First, we need to calculate the first day of the lunar new year for a specific year. Enter the year you wish to calculate, and then use some methods to determine the "fixed accumulated months" (the total number of months in that year) and the "leap remainder" (the additional days). If the leap remainder exceeds twelve days, that year is a leap year. Next, calculate the total number of days in that lunar year, and then use some methods to determine which day corresponds to the first day of the lunar new year. This part of the calculation is relatively complex; the original text explains: "Insert the year, seek externally, multiply by the fixed accumulated month, and add one for the accumulated years. The result is the fixed accumulated months; the remainder is the leap remainder. If the leap remainder is above twelve, the year has a leap. Use the common method to multiply the fixed accumulated months to get the hypothetical accumulated days; the full day method gives the fixed accumulated days, the remainder is the small remainder. Use the six cycles to subtract the accumulated days to get the large remainder, designate it by the inserted year, calculate externally, the sought year is exactly the first day of the eleventh month."
Next, how do we calculate the first day of the next month? It’s very simple; add 29 days to the number of days from the previous month's first day, and then add a small fraction (the exact amount depends on the situation). If this small fraction exceeds a certain value, this month is a "big month." The original text states: "To find the next month, add the large remainder twenty-nine, the small remainder seven hundred seventy-three; the small remainder full day method follows the large remainder. If the small remainder is six hundred eighty-four or above, that month is large."
Then, we calculate which day is the winter solstice. The method for calculating the winter solstice is similar to calculating the first day of the lunar new year; it also uses some methods to determine which day the winter solstice falls in that year. The original text explains: "Insert the year, seek externally, multiply by the remainder; the full year method gives the large remainder, the remainder is the small remainder. Use the six cycles to subtract it, designate it by the year, calculate externally, the result is the exact date of the winter solstice. For the winter solstice's small remainder, add the large remainder fifteen, the small remainder five hundred fifteen; the full two thousand three hundred fifty-six follows the large remainder, designate as per the method."
Next, determine the number of days in each month, as well as when the waxing moon, full moon, waning moon, and the new moon of the following month will occur. This part also involves some complex calculations, which must be adjusted according to the remainders and certain fixed values. The original text is: "Subtract the remainder from the chapter years, multiply by the years in the year, and a full chapter of the leap year is one month. If it's not complete, then half of the previous value is also considered one, with progress and retreat, without a middle month. Add seven to the larger remainder, and five hundred fifty-seven and a half to the smaller remainder; if the smaller remainder is like the day method from the larger remainder, follow the previous remainder to obtain the waxing moon. Add to get the full moon, add to get the waning moon, and add to get the new moon of the following month. The waxing and full moon are determined by a smaller remainder of four hundred and one or less; multiply by a hundred to obtain the day method. If not complete, seek the division, and calculate the day based on the closest solar term and the night not yet finished."
Next, let's calculate how many days after the winter solstice a solar eclipse will occur. This part requires the use of specific values and calculation methods. The original text is: "Integrate it into the calendar year, and seek the values externally; multiply by the remainder, the full calendar method yields the total, and add up the remainder to make it one. Multiply by the common to obtain the accumulation; the full method is the larger remainder, and if not complete, it is the smaller remainder. The larger remainder is calculated by the calendar; in the outer calculation, there is no sun after the winter solstice. To find the next solar event, add sixty-nine to the larger remainder and sixty-four to the smaller remainder; follow the method from the larger remainder, with no division being the end."
Finally, let's calculate when the stars and the moon appear in the sky and in what position. This part requires some astronomical knowledge and complex calculation methods. The original text is: "Multiply the day by the calendar method, subtract the full week from it, and divide by the calendar method; the result is the degree. Start from five degrees before the Ox, divide by the next constellation; if not complete, then the moon will be positioned at midnight. Seek the next day, add one degree, divide by the Dipper; if the division is small, subtract one degree as the calendar method, and add it. Multiply the month by the day, subtract the full week from it; the remainder is the degree in the full calendar method; if not complete, it is the minute. Follow the above method, then the moon will be positioned at midnight on the new moon. Seek the next month, add twenty-two degrees for a small month, two hundred fifty-eight minutes. For a large month, add one day, thirteen degrees, two hundred seventeen minutes; complete the method to obtain one degree. During the latter part of winter, the moon will be positioned in the constellations Zhang and Xin."
In conclusion, this passage describes an ancient astronomical calendar calculation method. The calculation process is complex and requires a certain level of mathematical and astronomical knowledge to understand.
First, let's discuss how to calculate the new moon (the first day of the lunar calendar) for each month. First, add the number of large divisions to the number of small divisions to get a total. If the total is not enough for a complete cycle, use small divisions to make up the difference. Then, add this total to the midnight degrees of the new moon (the first day of the lunar calendar) to determine the timing of the celestial alignment (when the moon and the sun are at the same longitude) on the new moon.
Next, calculate the new moon for the following month. Add 29 degrees to the previously calculated degrees, with a total of 312 large divisions and 25 small divisions. If the small divisions exceed one cycle, use small divisions to compensate for the large divisions; if the large divisions exceed one cycle, use large divisions to compensate for the degrees, and finally divide by the number of large divisions according to the Dipper, one of the twenty-eight lunar mansions.
To calculate the dates and degrees for the first quarter, full moon (the fifteenth day of the lunar calendar), and last quarter, add 7 degrees, 225 minutes, and 17.5 minutes respectively to the degrees of the new moon (these 7 degrees, 225 minutes, and 17.5 minutes include both large and small divisions, calculated in the same way as before). This will yield the degrees for the first quarter. Following this pattern, you can calculate the degrees for the full moon and the last quarter, as well as the new moon for the next month.
The method for calculating the moon's motion in degrees follows a similar approach. Add 98 degrees, 480 minutes, and 41 minutes to the degrees of the new moon (these 98 degrees, 480 minutes, and 41 minutes include both large and small divisions, calculated in the same way as before) to find the degrees for the first quarter moon. Likewise, continue this pattern to calculate the degrees for the full moon, last quarter moon, and the new moon for the next month.
To calculate the degrees of daylight and nighttime, use the solar calendar (the degrees of the sun's movement) and the lunar cycle (the moon's movement cycle), multiply by the night drips from the water clock during the most recent solar term's night drips, and then divide by 200 to get the daylight degrees. Subtract the sun's movement degrees from the calendar, and the lunar cycle from the moon's movement cycle; the remainder will be the degrees of darkness (night). Add the midnight degrees to both the daylight and darkness degrees to obtain the final degrees.
Set a date for a Yuan year, then subtract the Yuan year from the year you want to calculate, and multiply the remaining years by a cycle rate to get the total eclipses (solar or lunar eclipses). If there is a remainder, add 1. Then multiply the cycle rate by the month to get the total months; if there is a remainder, that is the remaining month. Multiply the leap month (intercalary month) by the remaining years to get the total leap months, then subtract the total months from the total leap months, and if there is still a remainder, begin calculations from the new moon.
To calculate the next solar or lunar eclipse, add five months to the last solar or lunar eclipse. The lunar surplus is 1635 days, and if it exceeds one cycle, it results in a month, which is the full moon (the 15th day of the lunar calendar). Based on the large and small surplus of the winter solstice, if the large surplus is relatively large, use double the large surplus, corresponding to the day of action of the Kan hexagram (one of the eight trigrams). If you add 175 to the small surplus and exceed the cycle of the Qian hexagram (one of the eight trigrams), it corresponds to the day of action of the Zhongfu hexagram (one of the eight trigrams).
To calculate the next hexagram, add six to the large surplus and one hundred three to the small surplus. The four primary hexagrams (Qian, Kun, Kan, Li) are calculated based on the middle date, multiplied by double the small surplus. Set the large and small surplus of the winter solstice, adding 27 to the large surplus and 927 to the small surplus. If it exceeds 2356, use the large surplus to offset, obtaining the day of action of the Earth. Add 18 to the large surplus and 618 to the small surplus to get the day of action of the Wood at the beginning of spring. Add 73 to the large surplus and 116 to the small surplus to get Earth again. Continue your calculations in the order of Earth, Fire, Metal, and Water.
Multiply 12 by the small surplus, and if it exceeds one cycle, you get a Chen (Earthly Branch), starting from Zi (the first of the Earthly Branches). The small surplus for the new moon, first quarter, and full moon needs to be calculated separately. Multiply 100 by the small surplus, and if it exceeds one cycle, you get a quarter of an hour; if there is a remainder, use one-tenth for calculations, and then based on the nearest solar term, start calculating from the night division. If the water level at night is not full, then it is based on the most recent situation.
Let's start by discussing the concept of advance and retreat. Advance and retreat mean adding and subtracting, right? Advance means adding, retreat means subtracting, and the final result is what you calculate. The difference in advance and retreat begins with two parts, reducing by four degrees with each rotation, halving the reduced amount each time, rotating again after three times, until the difference reaches three, going through a total of five degrees of rotation, and finally returning to the initial state.
Next is the topic of the moon's speed. The speed of the moon's movement varies, but overall, its cycle around the Earth is constant. To calculate the specific position of the moon's movement, various values from the cosmos are needed, then the remainder is squared, divided by the total number of weeks, and finally, the specific number of days the moon has traveled can be determined. The change in the speed of the moon's movement is actually a trend that affects the angle of the moon's position each day. By adding the impact of this trend to the moon's daily speed, the specific angle of the moon's movement each day can be calculated. The impact of this trend sometimes increases and sometimes decreases, ultimately leading to the moon's gain and loss. By multiplying the total degree of movement in half a month by a constant, and then dividing by the total number of weeks, the angle of the moon's position on the new moon day can be calculated.
Below is a specific calculation table. As you can see, the angle of the moon's movement each day, the reduction value, the profit and loss rate, the cumulative gain or loss, and the total degree of the moon's movement are all listed.
On the first day, the moon moved 14 degrees and 10 minutes, with a reduction of one, a profit and loss rate of an increase of 22 degrees, and a cumulative gain of 276 minutes. On the second day, the moon moved 14 degrees and 9 minutes, with a reduction of two, a profit and loss rate of an increase of 21 degrees, and a cumulative gain of 22 minutes, totaling 275 minutes. On the third day, the moon moved 14 degrees and 7 minutes, with a reduction of three, a profit and loss rate of an increase of 19 degrees, and a cumulative gain of 43 minutes, totaling 273 minutes... and so on.
Until the sixteenth day, the moon moved 14 degrees and 6 minutes. Here is a special case: the profit and loss rate decreases by 20 degrees, but because the decrease is insufficient, it should be adjusted to increase by 5 degrees. This is because the initially decreased value is 20 degrees, but in reality, the decrease is less than 20 degrees, so 5 degrees need to be added to make up for it. The final cumulative gain is 5 minutes, totaling 234 minutes.
On the 17th, 12 degrees and 8 minutes, three carryovers, the number reduced is 18, the amount reduced is 15, the final result is 236. On the 18th, 12 degrees and 11 minutes, four carryovers, the number reduced is 15, the amount reduced is 23, the final result is 239. On the 19th, 12 degrees and 15 minutes, three carryovers, the number reduced is 11, the amount reduced is 48, the final result is 243. On the 20th, 12 degrees and 18 minutes, four carryovers, the number reduced is 8, the amount reduced is 59, the final result is 246.
On the 21st, 13 degrees and 3 minutes, four carryovers, the number reduced is 4, the amount reduced is 67, the final result is 250. On the 22nd, 13 degrees and 7 minutes, four carryovers, the number of increases and decreases is the same, the amount reduced is 71, the final result is 254. On the 23rd, 13 degrees and 11 minutes, four carryovers, the number of increases and decreases is 4, the amount reduced is 71, the final result is 258. On the 24th, 13 degrees and 15 minutes, four carryovers, the number of increases and decreases is 8, the amount reduced is 67, the final result is 262. On the 25th, 14 degrees and 0 minutes, four carryovers, the number of increases and decreases is 12, the amount reduced is 59, the final result is 266.
On the 26th, 14 degrees and 4 minutes, three carryovers, the number of increases and decreases is 16, the amount reduced is 47, the final result is 270. On the 27th, 14 degrees and 7 minutes, three initial carryovers, plus three Sundays, the number reduced is 19, the amount reduced is 31, the final result is 273. Sunday 14 degrees (9 minutes), fewer than expected carryovers, the number of increases and decreases is 21, the amount reduced is 12, the final result is 275.
Sunday minutes: 3,333.
Weekly void total: 2,666.
Sunday total: 5,969.
Total for the week: 185,039.
Historical week: 164,466.
Fewer major totals: 1,101.
New moon total minutes: 11,801.
Minor totals: 25.
Half week: 127.
This section explains the calculation method, which is quite complex when expressed in modern terms. In simple terms, it involves calculating the date of the new moon (the first day of each lunar month) based on a series of values and operational rules. These numbers represent different astronomical parameters, and complex calculations are required to arrive at the final result. During the calculation, you first multiply the lunar month by the small fraction of the new moon. If the small fraction reaches thirty-one, you deduct it from the large fraction. If the large fraction reaches 164,466, you subtract that amount, and then divide the remainder by the weekly method; the quotient represents the number of days, while the remainder is the day remainder. The day remainder must be recorded separately, and the final result is the date when the new moon is incorporated into the calendar.
To calculate the date of the next month, you add one day to this date; the day remainder is 5,832, and the small fraction is 25. To calculate the dates of the first and last quarters (the fifteenth and thirtieth days of the lunar month), you add seven days to this date respectively; the day remainder is 2,283, and the small fraction is twenty-nine point five. Then, according to the previous rules, convert the fractions into days; if it reaches twenty-seven days, you subtract twenty-seven days, and the remainder becomes the weekly fraction. If it is insufficient to divide, subtract one day and add the weekly void.
Speaking of this calendar calculation, it is indeed very complicated! First, multiply the accumulated surplus and deficit values by the number of weeks to obtain an actual value. Then, use the common number multiplied by the daily surplus and deficit difference, and then multiply by the surplus and deficit rate to adjust the actual value; this is known as the overtime surplus and deficit. Calculate the difference in the moon's motion within a year, multiply it by half the number of weeks to obtain a difference value, and use this difference value to divide to get the surplus and deficit remainder, just like the daily surplus and deficit; the overtime of the new moon (the first day) occurs in the days before and after. The advance and retreat of the first and last quarters (the eighth and twenty-third days of the lunar month) are determined by the large remainder.
Next, multiply the year by the overtime surplus and deficit, and then divide by the difference value to obtain the full meeting number, representing the magnitude of the surplus and deficit. Adjust the positions of the sun and moon daily using the surplus and deficit values; if there is a deficiency, adjust the degrees according to the record method to ultimately determine the positions of the sun and moon. Next, multiply half the number of weeks by the new moon's small remainder, divide by the common number, and subtract this from the day remainder in the calendar. If the remainder is insufficient, add the number of weeks and then subtract that amount, and then go back one day. After going back, add the number of weeks and the fractions to arrive at the lunar calendar value at midnight.
To calculate the second day, go back one day and count up to twenty-seven days. If the remainder of the days fills the fraction of the week, subtract it; if not, add the fractional part of the week to the remainder. The remaining value is the day remainder for the second day of the calendar. Multiply the midnight day remainder by the profit and loss rate; if the result is a whole number multiple of the week, then that’s it; if there is a remainder, use it to adjust the profit and loss balance. If the remainder is insufficient for adjustment, use the week to adjust, which represents the profit and loss at midnight. A complete chapter year represents the degree, while an incomplete one is the fraction. Multiply the total value by the fraction and remainder; if the remainder is sufficient for the fraction of the week, the fraction is enough for the degree of the calendar. Add the profit numbers and subtract the loss numbers to adjust the degrees and remainders at midnight to determine the final degree.
Next, multiply the calendar day remainder by the decay value (a type of numerical value); if the result is a multiple of the week with no remainder, that’s it; if there is a remainder, it indicates the daily decay change. Multiply the fractional part of the week by the decay value, and use the result as a constant. After the calculations are complete, use it to adjust the decay changes. If it reaches the decay value, subtract it and transfer to the decay change of the next calendar. Use the decay change to adjust the turning minutes of the calendar days; the profit and loss fraction indicates the degree of the chapter year's transitions. Multiply the total value by the fraction and remainder, and add the daily midnight fixed degree to obtain the value for the next day. If the result of the calendar calculation is not a multiple of the week, subtract one thousand three hundred and eight, then multiply the total value by it; if it is a multiple of the week, add the remainder of eight hundred and thirty-seven, then add the lesser fraction of eight hundred and ninety-nine, and then add the decay change of the next calendar, then continue calculating as before.
Adjust the profit and loss rate by subtracting or adding the decay change to obtain the changed profit and loss rate, and then use it to adjust the profit and loss at midnight. After the calculations are complete, if the profit and loss is insufficient, subtract it instead and enter the next calendar; add and subtract the remainder as before. Multiply the monthly running fraction by the nightly decline of the nearest solar term, then divide by two hundred to obtain the clear value; subtract the clear value from the monthly running fraction to get the dim fraction. If the fraction is sufficient for the chapter year, it is the degree; multiply the total value by the fraction, and add the midnight fixed degree to obtain the dim and bright fixed degree. If the remainder exceeds half, take it; if not, discard it.
This passage describes the complex methods of ancient calendar calculations. Let’s break it down step by step and explain it in simpler terms.
First, "The lunar calendar has four phases, three transitions, intersecting and dividing the sky, calculated based on the lunar cycle to determine the days of the calendar." This sentence means that based on the four phases of the moon's movement (possibly referring to four different observational data) and the three stages of the moon's movement (possibly referring to new moon, full moon, and half moon), the number of days in each month is calculated.
"Multiplying the total number of days in a year by the number of conjunctions of new moons and full moons gives a value representing how often new moons and full moons occur in a year." This sentence is more abstract, probably saying that multiplying the number of days in a year by the number of new moons and full moons (a cycle from new moon to full moon and back to new moon) provides a value representing the frequency of these conjunctions.
"Multiplying the total by the number of conjunctions, then taking the remainder, which serves as a correction." This sentence is even more difficult to understand, meaning that multiplying a total number (possibly the total number of days in a year) by the number of conjunctions of new moons and full moons, then taking the remainder, and dividing it by the number of conjunctions gives a correction value.
"Calculating daily adjustments based on the lunar cycle, dividing the number of conjunctions by the number of days in a year gives a discrepancy rate." This sentence means that calculating the daily adjustments (possibly referring to minor changes in the number of days) based on the lunar cycle, and dividing the number of conjunctions by the number of days in a year yields a discrepancy rate.
The following are the specific calculation steps, recorded in a table form for each day's "reduction and gain" values. This part involves complex calendar calculations and is not easily translated into spoken language, so the original text is retained:
Yin and Yang Calendar Loss/Gain Rate Combined Number
Day 1: Reduce by 1, Gain 17, Initial Value
Day 2 (with a remainder of 1290, minor difference of 457.) This marks the previous limit
Reduce by 1, Gain 16, Seventeen
Day 3: Reduce by 3, Gain 15, Thirty-three
Day 4: Reduce by 4, Gain 12, Forty-eight
Day 5: Reduce by 4, Gain 8, Sixty
Day 6: Reduce by 3, Gain 4, Sixty-eight
Day 7: Reduce by 3 (Not enough reduction, reverse a loss to a gain, adding one, should reduce by three, for insufficient)
Gain 1, Seventy-two
Day 8: Add 4, Loss 2, Seventy-three
(When the moon travels halfway through its cycle, it should be adjusted for exceeding the limit.)
Day 9: Add 4, Loss 6, Seventy-one
Day 10: Add 3, Loss 10, Sixty-five
Day 11: Add 2, Loss 13, Fifty-five
Day 12: Add 1, Loss 15, Forty-two
Day 13 (with a remainder of 3912, minor difference of 1752.)
This marks the later limit
Add 1 (initially large calendar value, divided by days.) Loss 16, Twenty-seven
On the day of five thousand two hundred and three, subtracting lesser values results in sixteen major and eleven minor calculations, totaling four hundred seventy-three. Elapsed weeks total one hundred seventy-five thousand six hundred sixty-five. The difference rate is one thousand nine hundred eighty-six. The combination of new moons and divisions yields eighteen thousand three hundred twenty-eight. Micro division is nine hundred fourteen. The micro division method totals two thousand two hundred nine. This passage describes some parameters and methods used in calculations, such as "elapsed weeks," "difference rate," "combining new moons and divisions," "micro division," and so on, which are professional terms in ancient calendars that are difficult to explain concisely in modern spoken Chinese.
"By taking the accumulated months from the beginning of the lunar cycle, multiplying by the new moon and combining with the micro division, the micro division follows the combination, and the combination completes a week; those remaining that are less than a completed week are classified as solar calendar dates. Those that complete it are considered lunar calendar dates. The remainder is calculated based on the difference of one day per month, with the sought-after month combined with the new moon entering the calendar, not fully accounting for the remaining days." This passage describes how to determine whether a result is a solar or lunar calendar based on calculations, as well as how to calculate the remainder of days. In short, it involves complex calculations based on various parameters and formulas to ultimately determine the dates in the calendar.
"After adding two days, the remainder is two thousand five hundred eighty, with micro division nine hundred fourteen. Following the method to complete a day, subtracting thirteen, except for the remainder like the division of days. The solar and lunar calendars eventually interconvert, with the calendar coming first and the remainder following, the month progressing in the middle." This passage further explains the calculation results, detailing how to handle the remainder of days and the conversion between solar and lunar calendars.
"Each set of late and early historical full and short calculations, multiplying the number by the small division to get the micro division, adding and subtracting full and short to obtain the remainder of solar and lunar days. If the remainder is insufficient, advancing or retreating days as needed. Multiply the determined remainder by the gain and loss rate, combining the number with the month to determine the additional time." This final passage summarizes the entire calculation process, explaining how adjustments and corrections are made based on different circumstances to accurately determine the date. In conclusion, this passage describes an extremely complex ancient calendar calculation method, which relies on meticulous observation of lunar movements and clever application of mathematical models. It may be difficult for modern individuals to fully understand its meaning without a deep understanding of ancient astronomical calendars.
Let's first talk about how to calculate the new moon, also known as the first day of the month. First, multiply the differential rate by the remainder of the new moon, similar to calculus, to calculate a result. Then subtract this result from the calendar's day remainder. If the subtraction is not sufficient, add four weeks and then subtract one more day. After subtraction, add the resulting days to the fractional part. Simplify the fraction to obtain a simplified fraction, and you will get the calendar date of the new moon.
Next, let's calculate the second day. Add one day; the remainder is 31, and the fractional part is also 31. If the fractional part exceeds the maximum allowable value, subtract the number of weeks in a month. Then add one day and continue this process until reaching the end of the calendar. If the remainder exceeds the number of days in a month, subtract the number of days, which is the start date of the calendar. If the remainder does not exceed the number of days, use it directly, add 2720, and the fractional part is 31, giving you the date of the next calendar.
Then, multiply the total number by the surplus and remainder of the night of the late and fast calendar. If the remainder exceeds half of the week, it will be used as the fractional part. Add the surplus and subtract the reduction, then subtract from the remainder of the yin and yang days. If there is a surplus or deficiency in the remainder of the days, adjust the date with the number of weeks in a month. Next, multiply the determined remainder by the gain and loss ratio. If the result equals the number of weeks in a month, use the comprehensive value of gain and loss as a constant for the night.
Multiply the gain and loss ratio by the number of night leaks of the nearest solar term, then divide by 200 to get the numerical value of the bright hour. Subtract this value from the gain and loss ratio to get the numerical value of the dark hour. Then use the gain and loss number of the night as the constant for the dark and bright. If daylight saving time is in effect, divide the constant for the dark and bright by 12 to get the degrees; one-third of the remainder indicates less, less than one indicates strong, and two less indicate weak. This calculates the degree of the moon leaving the ecliptic. For the solar calendar, subtract the extreme degrees from the solar calendar's position, and for the lunar calendar, add. Positive for strong, negative for weak, add for the same name, subtract for different names. When subtracting, same names cancel each other out, while different names add up, with no mutual cancellation—two strong add, one less subtract, and one weak.
From the year Ji-Chou in the Upper Yuan to the year Bing-Xu in the eleventh year of Jian'an, a total of 7,378 years have passed:
- 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: Wood (Jupiter), Fire (Mars), Earth (Saturn), Metal (Venus), and Water (Mercury). The daily and annual degrees of each star must be simplified to derive the lunar and solar rates. Multiplying the annual chapter by the lunar rate yields the monthly method. The monthly chapter multiplied by the solar rate results in the monthly fraction. Dividing the monthly fraction by the monthly method gives the month number. Multiplying the total number by the monthly method results in the solar method. The constellation fraction multiplied by the lunar rate gives the constellation fraction. (The solar method uses the calendar method multiplied by the lunar rate, so here it also uses fractions.)
The new moon remainders of the five stars: large remainder and small remainder. By applying the total method to multiply by the month number and the solar method to divide it, we obtain the large remainder, while the leftover is the small remainder. Subtract the large remainder from 60 to find the small remainder.
The five stars enter the month and day, along with the day remainder. By applying the total method to multiply by the month remainder and the combined monthly method to multiply by the new moon small remainder, sum them up, simplify using the combined number, and then divide the final result by the solar method to obtain the result.
This text documents data on ancient astronomical calendars, similar to records of astrological calculations. Let's translate this sentence by sentence into contemporary spoken language while trying to maintain the original meaning.
First, the beginning of this text discusses the degrees of the five stars and their remainders. Specifically, it means to first subtract the integer degrees, and the remainder left is the degree remainder. Then, multiply the lunar rate by the remainder, and use the solar method to divide, obtaining the integer degree; the remainder left is the new remainder. If it exceeds the lunar rate, subtract the lunar rate and add the constellation fraction. This part sounds quite complex, somewhat like calculating the trajectory of celestial bodies.
Next, some numerical values are documented, such as "the total number of months in the year is seven thousand two hundred eighty-five," indicating the total number of months used for year recording; "leap month seven," referring to there being seven leap months; "total months two hundred thirty-five," likely indicating there are two hundred thirty-five total months (possibly a special unit of months); "twelve in a year," meaning there are twelve months in a year, which is easy to understand; "total method forty-three thousand twenty-six," "solar method one thousand four hundred fifty-seven," "combined number forty-seven," "lunar days two hundred fifteen thousand one hundred thirty," "constellation fraction one hundred forty-five," the precise meanings of these numbers are difficult to ascertain now; they may refer to parameters in a certain calculation method.
Then, the data for Jupiter was recorded. "Jupiter: Orbital period, 6722; Annual rate, 7341; Synodic month, 13; Lunar surplus, 64801; Synodic month calculation, 127718; Daily rule, 3959258; Major lunar excess, 23; Minor lunar excess, 1307; Entry day of the month, 15; Daily excess, 3484646; Synodic month difference, 150; Dou difference, 974690; Degree, 33; Degree remainder, 2509956." These dense numbers are overwhelming to look at, which are various calculations related to Jupiter's orbit, involving orbital period, annual rate, synodic month, surplus, and so on, with a high level of specialization that is difficult for ordinary people to understand. Next, the data for Mars was recorded, in the same format as the record for Jupiter. "Mars: Orbital period, 347; Annual rate, 7271; Synodic month, 26; Lunar surplus, 25627; Synodic month calculation, 64733; Daily rule, 2006723; Major lunar excess, 47; Minor lunar excess, 1157; Entry day of the month, 12; Daily excess, 973113; Synodic month difference, 300; Dou difference, 494015; Degree, 48; Degree remainder, 1991706." This is also a set of astronomical calculation results. Next, the data for Saturn was recorded. "Saturn: Orbital period, 3529; Annual rate, 3653; Synodic month, 12; Lunar surplus, 53843; Synodic month calculation, 6751; Daily rule, 278581; Major lunar excess, 54; Minor lunar excess, 534; Entry day of the month, 24; Daily excess, 166272; Synodic month difference, 923; Dou value, 51175; Degree, 12; Degree remainder, 1733148." It follows the same pattern, recording various data on Saturn's orbit.
Lastly, the data pertains to Venus, specifically recording the number of conjunctions. "Venus: Pi, 9,222; Day rate, 7,213; Number of conjunctions, 9." This record concludes here, which may be due to incomplete documentation or missing subsequent content. In summary, this document presents the results of ancient astronomical calendar calculations, with a wealth of data and a high level of specialization, making it quite difficult for modern people to understand.
One month has passed; the total is 152,293. Using the conjunction calculation method, the result is 171,418. Using the daily calculation method, the result is 5,313,958. The remainder for the new moon is 25. The remainder of the old moon is 1,129. The day of the month when it begins is the 27th. The day's remainder is 56,954. The virtual remainder is 328. The Dipper's portion is 1,318,190. The degree is 292. The degree's remainder is 56,954.
Venus: Pi is 11,561. Day rate is 1,834. Number of conjunctions is 1. In the next month, the moon's remainder is 211,331. The result of conjunction calculation is 219,659. The result of daily calculation is 6,809,429. The remainder of the new moon is 29. The remainder of the old moon is 773. The day of the month when it begins is the 28th. The day's remainder is 641,967. The virtual remainder is 684. The portion of the Dipper is 1,676,345. The degree is 57. The degree's remainder is 641,967.
First, calculate the values from the previous year and multiply them by Pi; if it can be evenly divided by the daily rate and results in one, it is referred to as an integral conjunction; the portion that cannot be evenly divided is termed the conjunction remainder. Divide the conjunction remainder by Pi; if it can be evenly divided by one, it is a conjunction in the previous year; if it can be evenly divided by two, it is a conjunction in the previous two years; if it cannot be divided evenly, then it is a conjunction in the current year. Subtract Pi from the conjunction remainder to determine the degree portion. For the integral conjunction of Venus and Mercury, odd numbers correspond to the morning, while even numbers correspond to the evening.
Using the number of months and the remaining months to multiply by the total, if the result can be evenly divided by the total number of months, then the month is obtained, and the remainder is the remaining months. Subtract the accumulated months from the total months, and the remaining is the entry months. Then multiply it by the leap month; if this is evenly divisible, you obtain a leap month, which is then subtracted from the entry months. The remainder is deducted from the year; this part is not included in the Tianzheng calculation. It represents the total month. If it is at the transition of the leap month, use the new moon to adjust.
Multiply the remaining months by the common method, multiply the new moon by the remaining small, and then divide by the number of meetings. If the result can be evenly divided by the daily method to obtain one, then it is the entry month. The remaining part is the remaining days; this part is not included in the new moon calculation.
Multiply the days by the minutes; if it can be evenly divided by the daily method to obtain one degree, the remaining part is the remainder. Use the method of the previous five cows to determine the degree.
The above is the method for calculating the entry month. Sum the total months and the remaining months; if it can be evenly divided by the total month, then it means in this year, if it can be evenly divided, subtract that from the total, considering the leap month. The remaining is the next year; if it is full, it will be in the next two years. Gold and water combine to form morning; adding evening to morning yields a new result.
First, let's calculate the lunar phases and their remainders. Add up the lunar phases and the remainder of the new moon; if it exceeds one month, then add another twenty-nine days (large remainder) or seven hundred and seventy-three minutes (small remainder). When the small remainder is full, calculate according to the algorithm of the large remainder; the method is the same as before.
Next, calculate the entry month and remaining days. Add up the entry month and remaining days; if the remainder is enough for one day, then count as one day. If the small remainder perfectly fills the gap during the new moon phase, then subtract one day; if the small remainder exceeds seven hundred and seventy-three, then subtract twenty-nine days; if not exceeded, subtract thirty days. The remaining will be calculated using the subsequent method to determine the entry month.
Finally, sum the degrees and their remainders; if the degree is enough for one day, then count as one degree.
Here are the operating data of Jupiter, Mars, Saturn, Venus, and Mercury:
Jupiter: Dormant (hidden) for 32 days, three hundred and forty-eight thousand four hundred and sixty-six minutes; Appearing (visible) for 366 days; Dormant running five degrees, two hundred and fifty-nine thousand nine hundred and fifty-six minutes; Appearing running forty degrees. (After retrograde twelve degrees, actually running twenty-eight degrees.)
Mars: lurks for 143 days and 973,313 minutes; appears for 636 days; lurks at 110 degrees, 478,998 minutes; appears and moves to 320 degrees. (After retrograding 17 degrees, it actually moves to 303 degrees.)
Saturn: lurks for 33 days and 166,272 minutes; appears for 345 days; lurks at 3 degrees, 173,148 minutes; appears and moves to 15 degrees. (After retrograding 6 degrees, it actually moves to 9 degrees.)
Venus: lurks in the east for 82 days at 11,398 minutes, then appears in the west for 246 days. (After retrograding 6 degrees, it actually moves to 246 degrees.) Lurks at 100 degrees, 11,398 minutes; appears in the east. (The daily degree is the same as in the west, lurks for 10 days, retrogrades 8 degrees.)
Mercury: lurks for 33 days in the morning, 6,012,555 minutes; appears in the west for 32 days. (After retrograding 1 degree, it actually moves to 32 degrees.) Lurks at 65 degrees, 6,012,555 minutes; appears in the east. (The daily degree is the same as in the west, lurks for 18 days, retrogrades 14 degrees.)
First, let's talk about the calculation method. First, subtract the remainder from the daily degree of the celestial body, then add the star's daily degree to the remainder. If the remainder is exactly equal to the daily degree, a complete cycle is obtained, as calculated earlier, to determine the degree difference between the star's position and that of the sun. Then, multiply the star's movement fraction (denominator) by this degree difference, and divide the remainder by the daily degree. If it cannot be divided evenly, add one if it exceeds half; then add this number to the star's movement degree. If the degree reaches the denominator, increment it by one degree. The methods for direct and retrograde movements are different. Multiply the current movement denominator by the previous fraction, and divide by the previous denominator to obtain the current movement fraction. The remainder inherits the previous number; for retrograde, subtract the remainder. If the degree is not sufficient, use the division method to calculate the fraction, using the movement denominator as a proportion. The fraction will increase or decrease, and the front and back must cooperate with each other. In summary, terms like "full," "approximately," and "complete" refer to precise division, while "leave," "reach," and "divide" pertain to exhaustive division.
Next, let's take a look at Jupiter's motion. Jupiter appears in the morning alongside the Sun, then hides away, moving forward, traversing 1,742,323 minutes in sixteen days, with the planet traveling 2 degrees and 3,234,607 minutes of arc. It then appears in the east, trailing the Sun. The forward motion is rapid, covering 11 minutes out of every 58 minutes in a day, and covering 11 degrees of arc in 58 days. Continuing forward, the speed slows down, moving 9 minutes per day, covering 9 degrees of arc in 58 days. It stays still for 25 days before moving again. During retrograde motion, it moves 1 minute out of every 7 minutes per day, retreating 12 degrees in 84 days. It remains stationary again for 25 days and then moves forward, covering 9 minutes out of every 58 minutes per day, and 9 degrees of arc in 58 days. Moving forward again, it accelerates, moving 11 minutes per day, covering 11 degrees of arc in 58 days, appearing in front of the Sun and disappearing into the western sky at dusk. In sixteen days, it covers 1,742,323 minutes, with the planet traveling 2 degrees and 3,234,607 minutes of arc, and then it aligns with the Sun. One cycle ends, totaling 398 days, covering 3,484,646 minutes, with the planet traveling 43 degrees and 2,509,956 minutes of arc.
In the morning, the Sun meets Mars, and Mars then hides away. It then begins to move forward for 71 days, covering 1,489,868 minutes, which is 55 degrees and 2,242,860.5 minutes of arc. After that, people can see it in the east behind the Sun in the morning. During its forward motion, Mars moves 14 out of 23 minutes per day, covering 112 degrees of arc in 184 days. Next, the forward speed slows down, moving 12 out of 23 minutes per day, covering 48 degrees of arc in 92 days. It then stops for eleven days. After that, it moves retrograde, covering 17 out of 62 minutes per day, retreating 17 degrees in 62 days. After stopping for eleven days again, it starts to move forward, covering 12 minutes per day, and 48 degrees of arc in 92 days. Moving forward again, it accelerates, covering 14 minutes per day, and 112 degrees of arc in 184 days. At this point, it moves in front of the Sun and can be seen hiding in the west in the evening. After another 71 days, covering 1,489,868 minutes, which is 55 degrees and 2,242,860.5 minutes of arc, it meets the Sun again. Thus, the entire cycle spans 779 days and 973,113 minutes, covering 414 degrees of arc and 478,998 minutes.
As for Saturn, it conjoins with the sun in the morning and then goes into concealment. It then moves forward, covering 1,122,426.5 minutes in 16 days, which is 1 degree and 1,995,864.5 minutes. In the morning, it can be seen in the east, positioned behind the sun. When moving forward, it travels 3/35 of a degree each day, covering 7.5 degrees in 87.5 days, moving 3 minutes per day. Then it stops and remains stationary for 34 days. After that, it goes retrograde, moving backward by 1/17 of a degree each day, moving back 6 degrees in 102 days. Another 34 days pass, and it starts moving forward again, covering 7.5 degrees in 87 days. At this point, it has moved in front of the sun, and in the evening, it can be seen in the west as it conceals itself. After another 16 days, it covers 1,122,426.5 minutes again, which is 1 degree and 1,995,864.5 minutes, and it conjoins with the sun once more. This entire cycle lasts 378 days and 166,272 minutes, covering 12 degrees and 1,733,148 minutes.
As for Venus, when it conjoins with the sun in the morning, it first "hides," which means it goes retrograde. After 5 days of retrograde, it moves back 4 degrees and can be seen in the east behind the sun. Continuing retrograde, it retreats 3/5 of a degree each day, moving back 6 degrees in 10 days. Then it "pauses," remaining stationary for 8 days. After that, it "rotates," meaning it starts moving forward at a relatively slow pace, covering 33 degrees in 46 days, and then it moves forward. After that, it speeds up, covering 15/91 of a degree each day, covering 166 degrees in 91 days. The speed continues to increase, covering 22/91 of a degree each day, moving back 113 degrees in 91 days, at which point it has moved behind the sun and can again be seen in the east in the morning. Finally, moving forward for 41 days and an additional 56,954 minutes, it covers 50 degrees and 56,954 minutes, and conjoins with the sun again. One conjunction lasts a total of 292 days and 56,954 minutes, and Venus covers the same degree.
When Venus conjoins with the Sun in the evening, it will first "hide," this time moving forward. It moves in direct motion for forty-one days and fifty-six minutes, covering fifty degrees and fifty-nine minutes, and it can be seen in the western sky in front of the Sun in the evening. Then it continues moving forward, speeding up to two degrees and ninety-one minutes each day, and after ninety-one days, it has moved a total of one hundred thirteen degrees. The speed then begins to slow down, moving one degree and fifteen minutes each day, totaling one hundred six degrees over ninety-one days, and then the direct motion slows down. Subsequently, it moves more slowly, covering three degrees and forty-six minutes per day, moving thirty-three degrees in forty-six days. Then it "stays" for eight days. Next, it "revolves," which means it starts to move in retrograde, moving back three-fifths of a degree each day, retreating six degrees in ten days, at which point it has moved ahead of the Sun, making it visible moving retrograde in the western sky during the evening. Continuing in retrograde, it speeds up, retreating four degrees in five days, and finally, it conjoins with the Sun again. Two conjunctions of Venus constitute one cycle, totaling five hundred eighty-four days and one hundred thirteen thousand nine hundred eight minutes, with Venus covering the same angular distance.
As for Mercury, when it conjoins with the Sun in the morning, it will first "hide," indicating it is in retrograde motion. It moves in retrograde for nine days, retreating seven degrees, and then it is behind the Sun, making it visible in the eastern sky during the morning. Continuing in retrograde, it speeds up, retreating by one degree each day. Then it "stays" for two days. Next, it "revolves," which means it starts to move in direct motion, moving at a slow pace of eight-ninths of a degree each day, and after nine days, it has moved a total of eight degrees and resumes direct motion. After that, it speeds up, moving one degree and a quarter each day, covering twenty-five degrees in twenty days, at which point it has moved behind the Sun again, making it visible in the eastern sky during the morning. Finally, in direct motion for sixteen days and six hundred forty-one minutes, it covers thirty-two degrees and six hundred forty-one minutes, and it conjoins with the Sun again. One conjunction totals fifty-seven days and six hundred forty-one minutes, with Mercury covering the same angular distance.
Speaking of Mercury, it sets with the sun and then hides, moving from west to east, covering about 32 degrees and 641,967,500 arcseconds in roughly sixteen days (such precision, wow). Then, in the evening, it can be seen in the western sky, just ahead of the sun. When it moves quickly, it can travel 1.25 degrees per day, covering 25 degrees in twenty days. When it moves slowly, it only covers 0.875 degrees per day, taking nine days to cover 8 degrees. If it stops, it stays still for two days. If it goes retrograde, it retreats one degree in a day, at which point it is ahead of the sun and then hides in the west by evening. When going retrograde, it also moves slowly, taking nine days to retreat seven degrees, ultimately reuniting with the sun. From its conjunction with the sun to the next conjunction, it takes a total of 115 days and 601,250,505.7 minutes; just looking at these numbers makes my head spin!
