From 135 BC to 195 AD, a total of 7,378 years have passed. (This sentence is based on the calculation result of the original text, not a direct translation.) The Gan method corresponds to 1,178 years, the Hui Tong method corresponds to 7,171 years, the Ji method corresponds to 589 years, the Zhou Tian method corresponds to 215,130 years, the Tong method corresponds to 43,026 years, the Tong number is 31, the Ri method corresponds to 1,457 years, the year midpoint is 12, the remainder is 390, the Zhang year corresponds to 19, the Mei method corresponds to 103, the Zhang leap corresponds to 7, the Hui number corresponds to 47, the Hui year corresponds to 893 years, the Zhang month corresponds to 235, the Hui rate corresponds to 1,882, the new moon conjunction number corresponds to 941, the Hui month corresponds to 1,145 years, the Ji month corresponds to 7,285 years, the Yuan month corresponds to 14,570 years, the month week corresponds to 7,874 years, and the small week corresponds to 254 years. Next is the method of calculation, which is somewhat complex. First, you need to divide the year you want to calculate by the Gan method (1,178); if it is not divisible, divide by the Ji method (589). If it is divisible, it indicates the outer Jiawu year; if it is not divisible, the remainder corresponds to the inner Jiazi year. Then, multiply the Zhang month (235) by the year of either the inner or outer Ji, then divide by the Zhang year (19) to obtain the fixed month, and the remainder corresponds to the leap remainder. If the leap remainder exceeds 12, then this year is a leap year. Next, multiply the Tong method (43,026) by the fixed month to obtain the false day, divide by the Ri method (1,457) to obtain the fixed day, and the remainder corresponds to the small remainder. Divide the accumulated days by 60 to obtain the large remainder. Depending on whether you are calculating for the inner Ji or the outer Ji, you can determine which day corresponds to the first day of the eleventh month of the year. To calculate the first day of the following month, add 29 to the large remainder, then add the small remainder (773); if the small remainder exceeds the Ri method (1,457), subtract it from the large remainder. If the small remainder is greater than 684, this month is considered a big month (30 days). To calculate the winter solstice, multiply the year of either the inner or outer Ji by the remainder (390), divide by the Ji method (589) to obtain the large remainder and small remainder. Divide by 60 to determine which day the winter solstice is. After calculating the winter solstice, add 15 to the large remainder, add the small remainder (515); if it exceeds 2,356, subtract it from the large remainder. To calculate the leap month, subtract the Zhang year (19) from the leap remainder, multiply the remaining number by the year midpoint (12); if it is greater than or equal to the Zhang leap (7), then it is a leap month; if it is less than the Zhang leap, but greater than half (3.5), it is also considered a leap month. To calculate the waxing moon, full moon, waning moon, and the first day of the next month, add 7, 15, 22, and 29 to the large remainder respectively; if the small remainder (557.5) exceeds the Ri method (1,457), subtract it from the large remainder. If the small remainder is less than 410, multiply by 100, divide by the Ri method (1,457) to get the time, then calculate the minutes, and finally determine the time of the solar terms.
In conclusion, this calculating method is very complicated and requires careful step-by-step calculations to obtain results. Modern people generally do not use this method to calculate dates anymore, as we have ready-made calendars available.
Let's start from the beginning to explain this algorithm. The meaning of the first sentence is: first determine a year, then multiply what you want to calculate (such as how many days you want to calculate) by a remainder. If the result is exactly a multiple of a fixed value (this multiple is called "calendar rule"), then it is recorded as "offset" (which means gone, canceled out). If there is a remainder, add this "offset" result, and together it counts as 1. Then multiply it by another number ("communicating"); if the result is exactly a multiple of another fixed value (this multiple is called "no rule"), then you get a "large remainder"; if it is not exactly a multiple, the remaining is a "small remainder." This "large remainder" is used to represent the year, and after calculation, you then calculate how many days have passed after the winter solstice.
Next, to calculate when the next "no" (offset) will be, you need to add 69 to the "large remainder" and add 64 to the "small remainder." If the result exceeds the corresponding "rule" (which is a multiple), subtract from the "large remainder"; if it is exactly a multiple, then it becomes zero. Then multiply the number of days you calculated by the "calendar rule," and divide the result by the number of days in a week (which is 360 degrees). The remainder is divided by the "calendar rule," and the result is the "degree." Starting from the fifth degree of the Ox star, divide the "degree" by the constellation degree (which is the number of degrees each constellation occupies); if it is not enough for a constellation, this indicates the sun's position at midnight on the first day of the lunar new year.
To calculate the situation of the second day, add one degree to the degree of the previous day, then divide it by the Dipper; if the share is insufficient, subtract one degree, then add it with the "calendar rule." To calculate the position of the moon, multiply the number of days in a month by the day, then subtract the number of days in a week. The remainder, if it is exactly a multiple of the "calendar rule," is the "degree"; if it is not a multiple, the remaining is the "minute." Using the same method as above, you can calculate the position of the moon at midnight on the first day of the lunar new year. To calculate the situation of the next month, add 22 degrees and 258 minutes for a small month, and add 1 day, 13 degrees, and 217 minutes for a large month; if it exceeds the "rule," then add one degree again. In the latter part of winter, the moon is typically near the Zhang and Xin constellations.
Next, multiply the number of days in a lunar year (章岁) by the small remainder of the new moon day (初一). If the result is exactly a multiple of the cycle (会数), this is referred to as "大分" (large division); if not, the remainder is referred to as "小分" (small fraction). Subtract the "fraction" (分) at midnight of the new moon from "大分". If the result is a multiple of the "纪法" (calendar system), it is deducted from "units" (度), following the same method as above, to calculate the time of the conjunction of the sun and moon (when the sun and moon appear in the same position). To calculate the next month's situation, add 29 degrees, 312 large divisions, and 25 small fractions. If the small fractions exceed the cycle, deduct them from the large divisions; if the large divisions exceed the "纪法", deduct them from the units, and then use the 斗宿 to divide the "大分".
To calculate the position of the first quarter (on the 7th, 15th, and 23rd), add 7 degrees, 225 minutes, and 17.5 small fractions to the degree of the conjunction. Using the same method, you can determine the position of the first quarter. Following this method, the positions of the full moon (15th), last quarter, and next month's conjunction can also be calculated. To calculate the position of the waxing moon, add 98 degrees, 480 large divisions, and 41 small fractions to the degree of the conjunction. Using the same method, you can determine the positions of the full moon, last quarter, and next month's conjunction.
To calculate the positions of sunrise and sunset (as well as moonrise and moonset), multiply the "纪法" by the number of time units (漏刻数) during the most recent solar term night, then divide by 200 to get the "明分" (bright division, indicating sunrise or moonrise). Subtract the number of days from "纪法", and subtract the number of months from the lunar cycle; the remainder is the "昏分" (dark division, indicating sunset or moonset). By adding the degrees of midnight, you can calculate the positions of sunrise, sunset, moonrise, and moonset.
Finally, set a starting year, then subtract it from the cycle year (会岁). Multiply the remaining years by the cycle rate (会率). If the result is exactly a multiple of the cycle year, it is called "积蚀" (eclipse, solar or lunar); if a remainder still exists, add the result of "积蚀" and count it as 1. Multiply the cycle month (会月) by it; if the result is exactly a multiple of the cycle rate, it is called "积月" (accumulated month); if not, the remainder is called "月余" (remaining month). Multiply the leap month (章闰) by the remaining years; if the result is exactly a multiple of the leap year (章岁), it is called "积闰" (accumulated leap), then deduct it from "积月". The remainder is then deducted from one year; if a remainder still exists, begin counting from the first day of the lunar calendar (天正).
Calculate the next solar or lunar eclipse by adding five months to the previous one; the remainder after adding five months is 1635. If the remainder exceeds this limit, add one month. The lunar eclipse occurs on the fifteenth day of the lunar month.
On the winter solstice, double the remainder value and then add another remainder value. This is the day the Kan hexagram applies. Then, add 175 to this value. If it reaches a certain threshold value associated with the Qian hexagram, subtract from the remainder. This is the day when the Zhong Fu hexagram is in effect.
Next, calculate the next hexagram by adding 6 to the large remainder and 103 to the small remainder. The calculation method for the four positive hexagrams is to double the remainder value on the day they are in effect.
Then, take out the remainder values from the winter solstice calculation and add 27 to the large remainder and 927 to the small remainder. If the total reaches 2356, subtract from the large remainder to get the day corresponding to Earth. Then add 18 to the large remainder and 618 to the small remainder to get the day corresponding to Wood at the beginning of spring. Add 73 to the large remainder and 116 to the small remainder to get another day corresponding to Earth. After calculating the day corresponding to Earth, calculate the corresponding days for Fire, Metal, and Water following a specific pattern.
Multiply the remainder value by 12. If it reaches a certain limit, you will get a Heavenly Stem and Earthly Branch starting from Zi (Rat). Special days like New Moon, First Quarter, and Full Moon require a separate calculation of the remainder value.
Multiply the remainder value by 100. If it reaches a certain limit, you will get a quarter. If the result is less than 10, calculate the decimal part. Based on the nearest solar term, start calculating from midnight. If the water level isn't full at midnight, use the most recent value instead.
During the calculations, there may be instances of advancement or retreat. Add for advancement and subtract for retreat. The difference between advancement and retreat starts at two degrees, decreasing by four degrees each time, with the reduction halving each time. After three reductions, stop when the difference reaches three. After five degrees, return to the initial state.
The speed of the moon's movement varies, sometimes fast and sometimes slow, repeating in cycles with unchanging rules. When calculating, various values from heaven and earth must be used, using the remainder rule in multiplication. If the result matches a specific value, a fraction of the cycle is obtained. By subtracting this fraction from the total days in a cycle and dividing by the moon's orbital period, the number of calendar days can be determined. The variations in the moon's speed are part of its movement rules. By adding the speed change values, one can obtain the degrees and minutes of daily movement. Adding the values of speed changes together gives the profit and loss rate. If it's a gain, it keeps accumulating; if it's a loss, it keeps decreasing, which is the accumulation of profit and loss. By multiplying half of a small cycle using a common method, if the result matches a specific value, subtract it from the historical cycle to obtain the fraction for the new moon day.
Here are the specific daily data: daily rotation degrees and minutes, decline, profit and loss rate, accumulation of gains and losses, and moon movement fractions.
Day 1: 14 degrees and 10 minutes, one retreat, gain 22, initial profit, 276
Day 2: 14 degrees and 9 minutes, two retreats, gain 21, profit 22, 275
Day 3: 14 degrees and 7 minutes, three retreats, gain 19, profit 43, 273
Day 4: 14 degrees and 4 minutes, four retreats, gain 16, profit 62, 270
Day 5: 14 degrees, four retreats, gain 12, profit 78, 266
Day 6: 13 degrees and 15 minutes, four retreats, gain 8, profit 90, 262
Day 7: 13 degrees and 11 minutes, four retreats, gain 4, profit 98, 258
Day 8: 13 degrees and 7 minutes, four retreats, loss, profit 102, 254
On September 9, moved 13 degrees and 3 minutes, retreated three times, added a loss of four, surplus one hundred and two, total of two hundred and fifty.
On October 10, moved 12 degrees and 18 minutes, retreated three times, added a loss of eight, surplus ninety-eight, total of two hundred and forty-six.
On October 11, moved 12 degrees and 15 minutes, retreated four times, added a loss of eleven, surplus ninety, total of two hundred and forty-three.
On October 12, moved 12 degrees and 11 minutes, retreated three times, added a loss of fifteen, surplus seventy-nine, total of two hundred and thirty-nine.
On October 13, moved 12 degrees and 8 minutes, retreated twice, added a loss of eighteen, surplus sixty-four, total of two hundred and thirty-six.
On October 14, moved 12 degrees and 6 minutes, retreated once, added a loss of twenty, surplus forty-six, total of two hundred and thirty-four.
On the fifteenth of October, traveled twelve degrees and five minutes, advanced once, reduced by twenty-one, resulting in a surplus of twenty-six, total of two hundred and thirty-three.
On the sixteenth of October, traveled twelve degrees and six minutes, advanced twice, reduced by twenty (since it was insufficient, five was subtracted to adjust the surplus, resulting in a surplus of five, which is referred to as a gain, while the initial reduction was twenty). This resulted in a surplus of five and a reduction of the initial number, total of two hundred and thirty-four.
On the seventeenth of October, traveled twelve degrees and eight minutes, advanced three times, surplus of eighteen, reduced by fifteen, total of two hundred and thirty-six.
On the eighteenth of October, traveled twelve degrees and eleven minutes, advanced four times, surplus of fifteen, reduced by twenty-three, total of two hundred and thirty-nine.
On the nineteenth of October, traveled twelve degrees and fifteen minutes, advanced three times, surplus of eleven, reduced by forty-eight, total of two hundred and forty-three.
On the twentieth of October, traveled twelve degrees and eighteen minutes, advanced four times, surplus of eight, reduced by fifty-nine, total of two hundred and forty-six.
On the twenty-first of October, traveled thirteen degrees and three minutes, advanced four times, surplus of four, reduced by sixty-seven, total of two hundred and fifty.
On the twenty-second of October, traveled thirteen degrees and seven minutes, advanced four times, increased losses, reduced by seventy-one, total of two hundred and fifty-four.
On the twenty-third of October, traveled thirteen degrees and eleven minutes, advanced four times, increased losses by four, reduced by seventy-one, total of two hundred and fifty-eight.
On the twenty-fourth of October, traveled thirteen degrees and fifteen minutes, advanced four times, increased losses by eight, reduced by sixty-seven, total of two hundred and sixty-two.
On the twenty-fifth of October, traveled fourteen degrees, advanced four times, increased losses by twelve, reduced by fifty-nine, total of two hundred and sixty-six.
On the twenty-sixth of October, traveled fourteen degrees and four minutes, advanced three times, increased losses by sixteen, reduced by forty-seven, total of two hundred and seventy.
On the twenty-seventh of October, traveled fourteen degrees and seven minutes, experienced initial advancement three times, added three Sundays, reduced by nineteen, reduced by thirty-one, total of two hundred and seventy-three.
Fourteen degrees on Sunday (nine minutes), less advancement, increased losses by twenty-one, reduced by twelve, total of two hundred and seventy-five.
Sunday total: three thousand three hundred and three.
Virtual week total: two thousand six hundred and sixty-six.
Sunday method total: five thousand nine hundred and sixty-nine.
Total for the week: one hundred eighty-five thousand thirty-nine.
Total for historical week: one hundred sixty-four thousand four hundred and sixty-six.
The Great Law, one thousand one hundred and one. The New Moon phase is one thousand eight hundred and one. The small division equals twenty-five. The half-month is one hundred and twenty-seven. This is a base for astronomical calculations; remember it. Next is the calculation method, which is a bit complex; I will try to explain it clearly. First, use the Upper Yuan multiplied by the month times the New Moon size and small division. When the small division is full, it transforms into a large division. When the large division is full, subtract one cycle; the remainder represents one day. If it is insufficient for one day, simply note the remaining portion. Set this remainder aside; our main focus is to calculate the New Moon's entry into the calendar, which means determining the date the new moon appears on the calendar.
To calculate the next month, add one day to the previous month's base. The daily remainder is five thousand eight hundred thirty-two, and the small division is twenty-five. To calculate the Full Moon (the fifteenth and thirtieth of the lunar month), add seven days to the previous month's base; the daily remainder is two thousand two hundred eighty-three, and the small division is twenty-nine point five. Next, convert these divisions into days according to the previously established rules; when it reaches twenty-seven days, subtract it, and handle the remainder by cycles. If it is insufficient to divide, subtract one day and add a week's imaginary unit.
Next, multiply the accumulated value of the calendar's adjustments by the number of cycles to get the real number. Then multiply the total number of days by the daily remainder division, and then multiply by the gain-loss rate. Use this result to adjust the real number; this represents the time adjustments. Subtract the annual value from the monthly movement division, and then multiply by the half-month to get a difference. Use this difference to divide, obtaining the gain-loss adjustment plus the small remainder, and handle the gains and losses according to the daily method. For the New Moon (the first day), if there is time gain, adjust the number of days forward or backward. The advance and retreat of the Full Moon is determined by the large remainder.
Multiply the year by the time gain and shrinkage, and use the difference method to divide, obtaining the full meeting number, which is the gain-loss of the large and small divisions. Add the gain-loss adjustment to the position of the moon on that day; if it is not enough, adjust the degrees according to the calendar method to determine the position and degrees of the sun and moon.
Multiply the half-month by the small remainder of the New Moon, divide by the total number of days, and then subtract from the daily remainder in the calendar. If it is insufficient to subtract, add the cycle number, then subtract, and finally subtract one day. After subtracting, add the cycle number and division to get the time of the midnight entry into the calendar. To calculate the second day, simply add one day; when the daily remainder reaches twenty-seven days, subtract one cycle. If it is not a full cycle, add the week's imaginary number; the remainder represents the daily remainder for the second day’s entry into the calendar.
Using the lunar calendar's midnight remainder multiplied by the gain and loss rate, divided by the number of cycles, if it is insufficient to divide, record the remainder. Use this remainder to adjust the accumulation of gains and losses; if the remainder is not enough for adjustment, use the total to divide, obtaining the midnight gain and loss, with a full year as the unit, and if insufficient, it is considered a fraction. Multiply the total number of days by the fraction and the remainder, treating the remainder according to the number of cycles; when the fraction is full, handle the degrees according to the calendar law, using the gains to increase and the losses to decrease the degrees at midnight and the remainder, to determine the final degree.
Multiply the remainder of the lunar calendar by the decay coefficient, divide by the number of cycles, and if it is insufficient to divide, record the remainder, which helps determine the daily variations and decay.
Multiply the weekly void by the decay coefficient, divide by the number of cycles to obtain a constant; at the end of the calendar, add this constant to the decay of variation, and when it reaches a full value, subtract it to adjust the decay for the next calendar.
This passage describes the calculation methods of ancient calendars, which are very complex, and we will interpret it sentence by sentence.
First paragraph: The length of time each day will change, with fluctuations, which explains the changes across the four seasons. To calculate the time each day, various factors, including the remainder, must be taken into account, and then these are summed to obtain the time for the next day. If the result of the calculation is not exactly one day, corrections must be made by multiplying by a specific number (1338); if the result exceeds one day, subtract a number (837), then divide by another number (899), and continue calculating until the result meets the requirements. This is like a complex mathematical formula, constantly adjusting to ensure the accuracy of the calendar.
Second paragraph: Based on the gains and losses of daily time, a correction factor is calculated to adjust the length of nighttime each day. If the result of the calculation is insufficient, it must be added back, using a method similar to the previous one. This section details the fine-tuning of day and night lengths to ensure the calendar matches actual observations.
Third paragraph: To calculate the time for each month, one must consider the solar terms and the duration of nighttime. Multiply the distance of the moon's orbit by the nighttime corresponding to the solar terms, then divide by 200 to obtain a value used to calculate sunset times. The calculation method is similar to the previous one and also requires corrections. This section explains how to calculate sunset times based on the moon's orbit and make necessary adjustments.
Fourth paragraph: There are four important moments in a month, and the time intervals between these moments are derived from complex calculations. This section introduces the calculation methods, including how to compute key values.
From the fifth paragraph to the last paragraph: Next are specific numerical examples, demonstrating how to calculate the daily profit and loss using the methods mentioned earlier. The title is "Lunar-Solar Calendar Decline Rate Combined," and the table presents the daily correction values, as well as some key parameters such as "limit balance," "differential," "lesser greater method," and so on. These values reflect the precision of ancient calendrical calculations and also reflect the standards of astronomical observation and mathematical calculation of the era. "Cycle of calendar, one hundred seventy-five thousand six hundred sixty-five. Rate of difference, one thousand nine hundred eighty-six. New moon conjunction, eighteen thousand three hundred twenty-eight. Differential, nine hundred fourteen. Differential method, two thousand two hundred nine." These represent the final results of the calculations, reflecting the complexity and precision of ancient calendrical calculations. These numbers represent calendrical cycles, correction rates, and other key parameters, showcasing the intricacies of ancient astronomical calendars.
In summary, this passage describes an extremely complex ancient calendrical calculation system, which involves a large amount of mathematical operations and astronomical observation data, reflecting the wisdom of ancient working people and their precise grasp of time. Although it is difficult for us to fully understand all the details behind it now, we can appreciate its subtlety.
Let’s begin by discussing how to calculate the months from the start of the lunar year to now. I first multiply the conjunction and differential values of the new moon, and deduct according to the method when the differential is full, deduct one week when the conjunction is full, and the remaining days that are less than a week represent the days counted in the solar calendar; deduct when full, the remaining days are the days entered into the lunar calendar. The remaining days are calculated according to the lunar cycle duration. After calculating, the days counted for the new moon conjunction are treated as remaining days if they fall short of a full day.
After adding two days, the remaining total is two thousand five hundred eighty, the differential is nine hundred fourteen, calculated into days following the prescribed method, deduct when it reaches thirteen, and calculate the remaining days as fractions. The final point at which the lunar and solar calendars intersect, the remaining days prior to entering the calendar are listed first, while those after entering are listed afterwards; this is when the moon reaches the midpoint.
Separately list the rate of change and the magnitude of gains and losses in the calendar, multiply by the number of meetings to obtain the differential, and adjust the remaining days of the lunar calendar by adding or subtracting the gains and losses. Adjust the number of days to determine if there are more or fewer remaining days. Multiply the determined remaining days by the profit and loss rate, calculate a one based on the number of days in a lunar cycle, and use the net value of gains and losses to determine the overtime hours.
Multiply the difference rate by the remaining days of the new moon, calculate a one based on the differential method, and use it to subtract from the remaining days of the calendar. If not enough, add another lunar cycle and then subtract one day. Add the fractional days to their fractional parts, simplify the differential by multiplication; this marks the time to begin the calendar at midnight.
To calculate the second day, add one day; the remaining days total thirty-one, and the fractional days are also thirty-one. Subtract the fractional days from the remaining days according to the multiplication, subtract a day when the remaining days reach a full lunar cycle. The calendar is complete; subtract the fractional days from the remaining days of the fraction; this is the beginning of the calendar. For days that are not full, simply add the remaining two thousand seven hundred and two; the fractional days are thirty-one, marking the time to enter the next calendar.
Multiply the total by the speed of entry into the calendar, the gains and losses in the middle of the night, and the remaining days. When the remaining days reach half a cycle, use it as a fraction; add the gains and losses to subtract the remaining days of the lunar calendar. If there are more or fewer remaining days, adjust the number of days with a lunar cycle. Multiply the determined remaining days by the profit and loss rate, calculate a one based on the number of days in a lunar cycle, and use the net value of gains and losses to determine the value in the middle of the night.
Multiply the profit and loss rate by the number of missed time markers during the recent solar terms; treat two hundredths as daytime, subtract it to calculate nighttime, then use the nighttime profit and loss value to establish the dusk value.
List the number of overtime hours or the determined number of hours, divide 12 by it to get the degree, one third of the remainder is considered 'less,' less than one is considered 'strong,' and two 'less' values are considered 'weak.' The result is the degree to which the moon deviates from the ecliptic. The number of days added in the solar calendar, minus the extreme in the ecliptic calendar, and the number of days subtracted in the lunar calendar is the degree of the moon leaving the extreme. Strong values are positive, while weak values are negative; add strong and weak values together, add those of the same kind, and subtract those of different kinds. When subtracting, subtract the same, add the different, cancel each other out if there is no corresponding match, add two strong and subtract one weak.
From the year of Ji Chou in the Shang Yuan calendar to the year of Bing Xu in the Jian'an eleven years, a total of seven thousand three hundred and seventy-eight years have been accumulated. 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—these are all years; I won't go into detail about these years. Next, I'll explain the calculation methods for the five elements and the movements of the five planets. It sounds a bit complicated, but I will try to explain it clearly.
Jupiter, Mars, Saturn, Venus, Mercury correspond to the five elements of wood, fire, earth, metal, and water respectively. They move at different speeds in the sky, so they need to be calculated separately. Terms such as "orbital period," "daily cycle," "annual cycle," "monthly cycle," etc. mentioned in the text are professional terms used to calculate the periods and positions of celestial bodies, which I won't explain in detail here, but they are used to calculate the positions of celestial bodies in the sky. The calculation method is quite complex, involving multiplication, division, and taking remainders into account. As you can see, terms like "common method," "daily method," "meeting number," etc. are coefficients used to assist in calculations. The final result indicates the positions of the celestial bodies in the sky and their respective periods. The calculation process is tedious and must be done step by step to arrive at the final result.
Next is the specific calculation regarding the five planets, such as "Shuo Dayu," "Shuo Xiaoyu," "Ru Yue Ri," "Ri Yu," and so on. These are intermediate results in calculating the movement of the five planets, and the ultimate goal is to determine their specific positions in the sky. The calculation method remains very complex, involving many multiplications, divisions, and fractions, requiring a solid mathematical foundation to understand. The text provides many specific values, such as "Ji Yue," "Zhang Run," "Zhang Yue," "Sui Zhong," "Tong Fa," "Ri Fa," "Hui Shu," "Zhou Tian," "Dou Fen," and so on, which are constants used in the calculation.
Then there is the specific calculation data for Jupiter, including its orbital period, daily rate, synodic month number, month remainder, synodic month method, daily degree method, Shuo Dayu, Shuo Xiaoyu, Ru Yue Ri, Ri Yu, Shuo Xu Fen, Dou Fen, degree number, degree remainder, and so on. These specific numbers are used to determine Jupiter's position and trajectory in the sky. The same calculation method also applies to Mars, with the text providing Mars' orbital period, daily rate, and a series of other calculation data. This detailed data is sufficient for complex astronomical calculations. These calculation results were used in ancient astronomy to predict astronomical phenomena, guide agricultural production, and even influence political decisions. It can be seen that ancient astronomers exerted tremendous effort to accurately calculate astronomical phenomena.
These numbers are quite overwhelming! Let's start by translating the first paragraph, which appears to be some kind of astronomical calculation record. "The remaining degrees are one million nine hundred and seventeen thousand six." "Earth: Orbital period, three thousand five hundred and twenty-nine." The orbital period of Saturn is three thousand five hundred and twenty-nine. "Daily rate, three thousand six hundred and fifty-three." The daily value of Saturn is three thousand six hundred and fifty-three. "Synodic month number, twelve." There are a total of twelve months. "Month remainder, fifty-three thousand eight hundred and forty-three." The remainder of the month is fifty-three thousand eight hundred and forty-three. "Synodic month method, sixty-seven thousand fifty-one." The value obtained from the total months method is sixty-seven thousand fifty-one.
"Daily calculation method: two hundred seventy-eight thousand five hundred eighty-one." The degree calculated by the daily method is two hundred seventy-eight thousand five hundred eighty-one.
"Larger remainder on the new moon day: fifty-four." The larger remainder on the new moon day (the first day of the lunar month) is fifty-four.
"Smaller remainder on the new moon day: five hundred thirty-four." The smaller remainder on the new moon day is five hundred thirty-four.
"Days into the new month: twenty-four." The number of days into the new month is twenty-four.
"Daily remaining value: one hundred sixty-six thousand two hundred seventy-two." The daily remaining value is one hundred sixty-six thousand two hundred seventy-two.
"Fractional part of the new moon: nine hundred twenty-three." The fractional part of the new moon is nine hundred twenty-three. (Fractional part refers to a certain subdivided unit)
"Dou unit: five hundred eleven thousand seven hundred five." The dou unit is five hundred eleven thousand seven hundred five. (Dou unit refers to a certain subdivided unit)
"Degrees: twelve." The degrees are twelve.
Next is the calculation regarding Venus:
"Remaining degrees: one million seven hundred thirty-three thousand one hundred forty-eight." The remaining degrees are one million seven hundred thirty-three thousand one hundred forty-eight.
"Venus's circumference: nine thousand twenty-two." The circumference of Venus is nine thousand twenty-two.
"Daily value of Venus: seven thousand two hundred thirteen." The daily value of Venus is seven thousand two hundred thirteen.
"Total months: nine." The total number of months is nine.
"Remaining monthly value: fifteen thousand two hundred ninety-three." The remaining monthly value is fifteen thousand two hundred ninety-three.
"Value calculated by the monthly method: one hundred seventy-one thousand four hundred eighteen." The value calculated by the monthly method is one hundred seventy-one thousand four hundred eighteen.
"Degree calculated by the daily method: five hundred thirty-one thousand three hundred ninety-eight." The degree calculated by the daily method is five hundred thirty-one thousand three hundred ninety-eight.
"Larger remainder on the new moon day: twenty-five." The larger remainder on the new moon day is twenty-five.
"Smaller remainder on the new moon day: one thousand one hundred twenty-nine." The smaller remainder on the new moon day is one thousand one hundred twenty-nine.
"Days into the new month: twenty-seven." The number of days into the new month is twenty-seven.
"Daily remaining value: fifty-six thousand nine hundred fifty-four." The daily remaining value is fifty-six thousand nine hundred fifty-four.
"Fractional part of the new moon: three hundred twenty-eight." The fractional part of the new moon is three hundred twenty-eight.
"Doufen is one million three hundred thirty-eight thousand one hundred ninety." Doufen is one million three hundred thirty-eight thousand one hundred ninety.
"The degrees are two hundred ninety-two." The degrees are two hundred ninety-two.
"The remaining degrees are fifty-six thousand nine hundred fifty-four." The remaining degrees are fifty-six thousand nine hundred fifty-four.
Finally, regarding the calculations for Mercury:
"Water: circumference is eleven thousand five hundred sixty-one." The circumference of Mercury is eleven thousand five hundred sixty-one.
"The daily rate for Mercury is one thousand eight hundred thirty-four." The daily value for Mercury is one thousand eight hundred thirty-four.
"The total number of months is one." The total is one month.
"The remaining value of the months is two hundred eleven thousand three hundred thirty-one." The remaining value of the months is two hundred eleven thousand three hundred thirty-one.
"The value obtained from the total month method is two hundred nineteen thousand six hundred fifty-nine." The value obtained from the total month method is two hundred nineteen thousand six hundred fifty-nine.
"The degrees obtained from the daily calculation method are six million eight hundred ninety-four thousand two hundred twenty-nine." The degrees obtained from the daily calculation method is six million eight hundred ninety-four thousand two hundred twenty-nine.
"The larger remainder for the new moon is twenty-nine." The larger remainder for the new moon is twenty-nine.
"The smaller remainder for the new moon is seven hundred seventy-three." The smaller remainder for the new moon is seven hundred seventy-three.
"The number of days entering the month is twenty-eight." The number of days entering the month is twenty-eight.
"The daily remaining value is six million four hundred one thousand nine hundred sixty-seven." The daily remaining value is six million four hundred one thousand nine hundred sixty-seven.
"The fractional part of the new moon is six hundred eighty-four." The fractional part of the new moon is six hundred eighty-four.
"Doufen is one million six hundred seventy-six thousand three hundred forty-five." Doufen is one million six hundred seventy-six thousand three hundred forty-five.
"The degrees are fifty-seven." The degrees are fifty-seven.
"The remaining degrees are six million four hundred one thousand nine hundred sixty-seven." The remaining degrees are six million four hundred one thousand nine hundred sixty-seven.
"Calculate the days from the beginning of the Yuan to the desired year, multiply by the lunar rate; if the result is a whole number of daily rates, this is referred to as accumulation; if not, the remaining part is called the remainder. Divide the remainder by the lunar rate; if the result is one, the stars aligned in the previous year; if the result is two, they aligned in the two years before; if there is no result, they align in the current year. Subtract the lunar rate from the remainder to get the fraction of a degree. The conjunction of Venus and Mercury: odd numbers indicate morning, while even numbers indicate evening.
This passage describes a complex calendar calculation method involving astronomical observations and mathematical calculations. The specific meaning requires in-depth research in conjunction with the knowledge of the calendar at that time.
First, let's calculate the moon. Multiply the values for each month and the remaining values individually, add the results together; if it is enough for a month, it is counted as one month; if not, it is counted as the remaining fraction of months. Then subtract the accumulated months from the total months; the remaining is the number of months entering the next calendar month. Next, consider the impact of leap months. If there is enough for a leap month, subtract it from the number of months entering the next calendar month, and the remaining value is used within the year, recorded outside astronomical calculations; this describes the method of month accumulation. If there is a leap month transition, adjust it according to the new moon.
Next, calculate the time of the stars aligning with the moon. Multiply the remaining months using the standard method, then multiply the remaining days of the new moon by the method of month accumulation, then add these two results and simplify by the number of conjunctions. If the result is enough for a daily degree, it means the day of the stars aligning with the moon has arrived. If not, the remainder is noted as the daily remainder, recorded outside the calculation of the new moon.
Then calculate the degrees of the weekly days. Multiply the weekly days by the fraction of a degree; count as one degree if it is enough for a daily degree, count the remaining degrees if not, starting from the five stars of the Ox constellation. The above is the method for calculating the alignment of stars."
Next, calculate the time span. Add up the months and the remaining months; if it adds up to a complete lunar month, count it as one year. If not, count it within the same year; subtract the excess if it is enough, and account for leap months. The remaining value will be for the next year; if it forms another complete lunar month, it counts towards the next two years. For Venus and Mercury, add morning to get evening, and add evening to get morning. (This part cannot be translated into colloquial language because it is astronomical terminology.)
Then calculate the remainder of the new moon. Add up the remainder of the new moon and the lunar month; if it exceeds a month, add a remainder of either 29 days or 773 minutes. If it is enough for a daily calculation, subtract from the larger remainder, using the same method as before.
Next, calculate the lunar and daily remainder. Add up the lunar and daily remainder; if the remaining amount is enough for a daily calculation, count it as one day. If the previous new moon remainder is enough for a leap month, subtract one day. If the following new moon remainder exceeds 773, subtract 29 days; if not, subtract 30 days instead. The remaining value will be the new moon day for the next complete new moon.
Finally, calculate the angular degrees. Add up the degrees and the remaining parts of the degrees; if it is enough for a daily calculation, count it as one degree.
Here are the specific data for Jupiter, Mars, Saturn, and Venus:
Jupiter: Retrograde motion for 32 days, 3,484,646 minutes; Direct motion for 366 days; Retrograde motion for 5 degrees, 2,509,956 minutes; Direct motion for 40 degrees (retrograde 12 degrees, actual motion 28 degrees).
Mars: Retrograde motion for 143 days, 973,113 minutes; Direct motion for 636 days; Retrograde motion for 110 degrees, 478,998 minutes; Direct motion for 320 degrees (retrograde 17 degrees, actual motion 303 degrees).
Saturn: Retrograde motion for 33 days, 166,272 minutes; Direct motion for 345 days; Retrograde motion for 3 degrees, 1,733,148 minutes; Direct motion for 15 degrees (retrograde 6 degrees, actual motion 9 degrees).
Venus: Morning retrograde motion in the east for 82 days, 113,908 minutes; Evening retrograde motion in the west for 246 days (retrograde 6 degrees, actual motion 240 degrees); Morning retrograde motion for 100 degrees, 113,908 minutes; Evening retrograde motion (daily motion same as west, retrograde for 10 days, retrograde 8 degrees).
Mercury: It appears in the morning after thirty-three days. It has traveled a total of 6,125,505 minutes. Then, it appeared in the west for thirty-two days. (Subtracting one degree of retrograde, the final calculation is thirty-two degrees.) Then, it moved forward by sixty-five degrees, still totaling 6,125,505 minutes. After that, it appeared in the east. The angle at which it appeared in the east is the same as that in the west; it remained hidden for eighteen days, then it moved back by fourteen degrees.
Calculate the number of degrees Mercury travels each day, plus the remaining degrees, and add the remaining degrees between the stars and the sun. If the remaining degrees equal a full day's worth, start calculating from the complete cycle, so you can determine the position of the stars alongside the sun and the number of degrees they travel. Multiply the denominator of the star's movement by the observed number of degrees; if the remaining degrees can be divided evenly by the daily calculation, the result is counted as one. If the denominator does not divide evenly and exceeds half, also consider it as one. Then add the fraction of its movement; if the fraction reaches a full day's worth of degrees, add one degree. The methods for retrograde and direct motion differ. Multiply the current denominator of movement by the original fraction; if the result equals the original denominator, then that is the current fraction of movement. The calculation of pauses, referring to when the planet stops, continues from the previous steps; subtract for retrograde motion. If the accumulated degrees are not enough, use the constellation "Dou" to adjust the fraction, with the denominator of movement as the basis; the fraction will increase or decrease, mutually restricting each other. Any reference to surplus or fullness indicates a need for precise division; "go" and "divide" both refer to exhaustive division.
As for Jupiter, in the morning, it appears alongside the sun, then it pauses and moves forward. After sixteen days, it has traversed 1,742,323 minutes, while the planet itself has traveled 2 degrees and 323,467 minutes. Then it appears in the east in the morning, behind the sun. When moving forward, it travels at a fast pace, covering eleven minutes out of fifty-eight each day, moving eleven degrees in fifty-eight days. Moving forward again, the speed is slow, traveling nine minutes each day, moving nine degrees in fifty-eight days. It pauses for twenty-five days, then turns. Moving backward, it travels one-seventh of a minute each day; after eighty-four days, it retreats twelve degrees. Pausing again, after twenty-five days, it moves forward, traveling nine minutes out of fifty-eight each day, moving nine degrees in fifty-eight days. Moving forward, it travels at a fast pace, covering eleven minutes each day, moving eleven degrees in fifty-eight days, in front of the sun, it sets in the west in the evening. After sixteen days, it has traversed 1,742,323 minutes, and the planet itself has traveled 2 degrees and 323,467 minutes, then it is back together with the sun. A complete cycle lasts 398 days, totaling 3,484,646 minutes, and the planet itself has traveled 43 degrees and 259,956 minutes.
Sun: In the morning when the Sun meets Mars, Mars goes into hiding. Then it starts moving forward for 71 days, covering a total of 1,489,868 minutes, equivalent to 55 degrees and 242,860.5 minutes. After that, in the morning, it can be seen rising in the east, positioned behind the Sun. While moving forward, it covers a total of 112 degrees over 184 days, moving 23 minutes and 14 minutes each day. The forward speed slows down, covering 48 degrees in 92 days, moving 23 minutes and 12 minutes each day. Then it stops for 11 days. It then starts moving backward, covering 17 degrees in 62 days, moving 17 minutes each day. It stops again for 11 days, then starts moving forward again, covering 48 degrees in 92 days, moving 12 minutes each day. Moving forward again, the speed increases, covering a total of 112 degrees over 184 days, moving 14 minutes each day. At this point, it is in front of the Sun and later slips away into the western horizon in the evening. After 71 days, covering a total of 1,489,868 minutes, equivalent to 55 degrees and 242,860.5 minutes, it meets the Sun again. The entire cycle lasts a total of 779 days and 973,113 minutes, covering a total of 414 degrees and 478,998 minutes.
Saturn: In the morning when the Sun meets Saturn, Saturn goes into hiding. Then it starts moving forward for 16 days, covering a total of 1,122,426.5 minutes, equivalent to 1 degree and 1,995,864.5 minutes. After that, in the morning, it can be seen rising in the east, positioned behind the Sun. While moving forward, it covers a total of 7.5 degrees over 87.5 days, moving 35 minutes and 3 minutes each day. Then it stops for 34 days. It then starts moving backward, covering 6 degrees in 102 days, moving 17 minutes each day. After another 34 days, it starts moving forward again, covering a total of 7.5 degrees over 87 days, moving 3 minutes each day. At this point, it is in front of the Sun and later slips away into the western horizon in the evening. After 16 days, covering a total of 1,122,426.5 minutes, equivalent to 1 degree and 1,995,864.5 minutes, it meets the Sun again. The entire cycle lasts a total of 378 days and 166,272 minutes, covering a total of 12 degrees and 1,733,148 minutes.
Venus, when it meets the sun in the morning, first "goes into hiding," which refers to its retrograde motion. In five days, it moves back four degrees, then it becomes visible in the east in the morning, at which point it is behind the sun. As it continues to retrograde, it moves three-fifths of a degree each day, which totals six degrees after ten days. Next comes the "pause" phase, during which it remains stationary for eight days. Then it "turns," indicating it begins to move forward, albeit at a slow pace, moving forty-six and one-third degrees each day, completing a total of thirty-three degrees over forty-six days. The speed picks up, moving one degree and fifteen ninety-firsts each day, covering a total of one hundred and six degrees in ninety-one days. The speed further increases, moving one degree and twenty-two ninety-firsts each day, completing one hundred and thirteen degrees over ninety-one days, at which point it is once again positioned behind the sun and can be observed in the east during the morning. Finally, it continues its forward motion for forty-one days and five hundred sixty-nine thousand nine hundred fifty-four fifty-sixths of a day, covering a total of fifty degrees and five hundred sixty-nine thousand nine hundred fifty-four fifty-sixths of a degree, before it meets the sun once more. The time from one conjunction to the next totals two hundred ninety-two days and five hundred sixty-nine thousand nine hundred fifty-four fifty-sixths of a day, with Venus traversing the same distance.
When Venus meets the sun in the evening, it first "hides"; this time, it is in direct motion. It travels fifty degrees and fifty-four minutes over five hundred sixty-nine thousand nine hundred fifty-four days, and in the evening, it can be seen in the west, ahead of the sun. It then continues in direct motion, accelerating to one degree and twenty-two minutes per day, covering one hundred thirteen degrees in ninety-one days. The speed slows down again, moving at one degree and fifteen minutes per day, covering one hundred sixty degrees in ninety-one days. Then the speed slows down again, moving at forty-six minutes and thirty-three degrees per day, covering thirty-three degrees in forty-six days. It "pauses" for eight days. Then it begins retrograde motion, moving at five minutes and three degrees per day, moving back six degrees after ten days; at this point, it moves ahead of the sun, and in the evening, it can be seen in the west. The retrograde motion accelerates, moving back four degrees after five days, and then it meets the sun again. From one conjunction to the next, the total duration is five hundred eighty-four days and eleven thousand three hundred ninety-eight days; Venus also covers the same number of degrees.
When Mercury meets the sun in the morning, it first "hides", which indicates it is in retrograde motion. It moves back seven degrees after nine days of retrograde, and then in the morning, it can be seen in the east, behind the sun. It continues to move in retrograde, speeding up, moving back one degree each day. It "pauses" for two days. Then it "spins", indicating it begins direct motion, at a relatively slow speed, moving eight minutes and nine degrees per day, covering eight degrees over nine days. Then the speed increases, moving one degree and a quarter per day, covering twenty-five degrees in twenty days; at this point, it moves behind the sun again, and in the morning, it can be seen in the east. Then it continues in direct motion, covering thirty-two degrees over six hundred forty-one thousand nine hundred sixty-seven days, and then it meets the sun again. From one conjunction to the next, the total duration is fifty-seven days and six hundred forty-one thousand nine hundred sixty-seven days; Mercury also covers the same number of degrees.
Wow, what is this saying? Let me explain it to you sentence by sentence.
The first sentence, "Water: evening with the sun, hidden, following," means that Mercury, when it appears simultaneously with the sun on the western horizon, seems to hide and then starts moving along the ecliptic in the same direction as the Earth. The word "hidden" refers to Mercury moving near the sun, being obscured by the sun's light and difficult to observe. "Following" refers to the direction of Mercury's orbit around the sun, similar to that of the Earth.
Next, "On the sixteenth day, six hundred forty-one million nine hundred sixty-seven minutes, the planet moves about thirty-two degrees and six hundred forty-one minutes, and is seen in the west in front of the sun," this sentence is a bit technical. Translated into plain language, it means: Mercury takes about 16 days (more precisely, 16 days plus a few minutes) to move about 32 degrees (again, precise to the minute). Then, in the evening, you can see it in the west, ahead of the sun.
"Following, swift, moving one and a quarter degrees per day, moving twenty-five degrees in twenty days," this means that when Mercury is moving forward, it moves quickly, covering one and a quarter degrees per day, and in twenty days it covers twenty-five degrees.
"Slowing down, moving eight-ninths of a degree per day, moving eight degrees in nine days," this time Mercury slows down, moving only eight-ninths of a degree per day, taking nine days to cover eight degrees.
"Pausing, not moving for two days," this is even more surprising, as it simply stops for two days, not moving at all.
"Reversing, retrograde, moving back one degree per day, in front of the sun, hiding in the west in the evening," oh, now it’s going in reverse! It starts to move backwards, retreating one degree per day, still in front of the sun, visible in the west in the evening, and then it "hides" again, obscured by the sun's light.
"Retrograde, slowing down, moving back seven degrees in nine days, and coming back together with the sun," when moving backwards, it also slows down, taking nine days to move back seven degrees, and finally comes back together with the sun.
The last sentence, "When they align again, 115 days, 601,200 minutes and 5,505 minutes, the planets follow a similar pattern as well," indicates that from one conjunction of Mercury and the Sun to the next, it takes a total of 115 days (plus a few extra minutes), and other planets follow a similar pattern as well. In summary, this passage details the observations made by ancient astronomers regarding Mercury's orbit, employing specialized ancient astronomical terminology, which can seem somewhat perplexing to modern readers.
