "pauses," remaining stationary for eight days. Next, it "rotates," or retrogrades, moving back three-fifths of a degree each day, for a total of six degrees in ten days, at which point it is in front of the sun and can be seen retrograding in the west in the evening. The speed increases, moving back four degrees in five days, ultimately meeting the sun. From two conjunctions, a complete cycle lasts five hundred and eighty-four days and eleven thousand three hundred and ninety-eight minutes, with the same total movement for the planet.
Mercury, when it meets the Sun in the morning, first "sinks" and goes into retrograde motion. It moves retrograde seven degrees in nine days, and then it can be seen in the morning sky in the east, positioned behind the Sun. As it continues retrograding, its speed increases, moving back one degree per day. Then it "pauses," remaining stationary for two days. Next, it "revolves," which means it resumes direct motion, moving at a slower pace, traveling eight-ninths of a degree each day, covering eight degrees in nine days, and it then begins to move forward. Its speed increases to one and a quarter degrees per day, covering 6,401,967 minutes in sixteen days, while the planet also travels thirty-two degrees 6,401,967 minutes, ultimately meeting with the Sun. The total duration from one conjunction to the next is fifty-seven days and 6,401,967 minutes, with the degrees traveled by the planet being identical.
Speaking of Mercury, it sets with the Sun and then remains hidden for a time. Specifically, after sixteen days, Mercury reaches the position of thirty-two degrees 6,401,667 minutes in ecliptic longitude, at which point it becomes visible in the evening sky in the west, positioned in front of the Sun. Its speed varies, at times quick and at times slow. When moving quickly, it can travel one and a quarter degrees in a day, covering twenty-five degrees in twenty days. When moving slowly, it travels only seven-eighths of a degree per day, covering eight degrees in nine days. Sometimes it also stops, staying still for two days. At times, it goes into retrograde, moving back one degree each day, during which it is also visible in the evening sky in the west, positioned in front of the Sun. During retrograde motion, its speed is also slow, taking nine days to retrograde seven degrees, ultimately meeting with the Sun again. This complete conjunction cycle lasts a total of 115 days and 6,012,505 minutes, and that is how Mercury moves.
In short, Mercury's orbital path is at times fast and at times slow, and it can even seem to move backwards. Its conjunctions with the Sun are quite regular, happening roughly every 100 days. This passage describes the ancient observations of Mercury's movement pattern. Viewed from a modern scientific perspective, it outlines Mercury's conjunction cycle and apparent motion, including changes in its speed and the retrograde motion. While it employs ancient astronomical terms, the reasoning behind it aligns with modern astronomical understandings of planetary motion.
