First, note the remaining days (small remainder) after the winter solstice, then add 15 days, and the small remainder now totals 515 days. Next, subtract these 515 days from the 2356 days (large remainder) as per the regulations. Then, use the leap remainder to subtract one year, and multiply the remaining days by a specific day of the year. If the result matches the number of days in a leap month, it counts as one month. If it is not an integer, if it exceeds half, it counts as a month; if it's less than half, it counts as no month, thus avoiding the issue of a "non-existent month." Then, add 7 days to the large remainder, bringing the small remainder to 557.5 days. Using the earlier method, subtract the small remainder from the large remainder to calculate the first quarter moon. Continuing this calculation allows you to determine the full moon, last quarter moon, and the new moon of the following month (the first day of the month). If the calculated small remainders for the first quarter, full moon, and last quarter are less than 410, multiply by 100 units (an ancient timekeeping measure); if the result matches the number of units in a day, it counts as one unit; if it is not an integer, take the approximate value and, based on the nearby solar term and the remaining time using the ancient timekeeping tool (water clock), determine the specific day. Keep track of the calculated number of years, then multiply the remainder by the number of years; if the result equals the number of years in the calendar, that represents the large remainder; if it isn't an integer, the result is the small remainder. By calculating the large remainder according to the calendar, you can determine how many days after the winter solstice are without sunlight (a certain day). To calculate the next day without sunlight, add 69 to the large remainder and 64 to the small remainder; if the result matches the number of laws, then subtract from the large remainder, and having no remainder indicates the calculation is complete. Multiply the calendar by the accumulated days (a certain day), then subtract the week (360 degrees), and the remaining number divided by the calendar gives you the degree measurement. Starting from the first five degrees of the Ox constellation, divide by the degree using the sequence of the constellations; if it's less than one constellation, that indicates the position of the sun at midnight on the first day of the lunar month (the first day of the month in the sky). To calculate the second day, add one degree to the total degrees, then divide by the division (degree unit) using the Dou constellation; if the division is insufficient, subtract one degree as the calendar, then add it back in.
Multiply the number of days in a lunar month by the accumulated days, and subtract the days of the week. The remaining value: if it's a whole number, it represents degrees; if not, it indicates minutes. Using this method, we can determine the moon's position at midnight on the first day of the lunar month.
To calculate the next month, add 22 degrees and 258 minutes for a small month, and add 1 day, 13 degrees, and 217 minutes for a full month. If the result is exactly a whole number, add one degree. During the latter part of winter solstice, the moon is near the Zhang and Xin constellations.
Multiply the number of years by the remainder when dividing by the first day of the lunar month. If the result is a whole number, it indicates large minutes; otherwise, the remainder represents small minutes. Subtract the large minutes from midnight on the first day of the month; if the result is a whole number, then subtract from the degrees. Using this method, we can calculate the position where the lunar new moon aligns in the sky and on Earth.
To calculate the next month, add 29 degrees, 312 minutes, and 25 seconds. If the small minutes are a whole number, subtract them from the large minutes; if the large minutes are a whole number, subtract from the degrees, then divide by the constellations.
To calculate the positions of the first quarter moon, full moon, and last quarter moon, add 7 degrees, 225 minutes, and 17.5 minutes to the conjunction degree (the degree of the sun and moon conjunction). Calculate the large and small minutes and degrees as previously described to determine the position of the first quarter moon. Continue the calculation to determine the full moon, last quarter, and the conjunction of the next month.
To calculate the positions of the first quarter moon, full moon, and last quarter moon, add 98 degrees, 480 minutes, and 41 minutes to the conjunction degree. Calculate the large and small minutes and degrees as previously described to determine the position of the first quarter moon. Continue the calculation to determine the full moon, last quarter, and the conjunction of the next month.
This text outlines an ancient astronomical calendar calculation method that is highly specialized. Let’s break it down sentence by sentence and explain it in contemporary language.
First, "Seek the brightness of the sun and moon; the sun follows the solar calendar and the moon follows the lunar calendar. Multiply the number of time intervals (leaks) during the night close to the solar term by 200 to obtain the length of the day (bright part). Subtract the length of the day from the standard time lengths of the solar and lunar calendars to determine the length of the night (dark part). Then add half of the night to each, according to the law." This passage describes how to calculate the length of day and night. Multiply the standard time length of day and night (solar and lunar calendars) by the number of leaks in the night close to the solar term, divide by 200 to get the length of the day (bright part), subtract the length of the day to get the length of the night (dark part), then add half of the night to each to get the exact length of day and night. This is similar to how we currently calculate sunrise and sunset times using formulas, but the method was more cumbersome at that time.
Next, "Establish the first year as a baseline, then calculate the years sought externally, subtract the accumulated years, and multiply the remaining years by the accumulated rate. If the accumulated years represent an eclipse, add one to the surplus. Multiply by the accumulated month; if the rate is an accumulated month, it is not a surplus month. Multiply by the intercalary month for the remaining years; the full years are the accumulated intercalary, subtract the accumulated month, and subtract it in the middle of the year, not completely, count from the winter solstice." This section deals with calculating the "accumulated years" and "accumulated months" in the calendar, which are errors accumulated over many years. It uses data from the first year as a baseline, then calculates based on the number of years and a value called the "accumulated rate" to determine the accumulated error (eclipse) and accumulated number of months (accumulated months). If there are remaining months, the situation of intercalary months must be considered, ultimately resulting in a corrected value. This part is very complex and is equivalent to correcting long-term errors in the calendar.
"Seek the next eclipse; add five months. If the remaining months reach one thousand six hundred and thirty-five, this is equivalent to adding a month, and the month will be expected." This section calculates the timing of the next solar or lunar eclipse. It states that based on known data, adding five months, if the remaining number of months reaches a certain value (1635), this is equivalent to adding a month. Then, based on this corrected number of months, the timing of the next lunar eclipse (expectation) is determined.
"Due to the large surplus of the winter solstice, double the small surplus; Kan is used for daily affairs. Add a small surplus of one thousand seventy-five; the Qian method is based on the large surplus, Zhongfu is used for daily affairs." This section discusses how to use "large surplus" and "small surplus" to calculate specific days, such as "Kan is used for daily affairs" and "Zhongfu is used for daily affairs," which may be related to ancient divination or Yin-Yang Five Elements theory. The specific meanings require an understanding of the historical context.
"To seek the next hexagram, add six to the remainder of the large division and three to the remainder of the small division. Multiply the small remainder by four and add it to the result. This continues the calculation of a certain hexagram's date, adjusting the large and small remainders separately and according to the four solar terms (spring equinox, summer solstice, autumn equinox, winter solstice).
Place the large and small remainders from the winter solstice, add twenty-seven to the large remainder and nine hundred twenty-seven to the small remainder. Subtract this result from two thousand three hundred fifty-six to derive the Earth-related date. Add eighteen to the large remainder and six hundred eighteen to the small remainder to determine the Wood-related date. Add seventy-three to the large remainder and one hundred sixteen to the small remainder to get Earth again." This section calculates the "related dates" of different solar terms based on the large and small remainders from the winter solstice, such as Earth-related dates and Wood-related dates, which are also related to ancient five elements theory.
Multiply the small remainder by twelve, divide by the method to get a Chen, starting from the Zi branch, and calculate the small remainder based on the new moon, first quarter moon, and full moon. This section describes the method of calculating the Heavenly Stems and Earthly Branches (Chen) based on the small remainder and explains how the new moon, first quarter moon, and full moon will affect the value of the small remainder.
Multiply the small remainder by one hundred, divide by the method to get a Ke, not rounding to Shi, seek the minute, learn the closest solar term, start the night and end it; if the water is not finished at night, use the nearest term." This section explains how to use the small remainder to calculate a more precise unit of time - a minute, and how to adjust the calculation results based on solar terms.
"Promote advancement or retreat, the outcome of adding advancement and subtracting retreat. There is a difference between advancement and retreat; after two divisions, adjust by increasing or decreasing four degrees, decrease by three for every half, stop when the difference is three, reduce by five degrees as before." This section explains that there will be adjustments of advancement and retreat during the calculation process, and the magnitude of the adjustment will change over time.
"The moon moves fast or slow; weekly advancement is constant. The calculations are based on the celestial and terrestrial cycles, multiplying the remainder by itself, as one of the meetings, is beyond the weekly division. Divide by the weekly cycle, subtract the days; slow or fast will decline, the change is the trend. Decrease by the decline and the rate of the moon's movement to convert to days. Add the decline left and right, as the loss or gain rate. Increase turns to increase, decrease turns to decrease, accumulation of gains and losses. Multiply by half a small week, as one of the common numbers, subtract from the weekly cycle, representing the new moon phase." This section describes the calculation method of the moon's varying speed of movement and how to adjust the calendar based on speed changes. This part involves many astronomical constants and complex calculations, which are difficult to explain in simple terms.
"Daily degrees, decay values, profit and loss rates, profit and loss accumulations, and lunar movement degrees.
On Day 1, the water level reading dropped by fourteen degrees and nine minutes of arc, reduced by twenty-one degrees, resulting in a water level of two hundred seventy-five.
On Day 2, the water level reading dropped by fourteen degrees and seven minutes of arc, reduced by nineteen degrees, resulting in a water level of two hundred seventy-three.
On Day 3, the water level reading dropped by fourteen degrees and four minutes of arc, reduced by sixteen degrees, resulting in a water level of two hundred seventy.
On Day 4, the water level reading dropped by fourteen degrees, reduced by twelve degrees, resulting in a water level of two hundred sixty-six.
On Day 5, the water level reading dropped by thirteen degrees and fifteen minutes of arc, reduced by eight degrees, resulting in a water level of two hundred sixty-two.
On Day 6, the water level reading dropped by thirteen degrees and eleven minutes of arc, reduced by four degrees, resulting in a water level of two hundred fifty-eight.
On Day 7, the water level reading dropped by thirteen degrees and seven minutes of arc, remaining unchanged, resulting in a water level of two hundred fifty-four.
On Day 8, the water level reading dropped by thirteen degrees and three minutes of arc, reduced by four degrees, resulting in a water level of two hundred fifty.
On Day 9, the water level reading dropped by twelve degrees and eighteen minutes of arc, reduced by eight degrees, resulting in a water level of two hundred forty-six.
On Day 10, the water level reading dropped by twelve degrees and fifteen minutes of arc, reduced by eleven degrees, resulting in a water level of two hundred forty-three.
On Day 11, the water level reading dropped by twelve degrees and eleven minutes of arc, reduced by fifteen degrees, resulting in a water level of two hundred thirty-nine.
On Day 12, the water level reading dropped by twelve degrees and eight minutes of arc, reduced by eighteen degrees, resulting in a water level of two hundred thirty-six.
On Day 13, the water level reading dropped by twelve degrees and six minutes of arc, reduced by twenty degrees, resulting in a water level of two hundred thirty-four."
On the fourteenth day, the water level decreased by twelve degrees and five minutes, increased by one degree, decreased by twenty-one degrees, resulting in a remaining water level of two hundred thirty-three.
On the fifteenth day, the water level decreased by twelve degrees and six minutes, increased by two degrees, decreased by twenty degrees (since the decrease was insufficient, it effectively increased by five degrees, making the total twenty degrees), resulting in a remaining water level of two hundred thirty-four.
On the sixteenth day, the water level decreased by twelve degrees and eight minutes, increased by three degrees, increased by eighteen degrees, decreased by fifteen degrees, resulting in a remaining water level of two hundred thirty-six.
On the seventeenth day, the water level decreased by twelve degrees and eleven minutes, increased by four degrees, increased by fifteen degrees, decreased by twenty-three degrees, resulting in a remaining water level of two hundred thirty-nine.
On the eighteenth day, the water level decreased by twelve degrees and fifteen minutes, increased by three degrees, increased by eleven degrees, decreased by forty-eight degrees, resulting in a remaining water level of two hundred forty-three.
On the nineteenth day, the water level decreased by twelve degrees and eighteen minutes, increased by four degrees, increased by eight degrees, decreased by fifty-nine degrees, resulting in a remaining water level of two hundred forty-six.
On the twentieth day, the water level decreased by thirteen degrees and three minutes, increased by four degrees, increased by four degrees, decreased by sixty-seven degrees, resulting in a remaining water level of two hundred fifty.
On the twenty-first day, the water level decreased by thirteen degrees and seven minutes, increased by four degrees, decreased by seventy-one degrees, resulting in a remaining water level of two hundred fifty-four.
On the twenty-second day, the water level decreased by thirteen degrees and eleven minutes, increased by four degrees, decreased by four degrees, decreased by seventy-one degrees, resulting in a remaining water level of two hundred fifty-eight.
On the twenty-third day, the water level decreased by thirteen degrees and fifteen minutes, increased by four degrees, decreased by eight degrees, decreased by sixty-seven degrees, resulting in a remaining water level of two hundred sixty-two.
On the twenty-fourth day, the water level decreased by fourteen degrees, increased by four degrees, decreased by twelve degrees, decreased by fifty-nine degrees, resulting in a remaining water level of two hundred sixty-six.
On the twenty-sixth, the calculation showed fourteen degrees and four minutes, adding sixteen, subtracting sixteen, and then subtracting forty-seven, resulting in two hundred seventy.
On the twenty-seventh, it was fourteen degrees and seven minutes; first, calculate a full week plus three days, then subtract nineteen, then subtract thirty-one, resulting in two hundred seventy-three.
On Sunday, it was fourteen degrees and nine minutes, adding less, subtracting twenty-one, subtracting twelve, resulting in two hundred seventy-five.
Overall, the water levels show a fluctuating pattern with notable decreases, particularly towards the later days.
Next are some astronomical data: Sunday fraction: 3,333; Zhou Xu: 2,666; Sunday law: 5,969; total Zhou: 185,039; historical Zhou: 164,466; small big law: 1,101; Shuo Xing large fraction: 1,801; small fraction: 25; half Zhou: 127.
How to use these data? First, multiply the yuan value by the lunar month, then apply the size of the Shuo Xing. If the small fraction reaches 31, deduct it from the large fraction. If the large fraction is full according to historical Zhou, subtract it. If the Sunday law is sufficient, count it as one day; if not, record the remainder. Set this remainder aside, focusing primarily on calculating the combination of Shuo and history.
Calculate the next month, add a day; the remainder is 5,832, and the small fraction is 25.
For calculating the crescent moon, add seven days to each; the remainder is 2,283, and the small fraction is 29.5. Convert these fractions into days, and subtract 27 when it reaches 27; the remaining is the same as Sunday. If it’s insufficient for division, subtract a day and add Zhou Xu.
Multiply the calculated profit and loss by the total Zhou to obtain a result. Then multiply the total number by the daily residual fraction, and multiply by the profit and loss rate; use it to adjust the result; this represents the adjustment for time profit and loss. Subtract the month from the chapter year, multiply by half Zhou, calculate a difference, use it for division, and obtain the net profit and loss. If there’s a profit or loss according to the day law, adjust the date of the Shuo, adjust the large remainder of the crescent moon, and determine the small remainder.
Multiply the chapter year by the time profit and loss, then divide by the difference to get the full number, which is the profit and loss size fraction; use it to adjust the position of the current day and month. If there is a profit or loss, adjust the degree fraction with the record law, and ultimately determine the positions of the day and month.
Multiply half Zhou by the small remainder of Shuo, divide by the total number, then subtract the residual day from history. If it is not enough to subtract, add Zhou law and then subtract, then subtract a day. After subtracting, add Sunday and its fraction; you can calculate the time of midnight for entering history.
Calculate the second day, add a day; if the daily residue exceeds the Sunday fraction, subtract it; if not enough, add Zhou Xu; the remainder is the daily residue for entering history on the second day.
Entering the remaining part of the calendar calculation in the late night, multiplying by the profit and loss rate, just like dividing by the number of days in a week, the remaining part is the remainder. Through the accumulation of profit and loss, the remaining part remains constant. The integer part is used as the calculation factor to decrease; this describes the method of managing profit and loss during the late night. Full chapter years are used as degrees, and the remaining part that cannot be divided is used as fractions. Multiply the total amount by the fraction and the remaining part; the remainder is derived from the fraction, similar to how it is with the number of days in a week. When the fraction is full, it comes from the degrees, using the surplus to increase and using reduction to decrease the original degree of the late night and the remaining part, finally obtaining a determined degree.
Next, multiply the remaining part of the calendar calculation by the column decay (a decreasing numerical sequence), just like dividing by the number of days in a week; the remaining part is the remainder. This way, you can know the decay pattern of the numerical values for each day. Multiply the week imaginary number by the column decay, just like calculating with the number of days in a week to get a constant. After the calendar calculation is finished, use it to increase the decay value; if the value of the column decay is full, subtract it and transition to the decay value of the next calendar.
Adjust the conversion of days to minutes with the increase or decrease of the decay value, surplus or shortage of fractions, and the inclusion and exclusion of chapter years. Multiply the total amount by the fraction and the remaining part; then add the day conversion to the determined degree at night to calculate the next day. If the result of the calendar calculation is not an integer number of days in a week, subtract 1338 from the total, then multiply the result; if it is an integer number of days in a week, add the remaining 837, then divide by the smaller number 899, add the decay value of the next calendar, and calculate as before.
Subtract or add the profit and loss rate from the decay value to get the changing profit and loss rate, then use it to adjust the profit and loss in the late night. After the calendar calculation is finished, if the profit and loss is insufficient, subtract it in reverse, enter the calculation of the next calendar, and add or subtract the remaining part just like the numbers above.
Multiply the running fraction of the historical month by the nighttime hour count of the nearest solar term, with 1/200 representing the bright minute. Subtract the monthly running fraction from it to get the dark minute. The fraction, like the chapter years, is used as degrees; multiply the total amount by the fraction, and add the determined degree at night to get the determined degree of dawn and dusk. If the remaining fraction exceeds half, round it up; if not, discard it.
Four tables of menstruation, three pathways of entry and exit, intersecting with the heavens, using the lunar cycle to calculate, to determine the number of days in the lunar calendar. Multiply the weeks by the lunar conjunction, similar to the combined lunar months, to calculate the conjunction. Multiply the total by the conjunction; the remaining portion corresponds to the conjunction number, to obtain the remainder. Use this to track the lunar month and week, to determine the daily progression. Dividing the conjunction number by one yields the difference rate.
Yin and Yang calendar: rates of decline, gain, and multiple counts.
On the first day, decrease by one and increase by seventeen, marking the beginning.
On the second day (limited to one thousand two hundred ninety, with a fraction of four hundred fifty-seven), this represents the previous limit.
Decrease by one, increase by sixteen, resulting in seventeen.
On the third day, decrease by three units, increase by fifteen, totaling thirty-three.
On the fourth day, decrease by four, increase by twelve, totaling forty-eight.
On the fifth day, decrease by four, increase by eight, resulting in sixty.
On the sixth day, decrease by three, increase by four, totaling sixty-eight.
On the seventh day, decrease by three (if the decrease is insufficient, the loss is reversed to a gain, indicating an increase of one, although three should be decreased due to insufficiency), resulting in an increase of one, totaling seventy-two.
On the eighth day, increase by four, decrease by two, totaling seventy-three.
(Exceeding the loss, meaning the moon travels half a week; when it has exceeded the extreme, it should be reduced.)
On the ninth day, increase by four, decrease by six, totaling seventy-one.
On the tenth day, add three days, subtract ten days, equaling sixty-five.
On the eleventh day, add two days, subtract thirteen days, equaling fifty-five.
On the twelfth day, add one day, subtract fifteen days, equaling forty-two.
On the thirteenth day (the limit is three thousand nine hundred and twelve, with a fraction of one thousand seven hundred and fifty-two), this is the final deadline.
Add one day (this marks the start of the calendar, divided by day). Subtract sixteen days, resulting in twenty-seven.
Divide by day (five thousand two hundred and three), subtract sixteen days from the smaller addition, resulting in eleven.
The lesser method results in four hundred seventy-three.
The calendar cycle spans one hundred seventy-five thousand six hundred and five days.
The difference rate is one thousand nine hundred and eighty-six.
The conjunction day and minute total eighteen thousand three hundred and twenty-eight.
The fraction totals nine hundred and fourteen.
The fraction method totals two thousand two hundred and nine.
Subtract the accumulated months from the previous year’s conjunction, then use the conjunction day and minute to multiply it separately; deduct the full fraction from the conjunction when the fraction reaches its full value, subtract when the conjunction is full. The remaining days that do not complete the calendar cycle will be counted towards the solar calendar; subtract when it is full, and what remains will be counted towards the lunar calendar. The rest are obtained as one day, calculated separately; the desired lunar conjunction and new moon are entered into the calendar, with the remainder representing the leftover days.
Adding two days results in a total of two thousand five hundred and eighty days, with a remainder of nine hundred and fourteen. Calculate the number of days using the method: subtract when it reaches thirteen, and handle any remaining days as fractions. The Yin and Yang calendars alternate in this way, with the entry limit before the remainder limit, and the exit limit after the remainder limit. The same applies to the midpoint of the monthly cycle.
List the delays and speeds of entries, and the sizes of excess and shortages in the calendar. Multiply the number of meetings by the fractions to get the remainder, subtract the excess, and add the Yin and Yang days. If the remainder is insufficient, adjust the calculation by postponing or advancing by one day to determine. Multiply the determined remainder by the profit and loss rate, as done in the monthly cycle, to obtain one, and combine the profit and loss for the added fixed number.
Multiply the difference rate by the remaining fraction of the new moon, as with the differential method, to get one, and use it to subtract the remainder of the entry calendar day. If insufficient, add the monthly cycle and subtract again, then subtract one day. Add the fractions together, simplifying the remainder into a fraction, which is the new moon night entry into the calendar.
To find the next day, add one day; the total is thirty-one, and the fraction is thirty-one. Subtract the fraction from the remainder as with the number of meetings; subtract the monthly cycle when the remainder is full, add one more day, and the calendar calculation is complete. When the remainder is full, subtract the fraction; this marks the beginning of the calendar. If the remainder is not full, then add two thousand seven hundred and two to it, with a fraction of thirty-one, which is the start of the next calendar.
Multiply the total by the delays and speeds of entries, as well as the excess and shortage from the calendar's night half, along with the remainder. When the remainder is full after a week and a half, it becomes a fraction. Add the excess and subtract the Yin and Yang days; if the remainder is insufficient, postpone or advance by one day using the monthly cycle. Multiply the determined remainder by the profit and loss rate, as with the monthly cycle, to get one, and use the profit and loss together as the fixed number for the night half.
Multiply the profit and loss rate by the nighttime duration of the nearest solar term; use one two-hundredth for clarity, and subtract it from the profit and loss rate to determine darkness. Then, use the profit and loss number for nighttime as the fixed value for both darkness and clarity.
This article discusses ancient astronomical calculations, which feels somewhat reminiscent of fortune-telling. First, it states that to calculate the moon's angular distance from the ecliptic, one must first calculate the additional time, then divide by 12, multiply the remainder by one-third, and if the remainder is less than one, it is termed "weak"; if it is greater than one, it is termed "strong." Two "weak" results are even weaker. The final result represents the moon's angular distance from the ecliptic. The solar calendar adds the distance of the moon to the ecliptic's apex, while the lunar calendar subtracts this distance to calculate the moon's angular distance from the apex. "Strong" is positive, "weak" is negative; same signs are added, different signs are subtracted. For subtraction, same signs cancel out, and different signs are added, with no cancellation occurring; two "strong" results together are stronger than one "weak."
From the year of the Metal Ox in the 1st year of the Yuan Dynasty to the 11th year of the Jian'an in the year of the Earth Dog, a total of 7,378 years have passed. The years during this period are:
- Metal Ox
- Earth Tiger
- Fire Rabbit
- Wood Dragon
- Wood Snake
- Metal Horse
- Water Goat
- Fire Monkey
- Metal Rooster
- Earth Dog
- Metal Pig
- Earth Rat
- Fire Ox
- Wood Tiger
Then it begins to explain the Five Elements: Wood is associated with the Planet of the Year (Jupiter), Fire is associated with the Wandering Star (Mars), Earth is associated with the Filling Star (Saturn), Metal is associated with the Bright Star (Venus), and Water is associated with the Star of the Morning (Mercury). The heavenly degrees (the degrees of a day) must be linked to the weekly motion of each star to calculate the weekly rate (degrees in a week) and the daily rate (degrees in a day). Multiply the annual rate by the weekly rate to obtain the monthly calculation (degrees in a month), then multiply the monthly rate by the daily rate to get the monthly fraction; dividing the monthly fraction by the monthly calculation gives the month number. Multiplying the month number by the monthly calculation yields the daily degree calculation (degrees in a day). The fraction associated with the Dipper constellation multiplied by the weekly rate results in the Dipper fraction. (The daily degree calculation uses the calendar law multiplied by the weekly rate, so here, fractions are also multiplied.)
Next, start calculating the major remainder and minor remainder of the five stars (major remainder refers to the first day of the lunar calendar). The specific method is: multiply the common method (a fixed numerical value) by the number of months, then divide by the daily method (a fixed numerical value); the result is the major remainder, and the remainder that cannot be further divided is the minor remainder, which is found by subtracting the major remainder from 60. Next, calculate the entry date of the five stars; the method is: multiply the common method by the month's remainder (the remainder of the moon), then add the result of multiplying the minor remainder of the first day by the common month calculation (a fixed numerical value). Add these two results together, then divide by the daily method, and the result is obtained. Finally, calculate the degree and degree remainder of the five stars; the method is: subtract the excess to obtain the degree remainder, then multiply the degree remainder by the weekly circumference (the number of degrees in a week), and then divide by the daily method. The result is the degree, and the remainder that cannot be divided is the degree remainder. If it exceeds the weekly circumference, subtract the weekly circumference and the dipper value.
Finally, many specific numerical values are given: recorded month number is 7285, leap month is 7, monthly cycle is 235, year is 12, common method is 43026, daily method is 1457, total number is 47, weekly circumference is 215130, dipper value is 145. Then the orbital period of Jupiter (6722), daily rate (7341), common lunar month count (13), month's remainder (64810), common month calculation (127718), daily method (3959258), major remainder of the first day (23), minor remainder of the first day (1370), entry month day (15), daily remainder (3484646), fictitious remainder (150), and dipper value (974690) are listed separately. These numbers can be quite overwhelming...
The degree value is thirty-three, and the remaining degree value is two million nine hundred and fifty-six thousand. For Mars: the orbital period is three thousand four hundred and seven, daily rate is seven thousand two hundred and seventy-one, common lunar month count is twenty-six, month's remainder is twenty-five thousand six hundred and twenty-seven, common month calculation is sixty-four thousand seven hundred and thirty-three, daily method is two million six hundred and twenty-three, major remainder of the first day is forty-seven, minor remainder of the first day is one thousand one hundred and fifty-seven, entry month day is twelve, daily remainder is ninety-seven thousand three hundred and thirteen, fictitious remainder is three hundred, and dipper value is forty-nine thousand four hundred and fifteen.
Next is the data for the next section. The degree is forty-eight, and the remaining degree is one million nine hundred and seventeen thousand six hundred. Earth: The weekly rate is three thousand five hundred and twenty-nine, the daily rate is three thousand six hundred and fifty-three, the total number of months is twelve, the remaining months are fifty-three thousand eight hundred and forty-three, the total month method is sixty-seven thousand fifty-one, the daily degree method is two million seven hundred and eighty-five thousand eighty-one, the major remainder is fifty-four, the minor remainder is five hundred and thirty-four, the entry month date is twenty-four, the daily remainder is one hundred and sixty-six thousand two hundred and seventy-two, the minor remainder is nine hundred and twenty-three, and the major remainder is five hundred and seventeen thousand five hundred.
Continuing, the degree is twelve, and the remaining degree is one million seven hundred and thirty-one thousand one hundred and forty-eight. Metal: The weekly rate is nine thousand two hundred and twenty-two, the daily rate is seven thousand two hundred and thirteen, the total number of months is nine, the remaining months are fifty-two thousand two hundred and ninety-three, the total month method is one hundred and seventy-one thousand four hundred and eighteen, the daily degree method is five hundred and thirty-one thousand three hundred and fifty-eight, the major remainder is twenty-five, the minor remainder is one thousand one hundred and twenty-nine, the entry month date is twenty-seven, the daily remainder is fifty-six thousand nine hundred and fifty-four, the minor remainder is three hundred and twenty-eight, and the major remainder is one hundred and eighty-one thousand.
Finally, here are the data for water. The degree is two hundred and ninety-two, and the remaining degree is fifty-six thousand nine hundred and fifty-four. Water: The weekly rate is eleven thousand five hundred and sixty-one, the daily rate is one thousand eight hundred and thirty-four, the total number of months is one month, the remaining months are twenty-one thousand one hundred and thirty-one, the total month method is twenty-one thousand nine hundred and fifty-nine, the daily degree method is six hundred eighty-nine thousand four hundred and twenty-nine, the major remainder is twenty-nine, the minor remainder is seven hundred and seventy-three, the entry month date is twenty-eight, the daily remainder is six hundred forty-one thousand nine hundred and sixty-seven, and the minor remainder is six hundred and eighty-four.
First, we need to calculate some numbers. One million six hundred seventy-four thousand three hundred forty-five, what number is this? The degrees are fifty-seven, the degree remainder is six hundred forty-one thousand nine hundred sixty-seven. Next, we need to use the number of years in the Yuan (an ancient calendar era), multiplied by pi (π), to calculate a "total"; the leftover part is the "remainder." Then divide the remainder by pi to get a number representing the year of the celestial conjunction. If it does not divide evenly, take the whole number, and the remainder is the year. Subtract pi from the remainder to get the degrees. For Venus and Mercury, odd products indicate the morning, while even products indicate the evening.
Next, we need to multiply the number of months and the remaining months by the total to calculate the total months; the leftover part is the remaining months. Subtract the total months from the calendar month, the remainder is the entry month. Multiply by the leap month (an extra month in the ancient calendar); a full leap month gets one leap, subtract the entry month, and the remaining part is deducted from the year to calculate the total months. If it is at the transition of the leap month, use the new moon (first day of the lunar month) to adjust.
Next, apply the common method to the remaining months, multiply the new moon method by the remaining new moon, then divide by the number of meetings to get the full-day method, which is the date of the celestial alignment. The leftover part is the remainder, noted separately from the new moon calculation. Multiply by the circumference (360 degrees) to get the degrees; a full day method gets one degree, the leftover part is the remainder, noted in the "five front of the ox" counting method. This is the method of finding the celestial conjunction.
Next, add up the number of months and the remaining months; if the full lunar month method is achieved, it is one month. If not, it is recorded in the year, and if full, subtract it. Take into account if there is a leap month; the remaining is the number of the following year; if it is full again, it is the number of the following two years. Venus and Mercury, morning added together is evening, evening added together is morning.
Add the size remainder of the new moon and the size remainder of the total month; if it exceeds a month, then add another large remainder of twenty-nine, small remainder of seven hundred seventy-three. If the small remainder is full in the day method, subtract from the large remainder, the method is as described above. Add the entry month and the remainder of the day together; if the full day method is achieved, it is one day. If the previous new moon's smaller remainder is full, subtract one day; if the later small remainder exceeds seven hundred seventy-three, subtract twenty-nine days; subtract thirty days if not full; the remainder is the entry day of the next conjunction.
Finally, add up the degrees, and the degree remainder; if the full day method is achieved, it is one degree.
Here is the following data for Jupiter: retrograde for 32 days, 348,464 minutes; direct motion for 366 days; retrograde movement of 5 degrees, 250,956 minutes; direct movement of 40 degrees, which includes a retrograde of 12 degrees, resulting in an actual movement of 28 degrees. Mars: retrograde for 143 days and 97,313 minutes, and direct motion for 636 days; retrograde movement of 110 degrees, 47,898 minutes; direct movement of 320 degrees, which includes a retrograde of 17 degrees, resulting in an actual movement of 303 degrees. Saturn: retrograde for 33 days, 166,272 minutes, and direct motion for 345 days.
"Observe the East. (Moves like in the West. Hides for ten days and then retreats by eight degrees.)" Venus appears in the east. The parentheses explain its movement: it travels the same degrees as in the west, hides for 10 days, and then retreats 8 degrees.
"Mercury: Appears in the morning and lasts for 33 days, traveling 6,120,055 'minutes'."
"Observe the West. For thirty-two days. (Excluding one degree of retrograde, it travels a total of thirty-two degrees.)" Mercury appears in the west, lasts for 32 days. Subtracting 1 degree of retrograde, it actually travels 32 degrees.
"Travels 65 degrees, totaling 6,120,055 'minutes'." Mercury travels 65 degrees, totaling 612005 'minutes'.
"Observe the East. Moves like in the West, hides for eighteen days, and then retreats by fourteen degrees." Mercury appears in the east, travels the same degrees as in the west, hides for 18 days, and then retreats 14 degrees.
The following sentences describe the calculation method, which is quite technical. We will try to explain it in a more colloquial way:
"Use a method to calculate the days a planet is hidden and the remaining degrees. Add the degrees left when the star aligns with the sun; if this reaches a certain threshold, continue with the previous calculations to find out when the star will appear and its travel distance." This is a complex algorithm that can only be accurately expressed in modern mathematical language.
"Multiply the star's movement denominator by the observed degrees. If the remainder is close to one day, even if it doesn’t divide evenly, count it as one; then, add the movement 'minutes'. When these 'minutes' reach a specific quantity, they convert to one degree. Note that retrograde and prograde movements follow different methods; multiply the current movement's denominator by the previous 'minutes'. If the result matches the previous denominator, you get the current 'minutes'."
"Those who remain continue with the previous; those who go against it decrease it; if unable to complete the measure, dividing by the passage of time, using the primary reference as the standard, the gains and losses influence each other." This sentence explains some special situations: if a planet stops moving (remains), it inherits the previous value; if it moves backwards, the value is decreased; if the hidden days are not enough to complete a cycle of movement, a certain method (dividing by the passage of time) is used to calculate the "increments," and the increments are adjusted according to the denominator of the movement, with the values influencing each other.
"All terms that are full and complete refer to precise divisions; removal and division are seeking the complete division." This sentence explains some terms in calculations: "full and complete" are precise divisions; "removal and division" is seeking the complete division.
"Jupiter: in the morning, it aligns with the sun, remains, follows, sixteen days one hundred seventy-four thousand two hundred thirty-three increments, planet two degrees three hundred twenty-three thousand four hundred six increments, and in the morning it is seen in the east, after the sun. Following, rapid, the sun travels eleven out of fifty-eight increments, fifty-eight days move eleven degrees. Continuing forward, but delayed, the sun moves nine increments, fifty-eight days move nine degrees. Remaining, not moving for twenty-five days then rotates. In retrograde, the sun moves one-seventh of a division, eighty-four days retreat twelve degrees. Once again remaining still, after twenty-five days it resumes movement, the sun moves fifty-eight increments out of nine, fifty-eight days move nine degrees. Following, rapid, the sun moves eleven increments, fifty-eight days move eleven degrees, in front of the sun, at dusk it is hidden in the west. Sixteen days one hundred seventy-four thousand two hundred thirty-three increments, planet two degrees three hundred twenty-three thousand four hundred six increments, and aligns with the sun. In total, three hundred ninety-eight days three hundred forty-eight thousand four hundred sixty-four increments, planet forty-three degrees two hundred fifty-nine thousand nine hundred fifty-six increments." This passage describes the movement of Jupiter, which is very complex, involving various situations such as forward movement, retrograde movement, remaining still, as well as the degrees of movement each day and every 58 days. It is not translated word for word here, as it requires astronomical knowledge to understand its meaning. In summary, this passage outlines the ancient methods of astronomical observation and calculation, which are highly specialized and intricate. This reflects the sophisticated calculation skills of ancient astronomers and their profound understanding of celestial movement patterns.
In the morning, when the sun and Mars met, Mars went into a period of invisibility. Then it started to move forward, moving for 71 days, covering a total of 1,489,868 minutes, which corresponds to a movement of 55 degrees and 242,860.5 minutes along its orbit. After that, people could see it in the east after sunrise. While in direct motion, Mars traveled 14 minutes and 23 seconds each day, covering 112 degrees in 184 days. Next, the speed of direct motion decreased, moving 12 minutes and 23 seconds each day, covering 48 degrees in 92 days. Then it stopped and remained motionless for 11 days. Then it started to move backward, traveling 17 minutes and 62 seconds each day, moving back 17 degrees in 62 days. It stopped again for 11 days, then started moving forward again, traveling 12 minutes each day, covering 48 degrees in 92 days. After that, the forward speed increased, moving 14 minutes each day, covering 112 degrees in 184 days. At this point, it ran in front of the sun, so people in the west could see it visible at night. After another 71 days, during which it covered 1,489,868 minutes and moved 55 degrees and 242,860.5 minutes along its orbit, it finally met the sun again. One cycle ended, taking a total of 779 days and 973,113 minutes, traversing 414 degrees and 478,998 minutes along its orbit.
In the morning, when the sun and Saturn met, Saturn went into a period of invisibility. Then it started to move forward, moving for 16 days, covering a total of 1,122,426.5 minutes, which corresponds to a movement of 1 degree and 1,995,864.5 minutes along its orbit. After that, people could see it in the east after sunrise. While in direct motion, Saturn traveled 3 minutes and 35 seconds each day, covering 7.5 degrees in 87.5 days. Then it stopped and remained motionless for 34 days. Then it started to move backward, traveling 1 minute and 17 seconds each day, moving back 6 degrees in 102 days. After another 34 days, it started moving forward again, traveling 3 minutes each day, covering 7.5 degrees in 87 days. At this point, it ran in front of the sun, so people in the west could see it visible at night. After another 16 days, during which it covered 1,122,426.5 minutes and moved 1 degree and 1,995,864.5 minutes along its orbit, it finally met the sun again. One cycle ended, taking a total of 378 days and 166,272 minutes, traversing 12 degrees and 1,733,148 minutes along its orbit.
Venus, when it meets the Sun in the morning, first disappears, then retrogrades, moving back four degrees over five days. Then it becomes visible in the east behind the Sun at dawn. After that, it remains motionless for eight days. Then it shifts to direct motion, slowing down, traveling forty-six and one-third degrees in a day and thirty-three degrees over forty-six days, then goes direct. As it continues in direct motion, its speed increases, traveling one degree and fifteen ninety-firsts in a day and one hundred and thirteen degrees in ninety-one days. Continuing direct motion, the speed increases further, traveling one degree and ninety-one twenty-seconds in a day and one hundred and thirteen degrees in ninety-one days. At this point, it is positioned behind the Sun, appearing in the east at dawn. In direct motion, over forty-one days, it travels fifty degrees and fifty-four thousandths, and the planet also travels fifty degrees and fifty-four thousandths, then meets the Sun again. One conjunction, totaling two hundred ninety-two days and fifty-four thousandths, with the planet traveling the same number of degrees.
When Venus meets the Sun in the evening, it first disappears, then goes direct, over forty-one days, traveling fifty degrees and fifty-four thousandths, and the planet travels fifty degrees and fifty-four thousandths, then in the evening, you can see it in the west, in front of the Sun. Going direct, speeding up, it travels one degree and ninety-one twenty-seconds in a day and one hundred and thirteen degrees in ninety-one days. Continuing direct motion, the speed slows down, traveling one degree and fifteen one-hundredths in a day and one hundred and six degrees in ninety-one days, then goes direct. Slowing down, it travels forty-six and one-third degrees in a day and thirty-three degrees over forty-six days. Then it stops, remaining motionless for eight days. Then it shifts to retrograde motion, moving three-fifths of a degree per day, retreating six degrees in ten days. At this point, it is positioned in front of the Sun, appearing in the west in the evening; it goes into retrograde motion and speeds up, retreating four degrees in five days, then meets the Sun again. Two conjunctions, totaling five hundred eighty-four days and eleven thousandths, with the planet traveling the same number of degrees.
Mercury, when it meets the sun in the morning, first disappears, then retrogrades, moving back seven degrees over nine days. Then, it can be seen in the east behind the sun in the morning. It continues to retrograde, speeding up, moving back one degree each day. It stops for two days. Then it turns direct, slowing down, moving eight-ninths of a degree each day, moving back eight degrees in nine days, then turning direct. Speeding up, it moves one and a quarter degrees per day, covering twenty-five degrees in twenty days, then appearing in the east behind the sun. Going direct, it moves 641 millionths of a degree in sixteen days, while the planet covers thirty-two degrees and 641 millionths of a degree, then meets the sun again. In total, it takes fifty-seven days and 641 millionths of a degree for the planet to cover the same distance.
Speaking of Mercury, when it sets with the sun, this alignment is known as "inferior conjunction." Its trajectory, sometimes direct, allows it to move three hundred twenty-six thousand four hundred sixteen millionths of a circle in sixteen days, allowing it to be seen in the west in the evening, positioned in front of the sun. When moving direct, it runs fast, moving one and a quarter degrees per day, able to cover twenty-five degrees in twenty days.
Sometimes it slows down, moving seven-eighths of a degree per day, taking nine days to move eight degrees. Sometimes it simply stops, not moving for two days. And sometimes it retrogrades, moving back one degree per day, where it can be seen in the west in the evening, in front of the sun. When retrograding, it is slow, taking nine days to move back seven degrees, until it meets the sun again.
The cycle from one conjunction with the sun to the next takes a total of 115 days and 601 millionths of a day. And that's how Mercury travels through the sky.
