Let's first calculate the leap year remainder by subtracting it from the standard year (one year). The remainder is then multiplied by the days in the year. If it totals a complete leap month (the number of days in the leap month), it counts as one month. If not, if it exceeds half, it counts as one month. This calculation method allows for adjustments, avoiding the situation of having no months.
Next, add the large remainder (a larger remainder) 7, and the small remainder, which is 557.5. The small remainder is subtracted from the large remainder based on daily calculations, and the remaining value continues to be calculated using the previous method to determine the first quarter moon. Further additions will enable the calculation of the full moon, last quarter moon, and the new moon of the following month. If the small remainder for the quarter and full moons is below 410, it is multiplied by 100 kè (an ancient unit of time). If it reaches a full day's kè, it counts as one kè; if not, we determine a fraction, and based on the situation of the incomplete kè at night during the recent solar terms, we round up to count as one day.
The calculated results are incorporated into the year count. The previously calculated remainder is multiplied by the year count, and if it meets the legal count of the year (a cycle), it is called accumulation; if there is a remainder, it is added to the accumulation and counted as one. Then, multiply by the cycle to derive the large remainder; if it meets the legal count, we derive the large remainder, and if not, we get the small remainder. Use the large remainder to determine the year count, and then calculate the number of days after the winter solstice (how many days after the winter solstice).
Next, calculate the subsequent number of days; add the large remainder 69, and the small remainder 64. If it reaches its legal number, subtract it from the large remainder; if there is no fraction, this indicates the end of the calculation.
Multiply the legal count of the year by the accumulated days (total days accumulated), and if it reaches the full circle (360 degrees), subtract it. The remainder is then divided by the legal count of the year, and the result is the degree. Starting from the first five degrees of the Ox constellation, divide by the order of the constellations. If it does not reach a full constellation, that is the position of the first day of the celestial calendar at midnight.
For the second day, add one degree and use the Dipper constellation to divide the fraction. If the fraction is small, subtract one degree as the legal count of the year and add it in.
Multiply the month days (the number of days in a month) by the accumulated days; if it reaches a full circle, subtract it. The remainder, if it reaches the legal count of the year, is the degree; if not, it is a fraction. Using this method, we can determine the moon's position at midnight on the first day of the celestial calendar.
For next month, add 22 degrees to the small month and 258 as the fraction; for the large month, add one day, making it 13 degrees and 217 as the fraction. If it reaches the legal number, we obtain one degree. In late winter, the moon is positioned near the Zhang and Xin constellations.
Using the remainder of the product of the number of years and the number of days since the new moon, if it completes a cycle, it is a major fraction; if not, it is a minor fraction. Subtract the major fraction from the minutes counted since midnight of the first day, and if it completes the annual cycle, subtract it from the degrees. Following the previous algorithm, you can calculate the position of the conjunction of the sun and moon on the first day of the month.
To calculate the next month, add 29 degrees, 312 major fractions, and 25 minor fractions. If the minor fraction completes a cycle, subtract it from the major fraction. If the major fraction completes the annual cycle, subtract it from the degrees. Divide by the major fraction.
Calculate the position of the first quarter moon, add 7 degrees from the conjunction, 225 minutes, and 17.5 minor fractions. Following the previous algorithm, you can calculate the position of the first quarter moon. From there, you can also find the positions of the full moon, last quarter moon, and the conjunction for the following month.
Calculate the position of the first quarter moon, add 98 degrees from the conjunction, 480 major fractions, and 41 minor fractions. Following the previous algorithm, you can calculate the position of the first quarter moon. From there, you can also find the positions of the full moon, last quarter moon, and the conjunction for the following month.
This text outlines an ancient method for calculating astronomical calendars, and it's quite technical. Let's explain it in modern language sentence by sentence.
First, it explains how to determine the lengths of day and night. "Calculate the duration of daylight and nighttime" means calculating how long the day and night are. It says to use the number of days, months, and solar terms to calculate. Two hundred time units make up a "bright unit" (daytime), and the rest is the "dark unit" (nighttime). Then add the time of day and night to midnight, and you can calculate the specific time. This is similar to how we use calendars and clocks today, although the method is a bit more complicated.
Next, it explains how to calculate the "return year" (presumably referring to the tropical year) and "return month" (presumably referring to the lunar month) in the calendar. "Assuming the starting year, calculate the return year..." This part talks about subtracting the starting year from the past years, and then using some ratio coefficient (return rate) to calculate the accumulated years and months. If there is a remainder, it is added to the next year or month. "Using the leap year to multiply the remaining years, completing the leap year is the accumulated leap year" refers to the calculation of leap years, calculating the leap month according to the rules of leap years, and then subtracting it from the accumulated months, leaving the remaining months of the year. This part is about calculating leap years and adjusting the calendar.
"To predict eclipses, add five months; if the remaining month exceeds 1635, you will receive an extra month, at which point it is the full moon." This part discusses the prediction of lunar eclipses. "Due to the large and small surplus values around the winter solstice, use them for calculating specific auspicious days..." This section begins to delve into deeper calendar calculations, combining the theories of yin, yang, and the five elements, using certain values (large surplus, small surplus) before and after the winter solstice to deduce specific days, such as "days for the Kan event" and "days for the Zhongfu event," which may be related to agriculture or rituals. "To seek the next hexagram, add six to the large surplus and one hundred three to the small surplus..." Continuing to use the large and small surplus to calculate hexagrams and auspicious days, while also incorporating the four solstices (spring equinox, summer solstice, autumn equinox, winter solstice) into the calculations. "Using the winter solstice's large and small surplus values, add twenty-seven to the large surplus and nine hundred twenty-seven to the small surplus..." Continuing to use the large and small surplus from the winter solstice and other values to calculate the days for the five elements: earth, wood, fire, metal, and water. "Multiply twelve by the small surplus to obtain a specific time, counting from the Rat..." Using the small surplus to calculate the heavenly stems and earthly branches (Chen), starting the count from the Rat. "Multiply one hundred by the small surplus to obtain a specific moment; if the calculation is not complete, look for the minutes..." Using the small surplus to calculate time, precise to moments and minutes. "To account for advancement and retreat, advancement adds and retreat subtracts the obtained..." This section discusses the application of advancement and retreat methods in calendar calculations, similar to what we now refer to as error correction. "The moon's motion varies in speed, and the weekly progression is consistent..." This part describes the changes in the moon's speed and how to calculate the moon's orbital period and speed. "Half of the small week multiplied by the common method, as in common numbers, reduces by the lunar week to calculate the new moon's time." This sentence calculates the specific time of the new moon (the first day of the lunar month). Finally, "The sun's rotation degrees and minutes, showing fluctuations, gains, and losses, the moon's movement in minutes, fourteen degrees per day, ten minutes, one retreat reduces, gains twenty-two, initial surplus two hundred seventy-six." This is a table listing daily degrees, speed changes, error corrections, and other data. In summary, this text describes a very complex ancient astronomical calendar calculation method, involving much knowledge of astronomy, mathematics, and the theories of yin, yang, and the five elements. Although we can now calculate these data more conveniently using modern tools, understanding the ancient calculation methods allows us to better appreciate the wisdom and achievements of ancient astronomers.
On the first day, the water level was 14.9 degrees, the ebb tide decreased by 21 degrees, the flood tide increased by 22 degrees, and the final water level was 275.
On the second day, the water level was 14.7 degrees, the ebb tide decreased by 19 degrees, the flood tide increased by 43 degrees, and the final water level was 273.
On the third day, the water level was 14.4 degrees, the ebb tide decreased by 16 degrees, the flood tide increased by 62 degrees, and the final water level was 270.
On the fourth day, the water level was 14 degrees, the ebb tide decreased by 12 degrees, the flood tide increased by 78 degrees, and the final water level was 266.
On the fifth day, the water level was 13.15 degrees, the ebb tide decreased by 8 degrees, the flood tide increased by 90 degrees, and the final water level was 262.
On the sixth day, the water level was 13.11 degrees, the ebb tide decreased by 4 degrees, the flood tide increased by 98 degrees, and the final water level was 258.
On the seventh day, the water level was 13.7 degrees, the ebb tide decreased, the flood tide increased, and the final water level was 254. This day was a bit special; the record only specified "loss" and "gain 102," without specifying the exact decrease and increase values.
On the eighth day, the water level was 13.3 degrees, the ebb tide increased and then decreased by 4 degrees, and the final water level was 250. This day was the same; the record only specified "loss 4" and "gain 102," without specifying the exact decrease and increase values.
On the ninth day, the water level was 12.18 degrees, the ebb tide increased, then decreased by 8 degrees, and the final water level was 246. Similarly, the record only specified "loss 8" and "gain 98."
On the tenth day, the water level was 12.15 degrees, the ebb tide increased, then decreased by 11 degrees, and the final water level was 243. The record indicated "loss 11" and "gain 90."
On the eleventh day, the water level was 12.11 degrees, the ebb tide increased, then decreased by 15 degrees, and the final water level was 239. The record indicated "loss 15" and "gain 79."
On the twelfth day, the water level was 12.08 degrees, the ebb tide increased, then decreased by 18 degrees, and the final water level was 236. The record indicated "loss 18" and "gain 64."
On the thirteenth day, the water level was twelve degrees six minutes. The ebb tide increased, resulting in a decrease of twenty degrees, and finally, the water level was two hundred and thirty-four. The record reads "loss of twenty" and "gain of forty-six".
On the fourteenth day, the water level was twelve degrees five minutes. The rising tide decreased, resulting in a decrease of twenty-one degrees, and finally, the water level was two hundred and thirty-three. The record reads "decrease by twenty-one" and "gain of twenty-six".
On the fifteenth day, the water level was twelve degrees six minutes. The rising tide decreased, resulting in a decrease of twenty degrees (because the initial decrease was insufficient, the value was adjusted from twenty to five, with a surplus of five degrees). The final water level was two hundred and thirty-four. This means that if the decrease is insufficient, we record a decrease of five degrees instead, and if there is a surplus of five degrees, we note it as an increase. The initial decrease was twenty degrees, hence it was deemed insufficient.
On the sixteenth day, the water level was twelve degrees eight minutes. The rising tide increased, resulting in a decrease of fifteen degrees, and finally, the water level was two hundred and thirty-six. The record reads "gain of eighteen, decrease of fifteen".
On the seventeenth day, the water level was twelve degrees eleven minutes. The rising tide increased, resulting in a decrease of twenty-three degrees, and finally, the water level was two hundred and thirty-nine. The record reads "gain of fifteen, decrease of twenty-three".
On the eighteenth day, the water level was twelve degrees fifteen minutes. The rising tide increased, resulting in a decrease of forty-eight degrees, and finally, the water level was two hundred and forty-three. The record reads "gain of eleven, decrease of forty-eight".
On the nineteenth day, the water level was twelve degrees eighteen minutes. The rising tide increased, resulting in a decrease of fifty-nine degrees, and finally, the water level was two hundred and forty-six. The record reads "gain of eight, decrease of fifty-nine".
On the twentieth day, the water level was thirteen degrees three minutes. The rising tide increased, resulting in a decrease of sixty-seven degrees, and finally, the water level was two hundred and fifty. The record reads "gain of four, decrease of sixty-seven".
On the twenty-first day, the water level was thirteen degrees seven minutes. The rising tide increased, resulting in a decrease of seventy-one degrees, and finally, the water level was two hundred and fifty-four. The record reads "loss of twenty, decrease of seventy-one".
On the twenty-second day, the water level was thirteen degrees eleven minutes. The rising tide increased, resulting in a decrease of seventy-one degrees, and finally, the water level was two hundred and fifty-eight. The record reads "loss of four, decrease of seventy-one".
On the twenty-third day, the water level was 13 degrees and 15 minutes; the high tide increased and then decreased by 67 degrees, and the final water level was 262. The record states "loss of eight, reduction of sixty-seven."
On the twenty-fourth day, the water level was 14 degrees; the high tide increased and then decreased by 59 degrees, and the final water level was 266. The record states "loss of twelve, reduction of fifty-nine."
On the twenty-sixth day, the result of the calculation was 14 degrees and 4 minutes; then, after three initial additions and subtractions, subtract 16 and then reduce by 47, and finally get 270.
On the twenty-seventh day, the result of the calculation was 14 degrees and 7 minutes; after three initial additions, add three Sundays, subtract 19, reduce by 31, and finally get 273.
On Sunday, it was 14 degrees (9 minutes); one addition less, subtract 21, reduce by 12, and finally get 275.
Next, here are some astronomical data: Sunday points, 3,303; Sunday virtual, 2,666; Sunday method, 5,969; through Sunday, 185,039; historical Sunday, 164,466; less big method, 1,101; new moon large points, 11,801; small points, 25; half Sunday, 127.
This data is used to calculate the movement of the moon. The specific method is: first, multiply the lunar month by the new moon large points. If the small points total 31, subtract this from the large points. If the large points are full, subtract the historical Sunday. If the remaining number is divisible by the Sunday method, then we get one day; the remaining part is the remainder of the day. Set aside the day remainder; the remaining result is the combined new moon and historical value that we need.
To calculate the next month, simply add one day to this base; the day remainder is 5,832, and the small points are 25. To calculate the first quarter moon, add seven days separately to this base; the day remainder is 2,283, and the small points are 29.5. Then, convert these points into days using the established method; if the days reach 27, subtract them; the remaining part is treated as Sunday points. If it's not enough to divide, subtract one day and add the Sunday virtual.
Calculate the profit-loss product derived from the previously calculated values, multiply it by the total weekly figure, and then multiply it by the remaining daily fraction; adjust the previous result; this is known as overtime profit-loss. Subtract the monthly travel fraction from the annual cycle, then multiply it by half a week to get a difference, use it to divide the previous result, and obtain the profit-loss size and remaining size. If there is a surplus with the daily method, adjust the new moon and time a few days before and after. The crescent moon moves forward and backward, used to determine the remaining size. Multiply the annual cycle by the overtime profit-loss, then divide it by the difference; the resulting total is the size of the profit-loss. Add the profit-loss values to the current day and month positions; if there is still a surplus, adjust the degree based on the calendar method, and finally determine the position and degree of the day and month. Multiply half a week by the remaining crescent moon, then divide by the remaining daily fraction; if the result is not enough, add the weekly method and then subtract, followed by subtracting one day. After subtracting, add the week and its fractions to determine the time of midnight in the calendar. To calculate the next day, simply add one day; if the remaining fraction exceeds the weekly fraction, subtract the weekly fraction. If it does not reach the weekly fraction, add the week void; the remaining is the remaining fraction of the next day into the calendar. Calculate the remaining segment of the calendar during the night, multiply it by the profit and loss rate; similar to dividing the weekly total by a specific number to yield 1, the remainder is the remainder. Then continuously accumulate the surplus and deficit with the profit and loss method; the remainder stays constant. Use the total as the divisor to divide the remainder; this is the method of midnight profit-loss. Treat a year as a unit; the remaining part is a fraction. Multiply the total by the fraction and the remainder, handle the remainder according to the weekly total; when the fraction is full, use the unit to handle the total, add the surplus, subtract the initial total and remainder of midnight, and get the final total. Next, multiply the remaining days in the calendar by the column decline factor; similar to dividing the weekly total by a specific number to yield 1, the remainder is the remainder, allowing you to understand the daily decline trends. Multiply the vacant portion of the weekly total by the column decline factor; use the weekly total as a constant, and after the calendar calculation, add it to the decline number; subtract when the decline number is full, and convert it to the decline number of the next calendar.
Using the decay value, add or subtract the daily increments in the calendar, and the surplus or deficit of fractions gives the degrees of deviation for the year. Multiply the total by the fractions and any remainders, then add the daily increments and the degrees determined at night to get the degrees for the next day. After the calendar is calculated, if the number of days is less than a week, subtract 138 and then multiply the total by 138; if the number of days exceeds a week, add the remainder of 837, then divide by the smaller number 899, and add the decay value for the next calendar, and calculate as before.
Subtract or add the profit and loss rate from the decay value to get the changing profit and loss rate, then use this changing profit and loss rate to calculate the increase or decrease at midnight. After the calendar is calculated, if there is a deficit in profit and loss, reverse the operation by subtracting and transition to the next calendar, adding or subtracting the remainder as in the previous numbers.
Multiply the monthly running fractions by the number of night watches of the recent solar terms; one two-hundredth represents the bright fraction. Subtract the monthly running fractions from it to get the dim fraction. The fractions, like a year, are units; multiplying the total by the fraction and adding the degrees determined at midnight gives the degrees determined for both dim and bright. If the remainder exceeds half, round it up to the nearest whole number; if not, discard it.
There are four tables for months, and three paths for discrepancies, which are distributed interleaved across the days. Use the monthly rate to divide and get the number of days in the calendar. Multiply the weekly days by the lunar-solar conjunction, to get the fraction of the lunar-solar conjunction. Multiply the total by the conjunction number; the remainder is like dividing the conjunction month number by 1 to get the retreat fraction. Based on the month and week, derive the fraction added each day. Dividing the conjunction month number by 1 gives the difference rate.
Yin and Yang calendar, decay, profit and loss rate, and combined number.
Day one: subtract one, gain seventeen, initial.
Day two (limited to a remainder of one thousand two hundred and ninety, minor fraction four hundred and fifty-seven.) This is the previous limit.
Subtract one, gain sixteen, making it seventeen.
Day three: subtract three, gain fifteen, making it thirty-three.
Day four: subtract four, gain twelve, making it forty-eight.
Day five: subtract four, gain eight, making it sixty.
Day six: subtract three, gain four, making it sixty-eight.
Day seven: subtract three (if insufficient, reverse loss to addition, meaning gain one; when subtracting three, it is insufficient).
Gain one, making it seventy-two.
Day eight: add four, lose two, making it seventy-three.
(If the limit is exceeded, it indicates that the month has completed half a week, and the degrees have exceeded the limit, so it should be reduced.)
Day nine: add four, lose six, making it seventy-one.
On the first day, after adding three days and subtracting ten days, sixty-five days remain.
On the second day, after adding two days and subtracting thirteen days, fifty-five days remain.
On the third day, add one day and subtract fifteen days, leaving forty-two days remaining. On the fourth day (the limit is 3,912, and the differential is 1,752). This is the deadline. Add one day (the start of the calendar, divided by day). Subtract sixteen days, and there are twenty-seven days left. Divide by day (5,203 days), subtract sixteen days from the lesser, and there are eleven days left. Lesser Law, four hundred seventy-three. Calendar cycle: 175,605. Difference rate: 1,986. New moon and conjunction: 18,328. Differential: 914. Differential law: 2,209. Subtract the accumulated months from the previous year's total, then multiply by the new moon and conjunction separately. Subtract the differential from the new moon and conjunction when it reaches its limit; subtract from the cycle when it reaches its limit. The remaining days that do not complete a full calendar cycle are counted towards the solar calendar; subtract when it reaches the limit, and the remaining days are entered into the lunar calendar. Each month is counted as a day, and the remaining days are counted separately. The sought-after month's new moon and conjunction are entered into the calendar, and any remaining days are counted as leftover days. Add two days, with two thousand five hundred eighty days remaining; the differential is nine hundred fourteen. Calculate the days using the method; subtract when it reaches thirteen, and manage the remainder using the divide-by-day method. The lunar and solar calendars alternate in this way, with the days entered into the calendar before the deadline, and the remaining days after the deadline. This is how the months proceed. Calculate the late and early historical surpluses and deficits, multiply by the count to get the differential, add the surplus and subtract the deficit, adjusting the days if necessary. Multiply the determined surplus by the profit and loss rate; if a month is treated as a week, use the profit and loss rate as an additional constant. Multiply the difference rate by the new moon's remainder; if it results in one, similar to the differential method, subtract from the historical days remaining; if not enough, add the month and then subtract, reducing by a day. Reduce the divide by day, add its division, simplify the count to the differential, which is the new moon entering the calendar at midnight. On the second day, add one day, resulting in a remainder of thirty-one days and a division of thirty-one. Subtract the division from the remainder like the count; subtract the week when the remainder is full, then add one day. The calendar is completed. Subtract the division when the remainder is full, marking the start of the calendar. If not full, keep the remainder, add the remaining two thousand seven hundred two, with a division of thirty-one, entering the next calendar.
Multiply the total number of cycles by the duration of the night to determine the remaining time. When the remainder is complete, it will be treated as a fraction. Use the sum of the full and the reduced values to adjust the remaining yin and yang days. If the remaining days are insufficient, adjust the total days in the lunar month. Multiply the calculated remaining days by the profit and loss ratio. If one lunar month is considered equivalent to one day, use the profit and loss ratio as the constant for nighttime calculations. Multiply the profit and loss ratio by the recent solar terms at night, with one two-hundredth of this value representing brightness. Subtract this value from the profit and loss ratio to determine the dimness, and use the profit and loss as the constant for dimness and brightness.
Now, let’s discuss how to calculate the days. If you know whether a certain day is in the lunar calendar or the solar calendar, divide twelve by the number representing the calendar type, then divide the remainder by three. A remainder of zero or one indicates a weak position, while a remainder of two indicates a strong position. This result indicates the angle of the moon relative to the ecliptic. For the solar calendar, subtract the ecliptic degree of the sun from the maximum angle; for the lunar calendar, add the maximum angle. A strong value is represented as positive, while a weak value is represented as negative. Add the positives and negatives together, combine two strong values and then subtract the weak value. Subtraction follows the same principle: subtract the same signs and add the different signs; there is no cancellation between values.
From the Ji Chou year in the Shang Yuan era to the Bing Xu year in the Jian An period, a total of seven thousand three hundred and seventy-eight years have passed. The specific years are:
- 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
The five elements are associated with the following celestial bodies: wood corresponds to the year star, fire to Mars, earth to Saturn, metal to Venus, and water to Mercury. Calculate the movement cycle and daily angles of each star to find the weekly and daily rates. Multiply the total cycles by the weekly rate to derive the monthly calculation; multiply the passage by the monthly rate to get the monthly fraction; divide the monthly fraction by the monthly method to get the number of months; multiply the number of months by the monthly method to get the daily method; multiply the Dipper constellation by the weekly rate to get the Dipper. (The daily method is multiplied by the record method by the weekly rate, so here we use fractions for multiplication.)
Next, calculate the major and minor remainders of the new moon for the five stars. (Use the common method to multiply by the number of months, then divide by it using the day method to get the major remainder; the remainder that cannot be evenly divided is the minor remainder. Then subtract the major remainder from sixty.) Then calculate the entry date of the five stars. (Multiply the month remainder by the common method, multiply by the new moon minor remainder using the combined month method, add them together, simplify the result, then divide by the day method to obtain the result.) Finally, calculate the degrees and their remainders for the five stars. (Subtract the excess to obtain the degree remainder, multiply it by the orbital period, simplify using the day method to get the degrees; the remainder that cannot be divided evenly is the degree remainder; subtract if it exceeds the orbital period, and add the Dou Fen.)
The Ji Yue value is 7,285, Zhang Run is 7, Zhang Yue is 235, Suizhong is 12, Tongfa is 43,026, Rifafa is 1,457, Huishu is 47, Zhoutian is 215,130, Dou Fen is 145. Taking Jupiter as an example: the orbital period is 6,722, the solar period is 7,341, the combined month number is 13, the month remainder is 64,801, the combined month method is 127,718, the day method is 3,959,258, the new moon major remainder is 23, the new moon minor remainder is 1,307, the entry date is 15, the day remainder is 3,484,646, the new moon virtual fraction is 150, and Dou Fen is 974,690. The degrees are 33, and the degree remainder is 2,500,956. For Mars: the orbital period is 3,407, the solar period is 7,271, the combined month number is 26, the month remainder is 25,627, the combined month method is 64,733, the day method is 2,006,723, the new moon major remainder is 47, the new moon minor remainder is 1,157, the entry date is 12, the day remainder is 973,113, the new moon virtual fraction is 300, and Dou Fen is 494,015.
Next is the calculation for Earth: the degree is forty-eight, and a remainder of one hundred ninety-nine thousand one hundred seventy-six. The circumference is three thousand five hundred twenty-nine, the daily value is three thousand six hundred fifty-three, totaling twelve months, and a remainder of fifty-three thousand eight hundred forty-three, totaling sixty-seven thousand fifty-one for the monthly method, two million seven hundred eighty-five thousand eight hundred one according to the daily method, with a remainder of fifty-four for the major lunar month, and a remainder of five hundred thirty-four for the minor lunar month, with twenty-four days per lunar month, and a remainder of one hundred sixty-six thousand two hundred seventy-two for the days, with nine hundred twenty-three for the apparent new moon, and fifty-one thousand seven hundred five for the constellations.
Then is the calculation for Metal: the degree is twelve, and a remainder of one hundred seventy-three thousand three hundred forty-eight. The circumference is nine thousand two hundred twenty-two, the daily value is seven thousand two hundred thirteen, totaling nine months, and a remainder of fifty-two thousand two hundred ninety-three, totaling one hundred seventy-one thousand four hundred eighteen for the monthly method, five hundred thirty-one thousand three hundred fifty-eight according to the daily method, with a remainder of twenty-five for the major lunar month, and a remainder of one thousand one hundred twenty-nine for the minor lunar month, with twenty-seven days per lunar month, and a remainder of fifty-six thousand nine hundred fifty-four for the days, with three hundred twenty-eight for the apparent new moon, and one hundred thirty-one thousand eight hundred ninety for the constellations.
Lastly is the calculation for Water: the degree is two hundred ninety-two, and a remainder of fifty-six thousand nine hundred fifty-four. The circumference is eleven thousand five hundred sixty-one, the daily value is one thousand eight hundred thirty-four, totaling one month, and a remainder of twenty-one thousand one hundred thirty-one, totaling twenty-one thousand nine hundred sixty-five for the monthly method, six hundred eighty-nine thousand four hundred twenty-nine according to the daily method, with a remainder of twenty-nine for the major lunar month, and a remainder of seven hundred seventy-three for the minor lunar month, with twenty-eight days per lunar month, and a remainder of six hundred eighty-four for the apparent new moon. This text records a series of astronomical calculations, including degrees, remainders, circumferences, daily values, etc., the specific meanings of which require knowledge of the historical calendar.
First, let's calculate the total, which is one million six hundred seventy-six thousand three hundred forty-five. The degree is fifty-seven, and a remainder of six hundred forty-one thousand nine hundred sixty-seven.
Next, multiply the value from the previous year by pi to obtain an integer, referred to as the "product sum," and the remainder is the "sum remainder." Divide the product sum by pi; if the result is 1, it corresponds to the conjunction of the previous year; if it is 2, it corresponds to the conjunction of the two previous years; if there is no quotient, it is the conjunction of the current year. Then, subtract pi from the sum remainder to obtain degrees and minutes. For Venus and Mercury, odd numbers indicate morning, while even numbers indicate evening.
Then, multiply the month and month remainder by the product sum respectively; if the result is an integer multiple of the "month sum method," calculate according to the month, and the remainder is the month remainder. Subtract the product month from the recorded month; the remainder is the recorded month. Multiply the leap month by the recorded month; if the result is a multiple of the leap month, subtract one leap month, then subtract from the year; this part is termed the conjunction of the month outside celestial calculations. If there is a leap month transition, adjust using the new moon.
Multiply the common method by the month remainder, multiply the month sum method by the new moon remainder, then simplify the result; if it is a multiple of the day method, it corresponds to the conjunction of the stars and the month. If it is not, the remainder becomes the day remainder; this part is calculated outside the new moon.
Multiply the degrees by the minutes; if the result is a multiple of the day method, it corresponds to one degree, and the remainder is processed using the method corresponding to the previous five cows.
The above is the method of finding the conjunction of stars. Next, add up the months and month remainders; if the result is an integer multiple of the month sum method, it is counted as a month. If it does not complete a full year, it is counted as the current year; if it is full, subtract, considering the leap month; the remainder is the next year; if it is full again, it corresponds to the next two years. For Venus and Mercury, morning added is evening, and evening added is morning.
Then, add the size of the new moon remainder and the size of the month remainder; if it exceeds a month, add twenty-nine to the large remainder and seven hundred seventy-three to the small remainder. If the small remainder reaches the threshold of virtual minutes, subtract from the large remainder, following the previous method.
Add the recorded month and the day remainder; if the result is a multiple of the day method, it corresponds to one day. If the previous new moon remainder is full of virtual minutes, subtract one day; if the later small remainder exceeds seven hundred seventy-three, subtract twenty-nine days; if it is not full, subtract thirty days; the remainder is the recorded day of the subsequent conjunction.
Finally, add up the degrees and the remainder of the degrees; if the result is an integer multiple of the day method, it corresponds to one degree. Jupiter:
For thirty-two days, it remained hidden, moving a total of three million four hundred eighty-four thousand six hundred forty-six units. It appeared for three hundred sixty-six days of visibility. It remained hidden five times, moving a total of two million nine hundred fifty-six thousand units. It appeared for forty days (excluding twelve days of retrograde, with a fixed movement of twenty-eight days). Mars: It remained hidden for one hundred forty-three days, moving a total of ninety-seven thousand three hundred thirteen units. It appeared for six hundred thirty-six days. It remained hidden for one hundred ten days, moving a total of four hundred seventy-eight thousand nine hundred ninety-eight units. It appeared for three hundred twenty days (excluding seventeen days of retrograde, with a fixed movement of three hundred three days). Saturn: It remained hidden for thirty-three days, moving a total of sixteen thousand six hundred seventy-two units. It appeared for three hundred forty-five days. This text describes ancient astronomical calculation methods, explained in modern spoken Chinese as follows: First, it describes the observed behavior of a celestial body (for now, we won't discuss which celestial body) as seen from Earth. It first remains hidden three times at a certain position on Earth, moving a total of one million seven hundred thirty-one thousand four hundred forty-eight units, then appears for fifteen degrees (subtracting six degrees of retrograde, the actual movement is nine degrees). Next, it details the movement of Venus. Venus appears in the east in the morning and lasts for eighty-two days, moving a total of one hundred eleven thousand nine hundred eight units. Then it appears in the west, lasting for two hundred forty-six days (subtracting six degrees of retrograde, the actual movement is two hundred forty degrees). In the morning, it appears in the east, with the same movement as in the west, remains hidden for ten days, and then retreats eight degrees. The movement of Mercury is similar. It appears in the east in the morning, lasting for thirty-three days, moving a total of six hundred twelve thousand five hundred five units. Then it appears in the west, lasting for thirty-two days (subtracting one degree of retrograde, the actual movement is thirty-two degrees). It remains hidden for sixty-five days, moving a total of six hundred twelve thousand five hundred five units, and then appears in the east. The observed movement in the east mirrors that in the west, remains hidden for eighteen days, and then retreats fourteen degrees.
This passage explains the method of calculating the movement of celestial bodies. It describes how to calculate the time and degree of appearance of a celestial body based on its hidden days, remaining degrees, and the degree remainder of the celestial body and the sun. The calculation involves the speed of the celestial body, retrograde and direct motion, waxing and waning, and division.
Finally, it presents the movement of Jupiter. Jupiter appears in the morning with the sun, then hides, moves in a direct motion for sixteen days, covering a distance of 1,742,323 units. The planet itself moves 2 degrees and 323,467 units, then reappears in the east after the sun. Its speed varies, sometimes pausing or even moving retrograde. It summarizes the movement of Jupiter over one complete cycle (398 days), covering a total distance of 3,484,646 units, with the planet itself moving 43 degrees and 250,956 units.
In conclusion, this text describes how ancient astronomers predicted the orbits of planets through complex calculations that involve numerical computations and astronomical concepts. While we may not fully grasp the methods used in ancient times, we can appreciate the ancient astronomers’ dedication to exploring the universe and their rigorous scientific methods.
When the sun rises in the morning, Mars and the sun are together; then Mars "lies low" and starts moving in direct motion. It moves in direct motion for 71 days, covering 1,489,868 minutes, equivalent to 55 degrees and 242,860.5 minutes. Then, in the morning, Mars becomes visible in the east, positioned behind the sun. While moving in direct motion, Mars moves 14/23 of a minute each day, covering 112 degrees in 184 days. Then its forward speed slows down, moving 12/23 of a minute each day, covering 48 degrees in 92 days. It then stops for 11 days. After that, it starts moving in retrograde, covering 17/62 of a minute each day, moving back 17 degrees in 62 days. It stops for another 11 days, then resumes direct motion, covering 12 minutes each day and 48 degrees in 92 days. Its forward speed increases, moving 14 minutes each day and covering 112 degrees in 184 days. At this stage, it passes in front of the sun, so it can be seen in the west at night. After 71 days, covering 1,489,868 minutes, equivalent to 55 degrees and 242,860.5 minutes, it reunites with the sun. Overall, this cycle lasts 779 days and 973,113 minutes, covering 414 degrees and 478,998 minutes.
As for Saturn, it also appears with the sun in the morning; then it "lies low" and starts moving in direct motion. It moves in direct motion for 16 days, covering 1,122,426.5 minutes, equivalent to 1 degree and 1,995,864.5 minutes. Then in the morning, it can be seen in the east, positioned behind the sun. While moving in direct motion, it moves 3/35 of a minute each day, covering 7.5 degrees in 87.5 days. Then it stops for 34 days. Subsequently, it moves in retrograde, covering 1/17 of a minute each day, moving back 6 degrees in 102 days. After another 34 days, it starts moving in direct motion again, covering 3 minutes each day and 7.5 degrees in 87 days. At this juncture, it passes in front of the sun, so it can be seen in the west at night. After 16 days, covering 1,122,426.5 minutes, equivalent to 1 degree and 1,995,864.5 minutes, it reunites with the sun. Overall, this cycle lasts 378 days and 166,272 minutes, covering 12 degrees and 1,733,148 minutes.
Venus, when it meets the sun in the morning, first goes retrograde, retreating four degrees in five days, and then it can be seen in the east, appearing behind the sun. It continues to move retrograde, covering 0.6 degrees each day, retreating six degrees over ten days. Then it "stays" for eight days, remaining stationary. Next, it "rotates," meaning it begins to go direct, moving slowly, covering thirty-three degrees in forty-six days, before starting to go direct. Then it speeds up, moving one degree and ninety-one minutes each day, covering one hundred sixty degrees in ninety-one days. It accelerates further, moving one degree and twenty-two minutes each day, covering one hundred thirteen degrees in ninety-one days, at which point it is behind the sun, appearing in the east in the morning. It continues to move direct, covering fifty-six thousand nine hundred fifty-four minutes of arc in forty-one days, and the planet also travels fifty degrees and fifty-six thousand nine hundred fifty-four minutes of arc, eventually meeting the sun again. One conjunction lasts for two hundred ninety-two days and fifty-six thousand nine hundred fifty-four minutes, with the planet covering the same number of degrees.
When Venus meets the sun in the evening, it first goes retrograde, covering fifty-six thousand nine hundred fifty-four minutes of arc in forty-one days, and the planet travels fifty degrees and fifty-six thousand nine hundred fifty-four minutes of arc, appearing in front of the sun in the west. It continues to move direct, speeding up, covering one degree and twenty-two minutes each day, covering one hundred thirteen degrees in ninety-one days. Then it slows down, moving fifteen minutes each day, covering one hundred sixty degrees in ninety-one days, while continuing to move direct. It slows down further, covering thirty-three degrees in forty-six days. Then it "stays," remaining stationary for eight days. It then "rotates," beginning to go retrograde, covering 0.6 degrees each day, retreating six degrees over ten days, at which point it is in front of the sun, appearing in the west in the evening. It goes retrograde, speeding up, retreating four degrees in five days, and finally meets the sun again. The two conjunctions count as one cycle, totaling five hundred eighty-four days and one hundred thirteen thousand nine hundred eight minutes, with the planet covering the same number of degrees.
Mercury, when it meets the sun in the morning, first is in retrograde, moving back seven degrees over nine days. Then it becomes visible in the east in the morning, positioned behind the sun. It continues its retrograde motion, speeding up to retreat one degree each day. It then "pauses," remaining stationary for two days. Then it "turns" and begins to move forward slowly, covering eight-ninths of a degree each day, traversing eight degrees over nine days, and starting to move forward. Then it speeds up, moving one and a quarter degrees per day, covering twenty-five degrees over twenty days, at which point it appears behind the sun in the eastern morning sky. It continues to move forward, traversing 641,967 minutes in sixteen days, with the planet also moving thirty-two degrees and 641,967 minutes, and finally meets the sun again. In total, one conjunction takes fifty-seven days and 641,967 minutes, with the planet covering the same distance.
Mercury, when it sets with the sun, is referred to as "retrograde." Then, it moves forward, covering thirty-two degrees and 641,967 minutes in sixteen days (that's an impressively precise figure!). At this time, you can see it in the west in the evening, and it is in front of the sun. When in direct motion, it travels relatively quickly, covering one and a quarter degrees per day, twenty-five degrees in twenty days. If it moves slowly, covering only seven-eighths of a degree per day, it takes nine days to cover eight degrees. If it stops moving, this is referred to as "pausing," remaining stationary for two days.
Then, it will also move in retrograde, which means it moves in the opposite direction, moving backward one degree each day. At this time, it can be seen in the west at sunset, positioned in front of the sun. During retrograde motion, it also moves slowly, taking nine days to move back seven degrees, and eventually it reunites with the sun. From one conjunction to the next, it takes a total of one hundred fifteen days and six hundred twelve thousand five hundred five minutes. The motion of Mercury is cyclical in nature. "At sunset, it aligns with the sun, moving forward on the sixteenth day, the planet travels thirty-two degrees in six hundred forty-one thousand nine hundred sixty-seven minutes, and at sunset, it is visible in the west, in front of the sun. Moving forward swiftly, the sun travels one and a quarter degrees; on the twentieth day, it travels twenty-five degrees and moves forward. When delayed, the sun travels eight-ninths; on the ninth day, it travels eight degrees. When stationary, it remains still for two days. During retrograde rotation, it retreats one degree each day; in front of the sun, at sunset, it is visible in the west. During retrograde motion, when delayed, on the ninth day it retreats seven degrees and aligns with the sun." This passage describes the laws governing Mercury's motion; isn't that fascinating?
