Increase the remainder by seven; the smaller remainder is 557.5. Subtract the smaller remainder from the larger remainder each day, following the previous algorithm, to determine the waxing crescent. Then calculate the gibbous moon, the waning crescent, and the next new moon. If the smaller remainder on the day of the gibbous moon is below 410, multiply the smaller remainder by 100. If the result is an integer multiple of the daily value, obtain 1 unit; if not, calculate the decimal part and add it to the remaining uncounted units from the previous solar term night to round up to the daily value.
Insert the year, multiply by the required value; the complete calculation is the product, and add the remainder to the product to get one. Multiply by the communication number; the complete calculation is the large remainder, and the remainder is the smaller remainder. Use the large remainder to determine the year and calculate the time after the winter solstice.
Calculate the next time of sunset by adding 69 to the large remainder and adding 64 to the small remainder. If the result is an integer multiple of the calculation, subtract from the large remainder, with no fractions.
Multiply by the law of the year, subtract the full week, and divide by the law of the year; the result is the angle. Start counting from the fifth degree before the constellation of the Ox, divide by the next constellation; if it does not complete a full constellation, it is the position of the sun at midnight on the day of the winter solstice.
Calculate for the next day by adding one degree and dividing by the Dipper; if the result is less than expected, subtract one degree to adjust the calculation, and then add it.
(Calculate the next day, add one degree to the calculation, divide by the number of minutes; if the number is not enough, subtract one degree and add the value of the years.)
Multiply the day of the month by the week, subtract the integer multiples of the number of days in a week, and the remainder represents the degree; if it cannot be evenly divided, it represents the minute. Following the above method, the position of the moon at midnight on the first day of the month can be determined.
To find the next month, add 22 degrees for a small month, 258 minutes. For a large month, add one day, 13 degrees, 217 minutes, and the method gives one degree. In the latter part of winter, the moon is between the constellations of Zhang and Xin.
(Multiply the age of the chapter by the remaining small month; if the result is a multiple of the full count, you get the large minute; if not, the remainder is the small minute. Subtract the large minute from the half-day of the new moon; if the result is a multiple of the value of the year, subtract from the degree. Following the above method, you can get the time of the conjunction of the sun and moon.)
To find the next month, add 29 degrees, 312 large minutes, 25 small minutes. If the small minutes are a multiple of the full count, subtract from the large minutes; if the large minutes are a multiple of the value of the year, subtract from the degrees, and divide the result by the number of minutes.
To find the position of the first quarter moon, add 7 degrees to the conjunction degree, 225 minutes, 17.5 small minutes; if the large and small minutes and degrees follow the previous calculations, then you can determine the position of the first quarter moon. Continuing with this method, you can also determine the positions of the full moon, last quarter, and the next conjunction.
To find the position of the waxing crescent moon, add 98 degrees to the conjunction degree, 480 large parts, and 41 small parts; calculate its position using the previously mentioned method for conjunctions and degrees. Similarly, calculate the positions of the full moon, last quarter moon, and the next conjunction moon.
Calculate the brightness of the sun and moon; multiply the number of years by the night hours of the nearest solar term, and divide by 200 to get the bright part; subtract the night hours from the number of years to get the dark part. Add the bright and dark parts to midnight and calculate the degree according to the above method.
First paragraph: In the second century BC, to calculate the days, first calculate how many years have passed, multiply the past years by a fixed ratio (the rate), and add any remaining accumulated eclipse (the variation in the length of the sun's shadow throughout the year) to the total. Then multiply this ratio by the number of months to calculate the accumulated months; if there is still a remainder, it is the month remainder. Then multiply the intercalary month (intercalary month rules) by the remaining years to calculate the accumulated intercalary month, subtract the accumulated months from the accumulated intercalary month, deduct the remaining part from the year, and if there is still a remainder, start counting from the starting point of the day.
Second paragraph: To calculate the next solar eclipse, add five months; the month remainder is 1635. When the rate reaches a value equal to one month, then determine based on the waxing or waning of the moon.
Third paragraph: Based on the large and small remainders at the winter solstice, the large remainder is the value of the sun's shadow length exceeding the standard value at the winter solstice, and the small remainder is the part less than the standard value. Twice the large remainder equals the small remainder; this marks the day governed by the Kan hexagram. Add 175 to the small remainder; when it reaches the standard of the Qian hexagram, use the large remainder; this is the day when the Zhong Fu hexagram is in command.
Fourth paragraph: Calculate the next hexagram; each hexagram adds 6 to the large remainder and 103 to the small remainder. For the four positive hexagrams, double the small remainder relative to the midpoint of the cycle.
Fifth paragraph:
Record the large and small remainders from the Winter Solstice, add 27 to the large remainder, add 927 to the small remainder, and when it reaches 2356, use the large remainder to determine the day of the Earthly Branches. Add 18 to the large remainder, add 618 to the small remainder, to determine the day associated with the Wood Earthly Branch for the Beginning of Spring. Add 73 to the large remainder, add 116 to the small remainder, to determine the Earthly Branch. Adding the numerical value of Earth, you get the Fire Branch; the Metal Branch and the Water Branch are calculated in the same way.
The sixth paragraph:
Multiply the small remainder by 12; when it reaches a certain value, you get a Chen (Earthly Branch), starting from Zi (the first Earthly Branch). This is another calculation method for determining the small remainder using the New Moon, First Quarter Moon, and Full Moon (Lunar calendar's first day, fifteenth day, and twenty-third day).
The seventh paragraph:
Multiply the small remainder by 100; when it reaches a certain value, you get a quarter of an hour. If there is a remainder, represent it as one-tenth, then calculate the fraction based on the recent solar terms, starting from midnight. The water continues to flow at night; use the nearest value to describe it.
The eighth paragraph:
Calculate with advance and retreat; add for advance, subtract for retreat; the result is the difference between advance and retreat. The difference between advance and retreat begins at two minutes, decreasing by four degrees each time. The decreasing value halves each time, then multiply by three, until the difference reaches three. After five degrees, it returns to the initial state.
The ninth paragraph:
The movement of the moon is sometimes fast and sometimes slow, repeating in cycles. The cycle (one cycle) is derived from the constant between heaven and earth; the remaining proportion is multiplied by itself, similar to the cycle, calculating the excess week (the part that exceeds one cycle). Subtract from the circle of the week (360 degrees), divide it by the monthly week (the time it takes for the moon to orbit the earth) to get the date (number of days in a month). The speed varies and decays; the trend reflects the patterns of change. Add the decay value to the monthly rate (the number of degrees the moon moves each day) to get the daily rotation degree minute (the number of degrees and minutes moved each day). Add the decay value on both sides to get the profit and loss rate. Add for profit, subtract for loss; it is the accumulation of profit and loss. Multiply half a small week (the time it takes for the moon to orbit half a cycle) by the common method (a fixed value), similar to the common number (a fixed value), then subtract it from the historical week (the time it takes for the moon to orbit a full cycle) to get the New Moon run (the degree of movement of the New Moon).
The tenth paragraph:
The table contains: daily rotation degree minute, column decay, profit and loss rate, profit and loss accumulation, monthly movement minute; specific values can be found in the original text. "One day fourteen degrees, ten minutes retreat, decrease, gain twenty-two, profit initial two hundred seventy-six," and so on; these are specific calculation results that are not translated into contemporary Chinese.
On the first day, the water level decreased by fourteen degrees and then by twelve degrees, and the remaining water level stood at two hundred and sixty-six degrees. On the second day, the water level decreased by thirteen degrees and fifteen minutes, decreased by eight degrees, and the remaining water level stood at two hundred and sixty-two degrees. On the third day, the water level decreased by thirteen degrees and eleven minutes, decreased by four degrees, and the remaining water level stood at two hundred and fifty-eight degrees. On the fourth day, the water level decreased by thirteen degrees and seven minutes, remained the same, and the remaining water level stood at two hundred and fifty-four degrees. On the fifth day, the water level decreased by thirteen degrees and three minutes, increased and then decreased by four degrees, and the remaining water level stood at two hundred and fifty degrees.
On the sixth day, the water level decreased by twelve degrees and eighteen minutes, increased by eight degrees, decreased by eight degrees, and the remaining water level stood at two hundred and forty-six degrees. On the seventh day, the water level decreased by twelve degrees and fifteen minutes, increased by eleven degrees, decreased by eleven degrees, and the remaining water level stood at two hundred and forty-three degrees. On the eighth day, the water level decreased by twelve degrees and eleven minutes, increased by fifteen degrees, decreased by fifteen degrees, and the remaining water level stood at two hundred and thirty-nine degrees. On the ninth day, the water level decreased by twelve degrees and eight minutes, increased by eighteen degrees, decreased by eighteen degrees, and the remaining water level stood at two hundred and thirty-six degrees. On the tenth day, the water level decreased by twelve degrees and six minutes, increased by twenty degrees, decreased by twenty degrees, and the remaining water level stood at two hundred and thirty-four degrees.
On the eleventh day, the water level decreased by twelve degrees and five minutes, increased by twenty-one degrees, decreased by twenty-one degrees, and the remaining water level stood at two hundred and thirty-three degrees. On the twelfth day, the water level decreased by twelve degrees and six minutes, increased by twenty degrees, decreased by twenty degrees (because of a shortfall, the decrease of five degrees became an increase of five degrees; thus, the net change is an increase of five degrees, while the decrease refers to the original twenty degrees). The remaining water level stood at two hundred and thirty-four degrees. On the thirteenth day, the water level decreased by twelve degrees and eight minutes, increased by eighteen degrees, decreased by fifteen degrees, and the remaining water level stood at two hundred and thirty-six degrees. On the fourteenth day, the water level decreased by twelve degrees and eleven minutes, increased by fifteen degrees, dropped by twenty-three degrees, and the remaining water level stood at two hundred and thirty-nine degrees.
On the fifteenth day, the water level dropped by twelve degrees and fifteen minutes, increased by eleven degrees, decreased by forty-eight degrees, and the remaining water level was two hundred and forty-three degrees. On the sixteenth day, the water level dropped by twelve degrees and eighteen minutes, increased by eight degrees, decreased by fifty-nine degrees, and the remaining water level was two hundred and forty-six degrees. On the seventeenth day, the water level dropped by thirteen degrees and three minutes, increased by four degrees, decreased by sixty-seven degrees, and the remaining water level was two hundred and fifty degrees. On the eighteenth day, the water level dropped by thirteen degrees and seven minutes, increased by zero degrees, decreased by seventy-one degrees, and the remaining water level was two hundred and fifty-four degrees. On the nineteenth day, the water level dropped by thirteen degrees and eleven minutes, increased by four degrees, decreased by seventy-one degrees, and the remaining water level was two hundred and fifty-eight degrees.
On the twentieth day, the water level dropped by thirteen degrees and fifteen minutes, increased by eight degrees, decreased by sixty-seven degrees, and the remaining water level was two hundred and sixty-two degrees. On the twenty-first day, the water level dropped by fourteen degrees, increased by twelve degrees, decreased by fifty-nine degrees, and the remaining water level was two hundred and sixty-six degrees. On the twenty-second day, the water level dropped by fourteen degrees and four minutes, increased by sixteen degrees, decreased by forty-seven degrees, and the remaining water level was two hundred and seventy degrees. On the twenty-third day, the water level dropped by fourteen degrees and seven minutes, increased by nineteen degrees (this is the third initial increase, adding three large Sundays), decreased by nineteen degrees, and the remaining water level was two hundred and seventy-three degrees.
On Sunday, the temperature was fourteen degrees and nine minutes, decreased by twenty-one degrees, and reduced by twelve degrees, totaling two hundred and seventy-five. This refers to the score on Sunday, which is three thousand three hundred and thirty. The virtual week is two thousand six hundred and sixty-six. The Sunday method is five thousand nine hundred and sixty-nine. The universal week is one hundred and eighty-five thousand thirty-nine. The historical week is one hundred and sixty-four thousand four hundred and sixty-six. The lesser method is one thousand one hundred and one. The new moon's major score is eleven thousand eight hundred and one. The small score is twenty-five. The half week is one hundred and twenty-seven.
These calculations pertain to various parameters for the moon's movement, using the product of the lunar element multiplied by the new moon's major score. If the small score reaches thirty-one, subtract it from the large score; if the large score is full, subtract it from the historical week. If there is enough to meet the Sunday method criteria, it counts as one day; otherwise, it is considered the remainder of the day. Let's set the remainder of the day aside, and focus on calculating the conjunction with the lunar calendar (the first day of the lunar calendar).
To calculate the next month, add one day; the remainder is five thousand eight hundred thirty-two days, and the fractional part is twenty-five. To calculate the lunar phases (the fifteenth and thirtieth of the lunar month), add seven days for each, and the remainder is two thousand two hundred eighty-three days, with a fractional part of twenty-nine point five. Convert these fractions into days using the specified method, subtract twenty-seven days for every complete cycle of twenty-seven days, consider the remainder as the weekly fraction, subtract one day for any remainder, and add the weekly virtual day. Calculate the calendar's surplus and shrinkage by using the common week to multiply it and set it as the base number. Then, use the common number to multiply by the remaining days, and then multiply by the profit and loss rate to adjust the base number; this represents the added time surplus and shrinkage. Subtract the monthly fraction from the total age, multiply by half a week to get the difference, divide by the difference, and obtain the surplus and reduction added and subtracted. If the surplus is insufficient, the new moon is added in the previous or following days. The lunar phases determine the small remainder. Multiply the total age by the added time surplus and shrinkage, divide by the difference, and obtain the total count, which represents the surplus and shrinkage fractions. Add the surplus and reduction to the current date and month; if the surplus is insufficient, use the record method to adjust the degrees and determine the exact degrees of the current date and month. Multiply half a week by the small remainder of the new moon, divide by the common number, then subtract the remaining calendar days. If the subtraction is not enough, add the weekly method and then subtract, then subtract one day. After subtracting, add the weekly day and its fraction to get the midnight entry into the calendar (midnight time lunar date). Calculate for the second day by adding one day; if the remaining days reach twenty-seven days, subtract the weekly day fraction. If not enough, add the weekly virtual to the remainder; the remainder represents the days remaining for the second day's entry into the calendar. Multiply the remaining days of the midnight entry by the profit and loss rate, divide by the weekly method; the remainder is utilized to adjust the surplus and shrinkage. If the remainder is not enough to adjust, use all as the method to adjust; this represents the midnight surplus and shrinkage. A complete chapter age counts as a degree; if insufficient, it counts as a minute. Multiply the common number by the minute and remainder, convert the remainder into minutes according to the weekly method; when the minutes are full, convert them into degrees according to the record method, subtract the surplus, add the reduction, subtract the current midnight degree and remainder, and determine the final measurement. Finally, multiply the remaining days of the midnight entry by the column decline, divide by the weekly method; the remainder indicates the daily changes and decline. Let's first see what this article is about; it seems complex, but let's break it down sentence by sentence.
The first paragraph states that using Zhou Xu to calculate the decay, just as Zhou Fa calculates a constant, after completing a cycle, add the variable decay, subtract the decay when full, then move on to the next cycle to calculate the variable decay. Then use the variable decay to add or subtract the historical day's conversion; if there are not enough or too many units, it is the degree of entry and exit in the calendar year. Multiply the total by the units, add the remainder, then add the day's conversion to the degree of the night to get the degree of the next day. If the calculation for a complete calendar cycle does not land on a Sunday, subtract 1338 from the total, then multiply it by the total; if it is Sunday, add the remainder of 837, then divide by the smaller number 899, add the variable decay of the previous calendar, and continue to calculate as before.
The second paragraph states that using the variable decay to subtract the profit-loss ratio, calculate the variable profit-loss ratio, and then use it to calculate the night's fullness and shrinkage. If there are not enough or too many units after the calendar is calculated, subtract or add in reverse, and handle the remainder the same way as before. Multiply the historical month by the closest solar term night, then divide by 200 to get the bright units. Subtract it from the historical month to get the dark units. The units are calculated in the same way as the calendar year: multiply the total by the units, add the fixed degree of the night, and get the dark and bright fixed degree. If the remainder exceeds half, round up; if not enough, discard.
The third paragraph states that the lunar cycle has four tables, three methods of entry and exit, distributed alternately in the sky. Divide the monthly rate by it to get the number of days in the calendar. Multiply the week by the conjunction of the new moon, just like the conjunction of the moon, to get the conjunction units. Multiply the total by the conjunction number; the remainder behaves like the conjunction number, yielding the retreat units. Use it to calculate the monthly week and get the daily progress units. Divide the conjunction number by one to get the difference rate.
Next is a table of data in tabular form; we will directly quote the original text:
Yin-Yang Calendar | Decay | Profit-Loss Ratio | Coefficient
First day | One minus | Gain seventeen | Initial
Second day (limited to a remainder of 1290, a slight difference of 457.) | This is the previous limit
One minus | Gain sixteen | Seventeen
Third day | Three minus | Gain fifteen | Thirty-three
Fourth day | Four minus | Gain twelve | Forty-eight
Fifth day | Four minus | Gain eight | Sixty
Sixth day | Three minus | Gain four | Sixty-eight
Seventh day | Three minus (not enough to subtract, reverse loss as addition, called gain by one; when subtracting three, for insufficient) | Gain one | Seventy-two
Eighth day | Four plus | Loss two | Seventy-three
(If the extreme loss is exceeded, and the monthly cycle progresses past half a week, adjustments should be made to reduce the degree.)
Ninth day | Four plus | Loss six | Seventy-one
Tenth day | Three plus | Loss ten | Sixty-five
Eleventh day | Two plus | Loss thirteen | Fifty-five
On the twelfth day, add 1, subtract 15, resulting in 42.
On the thirteenth day (limited to 3912, with a minor fraction of 1752).
This is the upper limit.
Add 1 (historically significant daily division), subtract 16, resulting in 27.
Daily division (5203) with fewer additions and subtractions results in 11.
Using the method of fewer additions and subtractions, we get 473.
Historical weeks total 17565.
Difference rate is 1986.
This table describes the daily decay, gain, and cumulative values in a certain historical calendar calculation, as well as some critical values and final results. In short, this is a very complex ancient calendar calculation method that is difficult for modern people to understand directly and requires professional knowledge of astronomy and calendars to interpret.
Let's first calculate the total, which is 18328.
Then the minor fraction is 914.
The method of minor differentiation is 2290.
Next, using the number of months from the first day of the month to this month, multiply by the conjunction and minor fraction. If the minor fraction exceeds the method of minor differentiation, subtract it from the conjunction; if the conjunction exceeds a week (360 degrees), subtract a week, and the remaining days are the days counted in the solar calendar; if it is less than a week, it is the days in the lunar calendar. The remaining days are calculated based on the number of days in each month, calculating the days of the conjunction into the calendar for the month, with any leftover days counted as the remainder.
Add two more days; the remainder is 2580, and the minor fraction is 914. Calculate the days according to the method, subtract 13 when it reaches 13, and calculate the remaining days based on fractions. This shows how the lunar and solar calendars influence each other, with the limits and remainders of the entry in the calendar at the front and the limits and remainders at the back, indicating that the moon has reached the middle position.
List the gains and losses of entering the calendar late, multiply the gains and losses by the number of months to get the minor fraction, add the gains and subtract the losses, and adjust the number of days if the gains are not enough. Multiply the determined remainder by the rate of profit and loss, calculate based on the number of days in a month, and calculate the total profit and loss to determine the additional fixed number.
Multiply the difference rate by the conjunction minor remainder, calculate 1 based on the minor differentiation method, subtract the remainder from the entry into the calendar; if not enough, add the number of days in a month and then subtract, which means one day less. Add the fractions together, simplify the minor fraction based on the number of months to get the minor fraction, which indicates the midpoint of the conjunction day and night in the calendar.
Starting from the second day, add one day; there are 31 days remaining and 31 minutes. Subtract the minutes based on the remainder. Subtract one month when the remainder is full for a month, then add one day. The calendar is complete when the minutes are full, marking the beginning of the new cycle. If the minutes are not full, retain them, then add the remainder of 2720 and 31 minutes, marking the start of the next cycle.
Multiply the total by the late-night adjustments and the remainder. When the remainder completes half a week, it is considered as minutes. Use the surplus and adjustments to account for the remaining yin and yang days. If the surplus is insufficient, adjust the days based on the month’s length. Multiply the determined remainder by the profit and loss rate, calculate it based on the month’s days, and derive the fixed number from the total profit and loss.
Multiply the profit and loss rate by the recent solar terms at midnight, considering 1/200 as daytime. Subtract the profit and loss rate to get the nighttime, and calculate the dusk and dawn fixed number using the profit and loss rate at midnight.
List the overtime and the fixed numbers for dusk and dawn, then divide by 12 to obtain the degree; one-third of the remainder is deemed less, anything below 1 is considered strong, and 2 less is regarded as weak. The result is the degree of the moon leaving the ecliptic. For the solar calendar, subtract the degree from the ecliptic; for the lunar calendar, do the same. Strong values are positive, weak values are negative; add the strong values together, combine the same, and subtract the different. When subtracting, subtract the same, add the different; if there is no corresponding value, cancel each other out, adding two strong values and subtracting one weak value.
From 178 AD to 211 AD, a total of 7378 years have been recorded; however, this is incorrect; it should be 7378 days. The intervening 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, and Bing Yin.
This passage addresses astronomical calculations related to the five elements: Jupiter, Mars, Saturn, Venus, and Mercury. Each has specific values for its daily speed and cycles in the sky. Terms like "weekly rate," "daily rate," "chapter age," "chapter month," "lunar method," "lunar minutes," "lunar number," "total number," and "daily degree method" are specialized terms used in ancient astronomical calculations to determine the trajectories and positions of planets. The specific calculation methods are too intricate to detail here, but they involve various multiplication and division operations.
Then calculate the data of "Shuo Dayu (new moon excess)", "Xiao Yu (small excess)", "Ru Yue Ri (day of entering the month)", "Ri Yu (day excess)", "Degrees", and "Degree Yu (degree excess)" of the five stars. These are results calculated based on different algorithms and parameters, used to predict the positions of planets. The calculation process involves numerous intermediate variables, such as "Tongfa (general method)", "Rifa (solar method)", "Huishu (meeting numbers)", "Zhou Tian (circumference)", "Dou Fen (斗分)", and so on; these numbers represent predefined constants or intermediate results of calculations.
Next, the following specific numerical values are provided:
- Ji Yue (calendar month): 7285
- Zhang Run (leap month): 7
- Zhang Yue (leap month days): 235
- Sui Zhong (mid-year): 12
- Tongfa: 43026
- Rifa: 1457
- Huishu: 47
- Zhou Tian: 215130
- Dou Fen: 145
These numbers correspond to different astronomical parameters. For example, Jupiter has a Zhou rate of 6722, a Ri rate of 7341, a He Yue Shu (combined month count) of 13, a Yue Yu (month excess) of 64810, a He Yue Fa (combined month method) of 127718, a Ri Du Fa (solar degree method) of 3959258, a Shuo Dayu of 23, a Shuo Xiao Yu of 1370, a Ru Yue Ri of 15, a Ri Yu of 3484646, a Shuo Xu Fen (new moon fractional) of 150, a Dou Fen of 974690, a Degrees of 33, and a Degree Yu of 2509956. These numbers are the results of the calculation of Jupiter's orbit.
Mars has a Zhou rate of 3447, a Ri rate of 7271, a He Yue Shu of 26, a Yue Yu of 25627, a He Yue Fa of 64733, a Ri Du Fa of 2006723, and a Shuo Dayu of 47. Data for other planets follows, with similar calculation methods to those used for Jupiter. In short, this passage describes an ancient astronomical calculation method, showing its complexity.
In a certain year B.C., I recorded astronomical observations. First is the observation data of Saturn: the Shuo Xiao Yu value is 1157, the Ru Yue Ri value is 12, the Ri Yu value is 97313, the Shuo Xu Fen is 300, the Dou Fen is 49415, the Degrees is 48, and the Degree Yu is 1991706. Saturn's Zhou rate is 3529, the Ri rate is 3653, the He Yue Shu is 12, the Yue Yu is 53843, the He Yue Fa is 67051, and the Ri Du Fa is 278581.
The following are the observation data for Venus: The large new moon surplus is fifty-four, the small new moon remainder is five hundred thirty-four, the day of entry into the month is twenty-four, the day remainder is one hundred sixty-six thousand two hundred seventy-two, the new moon fractional part is nine hundred twenty-three, the 斗分 (Dou Fen) is five hundred eleven thousand seven hundred five, the degree is twelve, and the degree remainder is one million seven hundred thirty-three thousand one hundred forty-eight. The orbital period of Venus is nine thousand twenty-two, the daily motion rate is seven thousand two hundred thirteen, the total number of lunar months is nine, the remaining days in the lunar cycle is one hundred fifty-two thousand two hundred ninety-three, the lunar conjunction method is one hundred seventy-one thousand four hundred eighteen, and the daily angular measurement method is five million three hundred eleven thousand three hundred fifty-eight.
Next are the observation data for Mercury: The large new moon surplus is twenty-five, the small new moon remainder is one thousand one hundred twenty-nine, the day of entry into the month is twenty-seven, the day remainder is fifty-six thousand nine hundred fifty-four, the new moon fractional part is three hundred twenty-eight, the 斗分 (Dou Fen) is one hundred thirty thousand eight hundred ninety, the degree is two hundred ninety-two, and the degree remainder is fifty-six thousand nine hundred fifty-four. The orbital period of Mercury is eleven thousand five hundred sixty-one, the daily motion rate is one thousand eight hundred thirty-four, the total number of lunar months is one, the remaining days in the lunar cycle is two hundred eleven thousand three hundred thirty-one, the lunar conjunction method is two hundred nineteen thousand six hundred fifty-nine, and the daily angular measurement method is six million eight hundred thousand four hundred twenty-nine.
Finally, an additional set of observation data for Mercury: The large new moon surplus is twenty-nine, the small new moon remainder is seven hundred seventy-three, the day of entry into the month is twenty-eight, the day remainder is six hundred forty-one thousand nine hundred sixty-seven, the new moon fractional part is six hundred eighty-four, the 斗分 (Dou Fen) is one hundred sixty-seven thousand six hundred forty-five, the degree is fifty-seven, and the degree remainder is six hundred forty-one thousand nine hundred sixty-seven. This data provides a detailed record of the movements of Saturn, Venus, and Mercury during that time.
First, let's calculate how many days there are in a year, then multiply this number by the year you want to calculate. If the result is an integer, write it down as the "cumulative total." If it's not an integer, the remaining part is called the "remainder." Then, divide the number of days in a year by the "cumulative total"; the quotient indicates how many years ago the constellation appeared. If the quotient is 2, then it appeared two years ago. If it doesn't divide evenly, it indicates that the constellation appeared during that year. Subtract the number of days in a year from the "remainder" to get a value called the "fraction of degrees." If the "cumulative total" of Venus and Mercury is odd, they appear in the morning; if it's even, they appear in the evening.
Next, multiply the number of months and the month remainder by the "cumulative total." If the result is an integer multiple of months, it means the constellation appeared in that month. If it's not an integer multiple, the remaining part is the month remainder. Then, subtract the total number of months from the accumulated months; the remainder is the "entry month." Multiply the leap month number by this result; if it's an integer multiple of the leap month, it means there was a leap month, so subtract it from the entry month. Subtract this remainder from the year; this result is called the "corrected day calculation in conjunction with the month." If there is an adjustment at the leap month transition, use the new moon day to adjust.
Multiply the number of days in a week by the "fraction of degrees." If the result is an integer multiple of the day calculation, it's a degree; if not, the remaining part is the remainder, noted as the "before Ox Five" period.
The above is the method for calculating the appearance of constellations.
Next, add up the number of months and the month remainder. If the result is an integer multiple of the total months, it means the constellation appeared in that month. If not, calculate within that year, considering leap months if applicable. The remaining part is the situation for the following year; another full month calculation gives the situation for the next two years. For Venus and Mercury, the total morning appearances equal the evening appearances, and vice versa.
Combine the remainder of the day of the new moon and the remainder of the day of the full moon. If the result exceeds one month, add another remainder of 29 days or 773 days. Subtract the full day's remainder from the new moon's remainder, as previously described.
Add the day and the remainder of the day of the month; if the result is an exact multiple of the day count, it indicates the specific date. If the previous remainder of the new moon has completely filled the virtual fraction, subtract one day. If the remainder exceeds 773 afterwards, subtract 29 days; if it is less than 773, subtract 30 days. The leftover part is the date of the next new moon.
Add up the degrees and the remainder of the degrees; if the result is an exact multiple of the day, you get a degree.
Jupiter: Retrograde for 32 days. 3484646 minutes. Direct for 366 days. Retrograde for 5 degrees. 2509956 minutes. Direct for 40 degrees. (Subtract 12 degrees retrograde, establish 28 degrees direct.)
Mars: Retrograde for 143 days. 973113 minutes. Direct for 636 days. Retrograde for 110 degrees. 478998 minutes. Direct for 320 degrees. (Subtract 17 degrees retrograde, establish 303 degrees direct.)
Saturn: Retrograde for 33 days. 166272 minutes. Direct for 345 days. Retrograde for 3 degrees. 1733148 minutes. Direct for 15 degrees. (Subtract 6 degrees retrograde, establish 9 degrees direct.)
Venus, it appears in the east in the morning, stays for a total of 82 days, having traveled a distance of 113980 minutes. After that, it can be seen in the west, for a total of 246 days. (Subtracting 6 degrees retrograde, the final distance is 246 degrees.) In the morning, it moves 100 degrees, having traveled 113980 minutes, then it can be seen in the east. (The daily movement in the east matches that in the west, stays for 10 days, then it moves back by 8 degrees.)
Mercury, it appears for 33 days in the morning, having traveled 612505 minutes. Then it can be seen in the west, for a total of 32 days. (Subtracting 1 degree retrograde, the final distance is 32 degrees.) It moves 65 degrees, having traveled 612505 minutes, then it can be seen in the east. (The daily movement in the east matches that in the west, stays for 18 days, then it moves back by 14 degrees.)
Next is the calculation method: using the remaining days and remaining degrees of motion, plus the remaining degrees of the conjunction of celestial bodies and the sun. If the remaining degrees equal a day's worth of degrees, calculate using the previously mentioned method to obtain the time and degree of appearance of the celestial body. Multiply the denominator of the celestial body's orbit by the degree of appearance; if the remaining degrees equal a day's worth of degrees, you get 1. If the denominator does not divide evenly and the result exceeds half, it is also counted as 1. Then, add the orbit fraction to the number of degrees in a day; if the fraction reaches the denominator, add one degree. Retrograde and prograde have different denominators. Multiply the current orbit denominator by the original fraction; if the result equals the original denominator, it is the current orbit fraction. The remaining days carry over from the previous calculation; subtract for retrograde. If the remaining days are not enough to reach a certain number of degrees, divide the fraction by the constellation, using the orbit denominator as a proportion. The fraction will fluctuate, influencing each other. Any mention of "如盈约满" refers to precise division; "去及除之，取尽之除也" refers to division by taking the full value.
As for Jupiter, it appears together with the sun in the morning, then stops, prograde, for 16 days, traveling a distance of 1,742,323 minutes, with the planet moving 2 degrees and covering a distance of 323,467 minutes. It can then be seen in the east behind the sun in the morning. Prograde, at a fast speed, it moves 11/58 degrees per day, taking 58 days to move 11 degrees. Then it continues prograde at a slow speed, moving 9 degrees in 58 days. It stops for 25 days without movement and then rotates. During retrograde, it moves 1/7 degrees per day, retreating 12 degrees over 84 days. After another 25 days of stopping, it resumes prograde, moving 9 degrees in 58 days at a speed of 9/58 degrees per day. Prograde, at a fast speed, it moves 11 degrees per day, taking 58 days to move 11 degrees, setting in the west in the evening. After 16 days, it has traveled a distance of 1,742,323 minutes, with the planet moving 2 degrees and covering a distance of 323,467 minutes, then it appears together with the sun. The entire cycle lasts 398 days, traveling a total distance of 3,484,646 minutes, with the planet moving 43 degrees and covering a distance of 2,509,956 minutes.
In the morning when the sun came out, Mars coincided with the sun, and then Mars went into hiding. Next, it moved forward for a total of 71 days, traveling 1,489,868 minutes, which is 55 degrees and 242,860.5 minutes. Then it could be seen in the east behind the sun in the morning. While moving forward, it was moving 14 minutes for every 23 minutes each day, traversing 112 degrees over 184 days. It then paused for 11 days. Then it went retrograde, traveling 17 out of 62 minutes each day, retreating 17 degrees in 62 days. It paused again for another 11 days. It then began to move forward again, traveling 12 minutes each day, covering 48 degrees in 92 days. Its speed increased again, moving 14 minutes each day, traversing 112 degrees over 184 days. At this point, it moved in front of the sun, and it became visible in the west at night as it concealed itself. In 71 days, it covered 1,489,868 minutes, which is 55 degrees and 242,860.5 minutes, and at this time, it coincided with the sun again. Throughout this entire cycle, it lasted 779 days and 973,113 minutes, traveling 414 degrees and 478,998 minutes.
As for Saturn, it also coincided with the sun in the morning and then went into hiding. Next, it moved forward for 16 days, traveling 1,122,426.5 minutes, which is 1 degree and 1,995,864.5 minutes. Then it could be seen in the east behind the sun in the morning. While moving forward, it was moving 3 minutes for every 35 minutes each day, covering 7.5 degrees in 87.5 days. It then paused for 34 days. After that, it went retrograde, traveling 1 out of 17 minutes each day, retreating 6 degrees in 102 days. After another 34 days, it began to move forward again, traveling 3 minutes each day, covering 7.5 degrees in 87 days. At this point, it moved in front of the sun, and it became visible in the west at night as it concealed itself. In 16 days, it covered 1,122,426.5 minutes, which is 1 degree and 1,995,864.5 minutes, and it coincided with the sun again. Throughout this entire cycle, it lasted 378 days and 166,272 minutes, traveling 12 degrees and 1,733,148 minutes.
Venus appears in the morning alongside the sun, and then it moves back, retreating four degrees over five days. In the morning, it can be seen in the east, behind the sun. It continues to move back, covering three-fifths of a degree in one day, retreating six degrees over ten days. Then it stops moving for eight days. After that, it moves forward slowly, covering three-fifths of a degree in one day, and over forty-six days, it travels thirty-three degrees. Then its speed increases, moving one degree and ninety-one minutes in one day, covering one hundred six degrees over ninety-one days. It speeds up again, moving one degree and ninety-one minutes and twenty seconds in one day, covering one hundred thirteen degrees over ninety-one days. At this point, it is again behind the sun, appearing in the east in the morning. Moving forward, it travels one fifty-six-thousand-nine-hundred-fifty-fourth of a degree over forty-one days, covering fifty degrees and one fifty-six-thousand-nine-hundred-fifty-fourth of a degree, and then it coincides with the sun again. The total for one coincidence is two hundred ninety-two days and one fifty-six-thousand-nine-hundred-fifty-fourth of a degree, and Venus travels the same number of degrees.
Venus appears in the evening alongside the sun, and then it moves forward, covering one fifty-six-thousand-nine-hundred-fifty-fourth of a degree over forty-one days, traveling fifty degrees and one fifty-six-thousand-nine-hundred-fifty-fourth of a degree, and then it is visible in the west, ahead of the sun in the evening. Then its speed increases, moving one degree and ninety-one minutes and twenty seconds in one day, covering one hundred thirteen degrees over ninety-one days. Its speed slows down again, moving one degree and fifteen minutes in one day, covering one hundred six degrees over ninety-one days. Then it slows down again, moving three-fifths of a degree in one day, covering thirty-three degrees over forty-six days. After that, it stops moving for eight days. Then it moves back, retreating three-fifths of a degree in one day, retreating six degrees over ten days; at this point, it is in front of the sun, appearing in the west in the evening. Then it moves back once more, speeding up, retreating four degrees over five days, and coincides with the sun again. The total for the two coincidences is five hundred eighty-four days and one hundred thirteen thousand nine hundred eighths of a degree, and Venus travels the same number of degrees.
Mercury, in the morning it appears with the sun, then it moves backward, retreating seven degrees over nine days. In the morning, it can be seen in the east, behind the sun. Then it continues to move back, faster, retreating one degree per day. After that, it stops, not moving for two days. Then it moves forward again, slowly, moving eight-ninths of a degree in one day, totaling eight degrees in nine days. Then the speed increases, moving one and a quarter degrees per day, totaling twenty-five degrees in twenty days. At this time, it is again behind the sun, appearing in the east in the morning. Moving forward, in sixteen days, it moves six hundred forty-one million nine thousand sixty-seven arcminutes, moving thirty-two degrees and six hundred forty-one million nine thousand sixty-seven arcminutes, and then it aligns with the sun again. One alignment, a total of fifty-seven days and six hundred forty-one million nine thousand sixty-seven arcminutes, the distance Mercury travels is the same.
Well, what on earth is this about? Let me break it down for you, sentence by sentence. The first sentence, "Mercury: in the evening it merges with the sun, retreats, follows," means that Mercury merges with the sun, then it retreats and starts moving in the same direction as the Earth. The term "retreat" indicates that Mercury moves close to the sun, covered by the sun's light, so we can't see it from Earth. "Follow" means that Mercury's direction of movement is the same as Earth's.
Next, "After sixteen days, moving thirty-two degrees and six hundred forty-one million nine thousand sixty-seven arcminutes, it can be seen in the west in the evening, positioned in front of the sun." Ah, this sentence is a bit complicated, but it roughly means that after sixteen days (specifically six hundred forty-one million nine thousand sixty-seven arcminutes of a day), Mercury moves 32 degrees (specifically six hundred forty-one million nine thousand sixty-seven arcminutes), and then in the evening, we can see it in the west, positioned in front of the sun. The ancient timekeeping and angle calculations were really detailed!
"When moving in the same direction, the speed is fast, moving one and a quarter degrees per day, after twenty days moving twenty-five degrees in the same direction." This means that when Mercury moves in the same direction, the speed is fast, moving one and a quarter degrees per day, and after twenty days, it can move twenty-five degrees, continuing in the same direction.
"Late, moving eight-ninths of a degree, it takes nine days to move eight degrees." This time Mercury slowed down, only moving eight-ninths of a degree, taking nine days to move eight degrees. "Stay, no movement for two days." "Stay" means stagnant, meaning that Mercury remained mostly stationary for two days. "Rotate, retrograde, retreat one degree a day, approaching the west in the evening." Then it began its retrograde motion! "Rotate" means rotating, here referring to retrograde motion. It retreats one degree a day, still in front of the sun, and in the evening, it retreats into the west. "Retrograde, late, retreat seven degrees in nine days, conjunct with the sun." During retrograde motion, it moves slowly as well, taking nine days to retreat seven degrees, and finally conjunct with the sun again. The last sentence, "Every conjunction lasts one hundred fifteen days and six million two thousand five hundred and five minutes, and the planets follow the same pattern." In summary, from one conjunction of Mercury and the sun to the next, it takes a total of one hundred fifteen days (precisely six million two thousand five hundred and five minutes), and other planets follow a similar pattern. These are simply records of ancient astronomical observations! The significance of these observations highlights the advanced understanding of astronomy in ancient times.
