On the day of the winter solstice, double the small remainder to create a large remainder; this marks the day governed by the Kan hexagram. Then add 175 to the small remainder, and the large remainder becomes complete; this marks the day when the Zhong Fu hexagram is in charge.
Next, calculate the subsequent hexagram: add 6 to each large remainder and add 103 to the small remainder. The four hexagrams correspond to the respective days mentioned, and the small remainder is doubled again.
Then retrieve the large and small remainders from the winter solstice: add 27 to the large remainder, add 927 to the small remainder; the total comes to 2356, derived from the large remainder. This is the day when Earth is in charge. Add 18 to the large remainder, add 618 to the small remainder; this marks the day when Wood governs the onset of spring. Add 73 to the large remainder, add 116 to the small remainder, and you get Earth again. If Earth continues to be added, Fire can be obtained, while Metal and Water can be set aside.
Multiply 12 by the small remainder; if the result is divisible, you get a Chen, starting from Zi; this is outside of the calculation. Use the new moon, waxing moon, and full moon to determine the small remainder. Multiply 100 by the small remainder; if the result is divisible, you get a moment; if not, calculate the fraction, refer to the nearest solar term, start counting from midnight, and continue until dawn. If the water level is not full before dawn, use the nearest value.
The calculation has both progress and retreat: progress is addition, retreat is subtraction. The difference between progress and retreat starts from 2 degrees, decreases by 4 degrees each time, and the decreasing amount is halved each time. After three times, decrease again until the difference reaches three; after five degrees, subtract back to the initial value.
The moon moves fast and slow, repeating itself. The counting derives from various numbers found in the cosmos; use the remainder to multiply by itself; if it matches the count, you determine the passage of the week. Divide the number of days in the week by the number of weeks in the month, and you get the historical days. There is decay in speed, and the changing pattern reflects the overall trend. Add the decay amount to the monthly rate to get the daily rotation degree. Add the decay amount on both sides to get the profit and loss rate. If it is profit, keep adding; if it is loss, keep subtracting; this represents the accumulation of gains and losses. Multiply half a small week by the common method; if it equals the common number, subtract it from the historical week number to get the new moon movement.
Below is the data table for daily rotation degree, decay, profit and loss rate, gain and shrinkage, and monthly movement:
Day One: 14° 10', one retreat, gain of 22, initial accumulation, 276
Day Two: 14° 9', two retreats, gain of 21, total gain of 22, 275
Day Three: 14° 7', three retreats, gain of 19, total gain of 43, 273
Day Four: 14 degrees 4 minutes, 4 steps back, gain 16, total 62, overall 270
Day Five: 14 degrees, 4 steps back, gain 12, total 78, overall 266
Day Six: 13 degrees 15 minutes, 4 steps back, gain 8, total 90, overall 262
Day Seven: 13 degrees 11 minutes, 4 steps back, gain 4, total 98, overall 258
Day Eight: 13 degrees 7 minutes, 4 steps back, loss of unspecified amount, total 102, overall 254
September 9th: walked 13 degrees, 3 minutes, 3 steps back, 3 steps forward, subtract 4, total 102, overall 250.
October 10th: walked 12 degrees, 18 minutes, 3 steps back, 3 steps forward, subtract 8, total 98, overall 246.
October 11th: walked 12 degrees, 15 minutes, 4 steps back, 4 steps forward, subtract 11, total 90, overall 243.
October 12th: walked 12 degrees, 11 minutes, 3 steps back, 3 steps forward, subtract 15, total 79, overall 239.
October 13th: walked 12 degrees, 8 minutes, 2 steps back, 2 steps forward, subtract 18, total 64, overall 236.
October 14th: walked 12 degrees, 6 minutes, 1 step back, 1 step forward, subtract 20, total 46, overall 234.
October 15th: walked 12 degrees, 5 minutes, 1 step forward, subtract 21, total 26, overall 233.
October 16th: walked 12 degrees, 6 minutes, 2 steps forward, subtract 20. Since there wasn't enough to subtract, I added 5 instead, resulting in a total of 234.
October 17th: walked 12 degrees, 8 minutes, 3 steps forward, add 18, subtract 15, total 236.
October 18th: walked 12 degrees, 11 minutes, 4 steps forward, add 15, subtract 23, total 239.
October 19th: walked 12 degrees, 15 minutes, 3 steps forward, add 11, subtract 48, total 243.
October 20th: walked 12 degrees, 18 minutes, 4 steps forward, add 8, subtract 59, total 246.
October 21st: walked 13 degrees, 3 minutes, 4 steps forward, add 4, subtract 67, resulting in a total of 250.
October 22nd, walked thirteen degrees and seven minutes, advanced four times, with additions and subtractions, subtracted seventy-one, total two hundred fifty-four.
October 23rd, walked thirteen degrees and eleven minutes, advanced four times, with additions and subtractions, subtracted seventy-one, total two hundred fifty-eight.
October 24th, walked thirteen degrees and fifteen minutes, advanced four times, with additions and subtractions, subtracted sixty-seven, total two hundred sixty-two.
October 25th, walked fourteen degrees, advanced four times, with additions and subtractions, subtracted fifty-nine, total two hundred sixty-six.
October 26th, walked fourteen degrees and four minutes, advanced three times, with additions and subtractions, subtracted forty-seven, total two hundred seventy.
October 27th, walked fourteen degrees and seven minutes, advanced three times in history, added three large cycles of Sundays, subtracted nineteen, subtracted thirty-one, total two hundred seventy-three.
On Sunday, fourteen degrees and nine minutes, less advanced, with additions and subtractions, subtracted twenty-one, subtracted twelve, total two hundred seventy-five.
Sunday minutes, three thousand three hundred three.
Cycle remainder, two thousand six hundred sixty-six.
Sunday law, five thousand nine hundred sixty-nine.
Total cycle, one hundred eighty-five thousand thirty-nine.
Historical cycle, one hundred sixty-four thousand four hundred sixty-six.
Less big law, one thousand one hundred one.
Shuo Xing Da Fen, eleven thousand eight hundred one.
Small points, twenty-five.
Half cycle, one hundred twenty-seven. This sentence is a number and does not need to be translated.
This next part means: Multiply the monthly accumulation of the Upper Yuan by the size of the Shuo day's operation. If the small points reach thirty-one, they are converted to minutes. If the big points reach a full cycle, they are subtracted, leaving the count of complete weeks, which equals one day. The remaining less than a day is the day remainder. For now, we will set aside the day remainder; what we are trying to find is the date of the new moon.
Calculate the date of the next month, add one day, the day remainder is five thousand eight hundred thirty-two, and the small points total twenty-five.
Calculate the date of the first quarter moon, which occurs on the fifteenth or sixteenth of the lunar month, add seven days respectively, the day remainder is two thousand two hundred eighty-three, and the small points total twenty-nine and a half. Convert the points to days using the method above, subtracting twenty-seven for a full cycle, leaving the week points. If not enough to divide, subtract one day, and add the cycle remainder.
Multiply the accumulated surplus and deficit values in the calendar by a value of one cycle as the base. Next, multiply the total number within a cycle by the remaining days, and then by the profit and loss rate (profit and loss rate: the ratio coefficient used when adjusting the calendar). Use this result to adjust the base; this is called overtime accumulation adjustment (overtime accumulation adjustment: adjusting the calendar to make it more accurate). Subtract the monthly operation minutes from the chapter year, multiply by half a week as the difference method, use it for division, and obtain the size of the remainder of surplus and deficit. According to the surplus and deficit of the daily law, the new moon day is adjusted by adding one day before and after. The waxing and waning determine the small remainder.
Multiply the chapter year by the overtime accumulation adjustment, divide by the difference method, and obtain the total profit and loss number (total profit and loss number: the total number of profits and losses within a cycle). This is the size difference of surplus and deficit; add the surplus and deficit to the position of the current day and month. If there is a surplus, apply the recording rules to adjust the degrees, determining the exact degrees and minutes of the day and month.
Multiply half a week by the small remainder of the new moon day, divide by the total number, then subtract the remaining days of the calendar. If the remainder is not enough, add the weekly standard and then subtract, followed by subtracting one day. Subtracting gives you the weekday along with its minutes, indicating the time of midnight entry into the calendar.
Calculate the second day, add one day, and the remaining days total twenty-seven. Subtract the number of full week days; if it is not an integer week day, add the incomplete part to the fictitious week. The remaining is the remaining days of the second day.
Multiply the remaining days of the midnight entry into the calendar by the profit and loss rate. If it can be evenly divided by the weekly standard to yield one, if it cannot be divided, then get the remainder. Use this remainder to adjust the accumulated surplus and deficit value; if the remainder cannot be used for adjustment, then divide by the total number to obtain the midnight surplus and deficit. A full chapter year is a degree; not full is a minute. Multiply the total number by minutes and remainder, convert the remainder into minutes according to the weekly standard, convert the full minutes into degrees according to the recording rules, subtract the surplus, and add the deficit to reduce the degrees and remainder of this midnight, determining the final degrees.
Multiply the remaining days of the calendar by the column decay (column decay: the decay of numerical values in the calendar). If it can be evenly divided by the weekly standard to yield one, if it cannot be divided, then get the remainder; this allows you to track the daily decay of numerical values.
Multiply the fictitious week by the column decay to obtain a constant. After the calendar concludes, apply this constant to account for the changing decay. Subtract the full column decay and convert it into the changing decay of the next calendar.
This text describes the ancient methods of calculating calendars, which are very complex. Let's interpret it sentence by sentence and express it in modern spoken language.
First paragraph: The length of each day varies; some days are longer and some are shorter, just like the changing seasons, with ups and downs. The calculation method is: first, accumulate the time for each day, and if it exceeds the standard 24-hour day, subtract 1338, then multiply the total time by 1338. If it does not exceed the standard 24-hour day, add 837 and then divide by 899, and then adjust according to these results and continue the calculation. This is like a complex mathematical formula, continuously iterating to ultimately obtain the accurate time.
Second paragraph: Based on the daily changes in time length, adjust the daily balance, thereby adjusting the length of time at night. If the final calculation result is insufficient, make up the difference using the same method as before. This part mainly discusses the calculation method for adjusting nighttime based on daily time changes to ensure accuracy in calculations.
Third paragraph: Calculate the monthly movement degrees, then multiply that by the number of hours for each solar term's nighttime, and then divide by 200 to get the daytime length. Subtract this daytime length from the monthly degrees to get the nighttime length. Multiply the total time by the length of daytime or nighttime, then add the standard nighttime to get the accurate time for both day and night. Any excess time is divided by two, taking the integer part. This section describes the calculation method for the lengths of daytime and nighttime, which is similarly complex and requires multiple calculations to arrive at the result.
Fourth paragraph: Calculating a month's calendar involves considering many factors, including the lunar orbit, etc. Through a series of calculations, the length of each day's time can ultimately be calculated. This section summarizes the overall calculation method for the lunar calendar, incorporating various technical terms like "new moon," "lunar conjunction," and "differential rate," all of which are important concepts in ancient calendrical systems.
From the fifth paragraph to the last paragraph: This section lists a table that details the daily values of "decrease" and "profit-loss ratio." These numbers represent the daily changes in time, as well as adjustments made based on these changes, such as "minus one plus seventeen," indicating a decrease of one unit on the first day and an increase of seventeen units, and so on. The table also provides calculation limits and differentials, as well as some important parameters, such as lunar weeks, rate differences, the new moon, differentials, and differential methods. These data are essential components of historical calendar calculations, reflecting the precision of ancient calendar calculations. "Lesser Grand Method, four hundred and seventy-three" refers to a coefficient of a certain calculation method.
In summary, this text describes a highly complex ancient calendar calculation method, involving a large amount of mathematical operations and astronomical knowledge. It is difficult for modern people to fully understand its meaning, requiring in-depth study of ancient astronomical calendars to fully grasp.
I first use the moment of the new moon, multiplied by the differential and the new moon, subtracting a full week's worth of differentials and new moon cycles, to determine the number of days in the solar calendar; if the number is complete, subtract it, and the remainder is the number of days in the lunar calendar. The remaining days are calculated based on the algorithm that one day is equivalent to the moon's orbiting for a week. In addition to this, the date of the new moon entering the calendar is calculated, with any remaining fraction representing the excess of the day.
Adding two days, the remainder is 2580, and the differential is 914, calculated as days according to the method, subtracting when reaching 13, with the remaining calculated as days based on fractions. The final boundary between the lunar and solar calendars is defined by the remainders before and after entering the calendar, indicating that the moon has reached its midpoint.
The fractions indicating early or late entries into the calendar, as well as the size of the calendar's profit and loss, are listed separately. Multiplying the fractions by the smaller values yields the differential, and adjusting the profit and loss based on the remainders from both the lunar and solar calendars, the number of days is increased or decreased accordingly. Multiplying the determined remainder by the profit-loss ratio, if the moon completes its weekly orbit, the overall profit and loss value is used as the fixed adjustment factor.
By multiplying the differential rate by the remainder of the moment of the new moon, using the method of differentiation to obtain one, subtract it from the remainder entering the calendar. If the remainder is insufficient, add a new moon, then subtract it, followed by subtracting one day. Subtract the fractional day, add its fraction, and simplify the differentiation with a numerical calculation to get the fraction; this marks the moment when the new moon is recorded in the calendar.
To determine the second day, add one day; the remainder becomes 31, and the fraction also becomes 31. Subtract the fraction from the remainder according to the numerical calculation, subtract a new moon when the remainder is full, then add one day. The calendar finally ends, and when the remainder is full of fractional days, subtract it; that is the initial moment of entering the calendar. If not full of fractional days, keep it, add the remainder 272, and fraction 31; that is the moment when the next calendar begins.
Multiply the total by the time delay or advance for entering the calendar, gains and losses at midnight, and the remainder. When the remainder equals half a week, treat it as the fraction; add for gains, subtract for losses, the remainder of the solar and lunar calendar. If the remainder is in surplus or insufficient, determine it by increasing or decreasing the number of days. Multiply the determined remainder by the profit and loss rate; if the moon completes a week of movement, use the comprehensive value of profit and loss as the midnight constant.
Multiply the profit and loss rate by the number of missing moments at the recent solar term at night; 1/200 is Ming. Subtract the profit and loss rate from it to get Hun, then use the profit and loss midnight number as the Hunming constant.
List the additional hours or constants of Hunming, divide 12 by it to get the degree; the remainder divided by 3 to get less. Less than 1 is strong, two less is weak; what is obtained is the angle of the moon's departure from the ecliptic. For the solar calendar, subtract the degree of leaving the pole from the ecliptic calendar where the sun is located; for the lunar calendar, subtract it, which is the angle of the moon's departure from the pole. Strong is positive, weak is negative; add the strong values together, add identical names, and subtract different names. When subtracting, subtract the same names, add the different names; if there are no opposites, add two strong and subtract one weak.
Starting from the year of Ji Chou in the Upper Yuan to the year of Bing Xu in the eleventh year of Jian'an, a total of 7,378 years accumulated. These are the years: 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, Bing Yin.
Next is the correspondence between the Five Elements and the constellations: Wood corresponds to the Star of the Year (Jupiter), Fire corresponds to the Fiery Star (Mars), Earth corresponds to the Earth Star (Saturn), Metal corresponds to the Bright Star (Venus), Water corresponds to the Morning Star (Mercury). Each constellation has a weekly rate and a daily rate, which is like the distance they travel each day and each year. Multiply the weekly rate by the number of years to get the lunar method; then multiply the lunar method by the number of days to get the monthly fraction; divide the monthly fraction by the lunar method to get the number of months. Then multiply the number of months by the universal method to get the daily method. The Dipper division is obtained by multiplying the Dipper division by the weekly rate. (The daily method uses the chronological method multiplied by the weekly rate, so here we also use division to multiply.)
Next is the calculation of the new moon and small remainder for the Five Stars. The specific method is to multiply the universal method by the number of months, then divide by the daily method to get the whole number obtained as the large remainder, and the remainder is the small remainder. Then subtract the large remainder from 60.
Next is the calculation of the Five Stars' entry day and daily remainder. The method is: multiply the universal method by the month remainder, then multiply the synodic month method by the new moon small remainder, add these two results together, then simplify, and finally divide by the daily method to get the entry day and daily remainder.
Finally, calculate the degrees and degree remainder of the Five Stars. The method is: first subtract the excess value to get the degree remainder in minutes, then multiply by the week days to get the degree remainder, then simplify by the daily method, the resulting whole number is the degree, and the leftover is the degree remainder. If the degree exceeds the week days, subtract the week days and the Dipper division.
The chronological month figure is 7285, the leap month is 7, the chapter month figure is 235, the mid-year is 12, the universal method is 43026, the daily method is 1457, the counting is 47, the week days is 215130, the Dipper division is 145.
Next, here are the specific data for Jupiter: the weekly rate is 6722, the daily rate is 7341, the synodic month is 13, the month remainder is 64810, the synodic month method is 127718, the daily method is 3959258, the new moon large remainder is 23, the new moon small remainder is 1370, the entry day is 15, the daily remainder is 3484646, the new moon virtual fraction is 150, the Dipper division is 974690, the degree is 33, and the degree remainder is 2509956.
Then, here are the data for Mars: the weekly rate is 3447, the daily rate is 7271, the synodic month is 26, the month remainder is 25627, the synodic month method is 64733, the daily method is 206723, the new moon large remainder is 47, the new moon small remainder is 1157, the entry day is 12, the daily remainder is 973113, the new moon virtual fraction is 300, the Dipper division is 494115, and the degree is 48.
Wow, all these numbers are making my head spin! Is this some sort of astronomical calendar calculation? Let's take it step by step.
First paragraph: "Remaining degrees, one million nine hundred ninety-nine thousand one hundred and seventy-six." This means: The remaining number of degrees is one million nine hundred ninety-nine thousand one hundred and seventy-six. This "degree" likely refers to some kind of astronomical unit, but I need to check the information to understand it.
"Earth: circumference, three thousand five hundred and twenty-nine. Rotation rate, three thousand six hundred and fifty-three. Total number of lunar months, twelve. Lunar surplus, fifty-three thousand eight hundred and forty-three. Total lunar method, sixty-seven thousand fifty-one. Daily degree method, two hundred seventy-eight thousand five hundred and eighty-one. Major lunar surplus, fifty-four. Minor lunar surplus, five hundred and thirty-four. Days in a lunar month, twenty-four. Daily surplus, one hundred sixty-six thousand two hundred and seventy-two. Lunar virtual division, nine hundred and twenty-three. Dou division, five hundred and seventeen thousand five hundred. Number of degrees, twelve. Degree surplus, one million seven hundred thirty-three thousand one hundred and forty-eight." This section is all about various astronomical data related to Saturn, such as circumference, rotation rate, lunar surplus, total lunar method, etc. I completely don’t understand this, so I can only reproduce the original text. This must be something that only ancient astronomers could understand!
Second paragraph: "Venus: circumference, nine thousand twenty-two. Rotation rate, seven thousand two hundred and thirteen. Total number of lunar months, nine. Lunar surplus, fifteen thousand two hundred and ninety-three. Total lunar method, seventeen thousand one hundred and eighteen. Daily degree method, five hundred thirty-one thousand three hundred and fifty-eight. Major lunar surplus, twenty-five. Minor lunar surplus, one thousand one hundred and twenty-nine. Days in a lunar month, twenty-seven. Daily surplus, fifty-six thousand nine hundred and fifty-four. Lunar virtual division, three hundred and twenty-eight. Dou division, one hundred thirty-eight thousand one hundred and ninety. Number of degrees, two hundred ninety-two. Degree surplus, fifty-six thousand nine hundred and fifty-four." This section presents the calculation results for Venus, similar to the section about Saturn, filled with overwhelming numbers. Ancient astronomical calculations are incredibly complex!
Third paragraph: "Water: The year rate, eleven thousand five hundred sixty-one. The daily rate, one thousand eight hundred thirty-four. Total months, one. Remaining days of the month, two hundred eleven thousand three hundred thirty-one. Combined monthly method, two hundred nineteen thousand six hundred fifty-nine. Daily degree method, six million eight hundred ninety-four thousand two hundred twenty-nine. Remaining days of the new moon, twenty-nine. Remaining days of the small new moon, seven hundred seventy-three. Days to enter the month, twenty-eight. Remaining days, six million four hundred one thousand nine hundred sixty-seven. New moon virtual division, six hundred eighty-four. Big Dipper division, one hundred sixty-seven thousand six hundred forty-five. Degree number, fifty-seven. Remaining degrees, six million four hundred one thousand nine hundred sixty-seven." This is the calculation for Mercury, and like before, it is a pile of astronomical numbers. I feel I need a book on ancient astronomical calendars to interpret these data.
Final paragraph: "Set the year sought for the beginning of the month, multiply by the year rate; if it divides evenly into the daily rate, it is called accumulation; if not, the remainder is called the remaining accumulation. By dividing the year rate by the remainder, one can calculate the positions of celestial bodies in the past few years. The conjunction of Venus and Mercury, odd numbers indicate morning appearances, even numbers indicate evening appearances." This section summarizes the calculation method, roughly meaning: first set a starting year, then multiply the year rate by the year to get a value; if it divides evenly into the daily rate, it is called accumulation; if not, the remaining part is the remaining accumulation. By dividing the year rate by the remaining accumulation, one can deduce the positions of celestial bodies in previous years. The conjunction of Venus and Mercury, odd numbers indicate morning appearances, even numbers indicate evening appearances. In summary, this entire process is very complex, and I completely cannot understand its intricacies.
First, let's calculate the moon's cycle. Multiply the total number of months by the remaining days, and if the result meets the standard for one month, it is counted as a month; if not, it is recorded as remaining months. Then, subtract the accumulated months from the total months, and the remainder is the number of months entering the next cycle. Multiply by the chapter leap (leap month), and if it meets the standard for a leap month, subtract it from the number of months entering the next cycle; the remainder is used within the year and noted outside of astronomical calculations; this is the combined month. If it coincides with the transition of a leap month, adjust using the new moon day.
Next, first, multiply the month’s remainder by the conventional method, then multiply the standard month by the remainder of the new moon day, then add them together and divide by the conjunction value. If the result is enough for a standard day, that is the day of the conjunction of celestial bodies and the moon; if not enough, the remainder is noted separately from the new moon day calculation.
Then, multiply the number of days in a week by degrees and minutes. If it is enough for a standard day, it is recorded as one degree; if not enough, it is recorded as a remainder, and degrees are recorded using the method of the first five significant figures.
The above is the calculation method for the conjunction of celestial bodies and the moon.
Next, calculate the year. Add the number of months together, and add the remainder of the months together. If it is enough for a standard month, it is recorded as one year; if not enough, it is noted for the current year. Subtract if it is enough, and include leap months in the calculation; the remaining part is considered the next year. If it is enough for two standard months, it is recorded as the following two years. Venus and Mercury, add morning to evening, add evening to morning. (This sentence does not need to be translated, quoted from the original text.)
Then, add the remainder of the new moon day and the conjunction remainder together. If it exceeds a month, add the remainder of the day; add 29 to the large remainder and 773 to the small remainder. If the small remainder is enough for a standard day, subtract it from the large remainder, as described earlier.
Add the days of the month and the remainder of the day together; if the remainder is enough for a standard day, it is recorded as one day. If the previous new moon day remainder perfectly fills the fractional part, subtract one day; if the remaining small remainder exceeds 773, subtract 29 days; if not enough, subtract 30 days; the remaining part is recorded as the next conjunction day.
Finally, sum the degrees together, and add the remainder of the degrees together; if it is enough for a standard day, it is recorded as one degree.
Below are the orbital data for Jupiter, Mars, Saturn, and Venus:
Jupiter: retrograde for 32 days, 3484646 minutes; direct for 366 days; retrograde for 5 degrees, 2509956 minutes; direct for 40 degrees. (Except retrograde 12 degrees, direct 28 degrees.)
Mars: retrograde for 143 days, 973113 minutes; direct for 636 days; retrograde for 110 degrees, 478998 minutes; direct for 320 degrees. (Except retrograde 17 degrees, direct 330 degrees.)
Saturn: retrograde for 33 days, 166272 minutes; direct for 345 days; retrograde for 3 degrees, 1733148 minutes; direct for 15 degrees. (Except retrograde 6 degrees, direct 9 degrees.)
Venus: morning retrograde in the east for 82 days, 113908 minutes; direct in the west for 246 days. (Except retrograde 6 degrees, direct 246 degrees.) Morning retrograde for 100 degrees, 113908 minutes; direct in the east. (Daily as in the west. Retrograde for 10 days, direct 8 degrees.)
Mercury, on the other hand, rises in the morning after thirty-three days, having traveled a distance of six million one hundred twenty-five thousand five minutes.
Then, it appeared in the western sky, remaining visible for thirty-two days. (Here, subtract one degree of retrograde motion, so in the end, it traveled thirty-two degrees.) After that, it moved forward sixty-five degrees, for a total of six million one hundred twenty-five thousand five. Next, it appeared in the eastern sky, moving the same number of degrees each day as it had in the west, remaining hidden for eighteen days and retrograding fourteen degrees.
The method to calculate the number of days Mercury was hidden and the remaining degrees is: add the degrees Mercury was hidden to the remaining degrees, then add this to the remaining degrees of the Sun. If the remaining part reaches one solar degree, recalculate using the previous method; this allows for the determination of the timing and degrees of Mercury's appearance. Multiply the denominator of Mercury's motion by the degrees it was visible, and if the remaining part can be evenly divided by the solar degree to get one, any remainder greater than half should also be counted as one; then add the resulting number to its fractional motion. If the fraction equals the full denominator, increase by one degree. The denominators for retrograde and direct motion are different; multiply the denominator it was using at that time by the remaining fraction. If the result equals the original denominator, that is the fraction it was operating under at that time. Retaining means inheriting the previous result, while retrograde means subtracting. If the degrees hidden are insufficient, use the斗 to divide by the fraction, using the motion's denominator as a ratio, which may result in increases or decreases in the fraction, with each affecting the other. Any references to fullness or emptiness pertain to division aimed at achieving precise results; both 'taking' and 'dividing' are methods for seeking completeness.
Jupiter, in the morning, it is together with the sun, then it goes into hiding, moving in a direct motion. After 16 days, it has traveled a total of 1,742,323 minutes, and the planet has moved 2 degrees and 323,467 minutes. Then, in the morning, it appears in the east, behind the sun. Moving forward at a fast speed, it traverses approximately 11/58 degrees each day, moving 11 degrees in 58 days. Then, it continues moving forward at a slower speed, traversing approximately 9/58 degrees each day, moving 9 degrees in 58 days. It then remains stationary for 25 days before starting to move again. In retrograde, it moves approximately 1/7 degrees each day, retreating 12 degrees after 84 days. It stops again, starting to move forward after 25 days, traversing approximately 9/58 degrees each day, moving 9 degrees in 58 days. Moving forward at a fast speed again, it traverses approximately 11/58 degrees each day, moving 11 degrees in 58 days, in front of the sun, then hiding in the western sky at dusk. After 16 days, it has traveled 1,742,323 minutes, and the planet has moved 2 degrees and 323,467 minutes, then it aligns with the sun again. The complete cycle lasts a total of 398 days and 3,484,646 minutes, during which the planet has moved 43 degrees and 250,956 minutes.
Sun: In the morning, the sun aligns with it, then it becomes obscured. Next is moving forward, lasting 71 days, a total of 1,489,868 minutes, during which the planet has traveled 55 degrees and 242,860.5 minutes, then in the morning, it can be seen in the east, behind the sun. During the forward movement, it moves approximately 14/23 degrees each day, traversing 112 degrees in 184 days. Moving forward again, at a slower speed, it traverses approximately 12/23 degrees each day, moving 48 degrees in 92 days. It then remains stationary for 11 days. Then in retrograde, it moves approximately 17/62 degrees each day, retreating 17 degrees after 62 days. It stops moving again for 11 days, then moves forward again, traversing approximately 1/12 degrees each day, moving 48 degrees in 92 days. Moving forward again, at a faster speed, it traverses approximately 1/14 degrees each day, moving 112 degrees in 184 days, at this point, it is in front of the sun, hiding in the western sky at dusk. After 71 days, a total of 1,489,868 minutes, the planet has traveled 55 degrees and 242,860.5 minutes, then it aligns with the sun again. The complete cycle is completed, lasting a total of 779 days and 97,313 minutes, during which the planet has traveled 414 degrees and 478,998 minutes.
Saturn: In the morning, Saturn and the Sun come together, then it disappears. Next is the direct motion, lasting for a total of 1,122,426.5 minutes. The planet travels 1 degree and 1,995,864.5 minutes, then it can be seen in the east behind the Sun in the morning. During direct motion, it moves 3/35 of a degree each day, totaling 7.5 degrees over 87.5 days. Then it stops moving for 34 days. It then goes into retrograde, moving 1/17 of a degree each day, retreating 6 degrees in 102 days. After another 34 days, it resumes direct motion, moving 1/3 of a degree each day, covering 7.5 degrees in 87 days; at this point, it is positioned in front of the Sun and disappears in the west by evening. After 16 days, totaling 1,122,426.5 minutes, the planet travels 1 degree and 1,995,864.5 minutes, then it comes together with the Sun again. A complete cycle lasts a total of 378 days and 166,272 minutes, with the planet traveling 12 degrees and 1,733,148 minutes.
Venus: In the morning when it comes together with the Sun, it first disappears, then goes into retrograde, moving back 4 degrees in 5 days; then it can be seen in the east behind the Sun. It continues retrograde, moving 3/5 of a degree each day, retreating 6 degrees in 10 days. Then it stops, not moving for 8 days. Then it starts moving direct, slowly, covering 33/46 of a degree each day, covering 33 degrees in 46 days. Speeding up, it moves 15/91 of a degree each day, covering 160 degrees in 91 days. In faster direct motion, the speed further increases, moving 22/91 of a degree each day, covering 113 degrees in 91 days; at this point, it is behind the Sun and appears in the east in the morning. During direct motion, it covers 56,954/56,954 days, traveling 50 degrees and 56,954/56,954 degrees, then it comes together with the Sun again. Each complete cycle lasts a total of 292 days and 56,954 minutes, with Venus traveling the same number of degrees.
When Venus meets the sun in the evening, it first conceals itself, then moves forward, for 41,954/56,954 days and a distance of 50 degrees, then can be seen in the evening to the west, positioned in front of the sun. Continuing forward, the speed increases, moving 1 degree and 54 minutes per day for 91 days, covering 113 degrees. The speed slows down, moving 1 degree and 15 minutes per day for 91 days, covering 16 degrees, then continues forward. The speed slows down further, moving 46 minutes and 33 seconds per day for 46 days, covering 33 degrees. It stops for 8 days, then starts moving in retrograde, moving 3 minutes and 53 seconds per day for 10 days, moving backward a total of 6 degrees. At this point, it is in front of the sun, appearing in the west in the evening, moving in retrograde motion, the speed increases, moving backward 4 degrees for 5 days, then meets the sun again. Two conjunctions complete one cycle, totaling 584/11,398/11,398 days and degrees, with Venus completing the same total distance.
Mercury, when meeting the sun in the morning, first conceals itself, then moves in retrograde, moving backward a total of 7 degrees for 9 days, then can be seen in the east in the morning behind the sun. Continuing in retrograde, the speed increases, moving back 1 degree per day. Then it stops for 2 days. Then it starts moving forward, slowly, moving 0.8 degrees per day for 9 days, covering 8 degrees. The speed increases, moving 1 degree and 15 minutes per day for 20 days, covering 25 degrees; at this point, it is behind the sun, appearing in the east in the morning. Moving forward, covering 32 degrees over a period of 641,967 days, then meets the sun again. One conjunction, totaling 57 days and 641,967 degrees, with Mercury covering the same degree of movement.
Speaking of Mercury, it sets alongside the sun, then appears to hide, and its motion is direct. Specifically, every sixteen days, it travels 32 degrees and 641,966 minutes and 40 seconds along the ecliptic. At this time, it can be spotted in the western sky during the evening, visible ahead of the sun. During its direct motion, it moves quite quickly, traveling a quarter of a degree each day, so in twenty days it covers 25 degrees.
At times, it slows down, moving just 7/8 of a degree per day, taking nine days to cover 8 degrees. Sometimes it even remains stationary for two days. Even more remarkably, it can also move in retrograde! It moves backward, retreating at a rate of one degree per day, and during retrograde, it can be seen in the western sky in the evening, ahead of the sun. During retrograde, its motion is slower, taking nine days to retreat seven degrees, then eventually it will catch up with the sun again.
From one conjunction with the sun to the next, the entire cycle lasts 115 days and 601,255.5 minutes, and each orbit of Mercury follows a similar path.
