This text describes ancient astronomical calculation methods, involving the movement patterns and calculation formulas of the five planets. Let's break it down sentence by sentence and explain it in terms we can understand today.
First, it defines the celestial bodies corresponding to the five elements: Jupiter is called the "Year Star," Mars is called the "Wandering Star," Saturn is called the "Filling Star," Venus is called the "Brilliant Star," and Mercury is called the "Star of the Morning." Their daily movement distances in the sky, as well as the number of weeks and days they travel in a year, all have fixed values. Then, it explains some calculation methods, such as how to determine the number of days in a month and how to calculate the distance a celestial body travels in a month, etc. These calculation methods involve many numbers, such as "the number of weeks in a year multiplied by a constant gives the calculation for that month," meaning that by multiplying the number of weeks in a given year by a constant, one can obtain the calculation method for that month. The subsequent "apply the method to calculate the number of months," and so on, are similar calculation steps, using various numbers for multiplication and division to ultimately obtain the movement data of the celestial bodies. These calculation processes are relatively complex and involve many astronomical constants. We don't need to get too caught up in the specific calculation methods; it's enough to know that it uses a particular method to calculate the trajectory of the celestial bodies.
Next, it begins calculating the specific data for the five planets. "The five planets' new moon degrees, large and small excess" refers to calculating the remaining degrees of the celestial bodies on the new moon day (the first day of the lunar month), divided into large excess and small excess. "The day the five planets enter the month and the remaining days" refers to calculating the number of days a celestial body enters a certain month and the remaining days. "Total degrees of movement and remaining degrees of the celestial bodies" refers to calculating the total degrees of movement and the remaining degrees, subtracting any excess over a week (360 degrees). These calculations use various constants mentioned earlier, such as "general method," "daily method," and "meeting numbers," which are coefficients established for ease of calculation.
Then, it lists various constants needed for the calculation, such as the year length of 7285, the leap year of 7, the month length of 235, the number of months in a year of 12, and so on. These numbers were all summarized by ancient astronomers through long-term observations. Specific calculation data for Jupiter and Mars are also provided, including their orbital periods, daily rates, synodic periods, lunar excess, synodic month method, daily method, large lunar excess, small lunar excess, entry day of the month, daily excess, fictitious lunar division, constellation division, degrees, and degree excess. These data were all derived based on the previously mentioned calculation methods, used to predict the positions of Jupiter and Mars. While these numbers may seem complex, they represent the ancient astronomers' comprehension and exploration of cosmic laws. Although these calculation methods may seem cumbersome, they reflect the meticulous observation of astronomical phenomena and the advanced mathematical skills of ancient civilizations. "The year length is seven thousand two hundred and eighty-five. The leap year is seven. The month length is two hundred and thirty-five. The number of months in a year is twelve. The general method is forty-three thousand twenty-six. The daily method is one thousand four hundred and fifty-seven. The number of conjunctions is forty-seven. The total days is two hundred and fifteen thousand one hundred and thirty. The constellation division is one hundred and forty-five." These figures serve as the foundation for the calculations.
Finally, it presented specific parameters for Jupiter and Mars for detailed calculations. This data illustrates how ancient astronomers utilized these formulas and constants to predict planetary orbits. While we now employ more advanced techniques for astronomical calculations, these ancient methods still hold significant historical value, highlighting the remarkable achievements of ancient Chinese astronomy. "Jupiter: circumference (周率), 6722, daily rate: 7341, synodic month: 13, remainder of the synodic month: 6481, synodic month method: 12718, daily method: 3959258, remainder of the new moon: 23, remainder of the small new moon: 1307, entry of the moon: 15, remainder of the day: 3484646, new moon virtual division: 150, Dipper division: 974690, degrees total: 33, remainder of the degrees: 2509956. Mars: circumference (周率), 347, daily rate: 7271, synodic month: 26, remainder of the synodic month: 25627, synodic month method: 64733, daily method: 206723, remainder of the new moon: 47, remainder of the small new moon: 1157, entry of the moon: 12, remainder of the day: 973113, new moon virtual division: 300, Dipper division: 49415, degrees total: 48, remainder of the degrees: 1991706." These are the detailed calculation results for Jupiter and Mars. The circumference (周率) of Saturn is 3529, running 3653 per day. When calculated, one month consists of 12 days, with a remainder of 53,843. Calculated by month, it totals 67,051; calculated by day, it is 2,785,881. The remainder of the new moon is 54, the small remainder is 534. Each month is counted as having 24 days, leaving a remainder of 166,272. The new moon virtual division is 923, the Dipper division is 51,175, the degrees total 12, with a remainder of 173,148.
The orbital period of Venus is 922, traveling 7,213 units each day. For one month, it amounts to 9, with a remainder of 152,293. When calculated monthly, the total is 171,418; when calculated daily, the total is 5,319,958. The new moon remainder is 25, with a remainder of 1,129. There are 27 days in a month, with a remainder of 56,954. The new moon remainder is 328, the Dipper fraction is 1,308,190, the degree is 292, with a remaining degree of 56,954.
The orbital period of Mercury is 11,561, traveling 1,834 units each day. For one month, it amounts to 1, with a remainder of 211,331. When calculated monthly, the total is 219,659; when calculated daily, the total is 680,429. The new moon remainder is 29, with a remainder of 773. There are 28 days in a month, with a remainder of 641,967. The new moon remainder is 684, the Dipper fraction is 1,676,345, the degree is 57, and the remaining degree is 641,967.
First, identify the reference point for the year you wish to calculate, then multiply it by the orbital period. If it is divisible by the daily rate, it is recorded as the accumulated total, and the accumulated remainder is the portion that cannot be divided. Divide the accumulated remainder by the orbital period, and if it is divisible, if the quotient is one, it is the previous year; if the quotient is two, it is the previous two years; if it cannot be divided, it is the current year. Subtract the orbital period from the accumulated remainder to determine the degree. For the accumulated values of Venus and Mercury, odd numbers correspond to the morning, while even numbers correspond to the evening.
First, let's calculate the lunar matters. Multiply the value of each month by the remaining months, add them up; if it is divisible by the lunar calculation method, count it as a complete month; if it is not enough to be divided, then it is the remaining months. Then subtract the accumulated months from the months that have passed, and the remaining months will be the number of months entering the next month. Multiply by the leap month; if it is divisible by the leap month method, you will have a leap month. Then subtract it from the number of months entering the next month; the remaining part will be used in the calculation within a year, this part is referred to as "Tianzheng Suanwai, He Moon." If there is a leap month transition, adjust it with the new moon day.
Next is the calculation of the conjunction of constellations. Multiply the remaining months using the standard method, then multiply by the conjunction method the remaining new moon days, add these two results together, and then simplify the result. If the result is divisible by the daily calculation method and you get 1, then that is the date of the conjunction of constellations with the new moon; if it is not enough to be divided, the remaining part is the day remainder, this part is called "New moon calculation outside." Then, multiply the week days by the degree minutes; if it is divisible by the daily calculation method, you will get one degree; if not enough to be divided, the remaining part is the remainder, and this remainder starts counting from the five degrees before the Ox. The above is the calculation method for the conjunction of constellations.
Next is the calculation of the year. Sum the months and the remaining months; if it is divisible by the conjunction method, you will get a month; if it is not enough to be divided, then see which year it is; if it is divisible, subtract it, consider the leap month, and the remaining part is the next year; then when it reaches the conjunction method, it will be the next two years. Venus and Mercury, add morning is evening, add evening is morning. (This refers to the times of appearance of Venus and Mercury).
Then, add the remainder of the new moon day and the remainder of the conjunction. If it exceeds one month, then add a large remainder of twenty-nine and a small remainder of seven hundred seventy-three. If the small remainder is divisible by the daily calculation method, then subtract from the large remainder; the method is the same as before. Add the new moon day and the day remainder; if it is divisible by the daily calculation method, you will get one day; if the small remainder of the previous new moon day is divisible by the imaginary part, then subtract one day; if the small remainder of the later new moon day exceeds seven hundred seventy-three, then subtract twenty-nine days; if it is not enough, then subtract thirty days; the remaining part will represent the new moon day of the next conjunction.
Finally, add up the degrees, add up the remainder of the degrees; if it is divisible by the daily calculation method, you will get one degree. Below are the specific data for Jupiter, Mars, Saturn, and Venus:
Jupiter was in retrograde for 32 days, totaling 3,484,646 minutes; it was visible for 366 days; it moved retrograde by 5 degrees, totaling 2,509,956 minutes; it moved visibly by 40 degrees. (Retrograde by 12 degrees, it ultimately covered 28 degrees.)
Mars was in retrograde for 143 days, totaling 973,113 minutes; it was visible for 636 days; it was retrograde for 110 degrees, totaling 478,998 minutes; it moved visibly by 320 degrees. (Retrograde by 17 degrees, it ultimately covered 303 degrees.)
Saturn was in retrograde for 33 days, totaling 166,272 minutes; it was visible for 345 days; it was retrograde for 3 degrees, totaling 1,733,148 minutes; it moved visibly by 15 degrees. (Retrograde by 6 degrees, it ultimately covered 9 degrees.)
In the morning, Venus was retrograde in the east for 82 days, totaling 113,908 minutes; it was visible in the west for 246 days. (Retrograde by 6 degrees, it ultimately covered 240 degrees.) In the morning, it was retrograde by 100 degrees, totaling 113,908 minutes; it was visible in the east. (Daily movement the same as in the west, retrograde for 10 days, retrograde by 8 degrees.)
Mercury appears in the morning after 33 days. It has traveled a total of 6,002,505 minutes. Then, it appeared in the west and lasted for 32 days. (Subtracting one degree of retrograde, it ultimately moved 32 degrees.) When it was retrograde, it moved 65 degrees, still totaling 6,002,505 minutes. After that, it appeared in the east. Its speed in the east is the same as in the west, retrograde for 18 days, and it moved back 14 degrees.
Calculate the daily movement of Mercury and the remaining degrees; add the remaining degrees of the star and the sun. If the remaining parts meet the daily movement standard, it is recorded as one. According to the previous calculation method, the appearance of the star next to the sun and the movement of degrees can be calculated. Multiply the denominator of the star's movement by the observed degrees; if the remaining part meets the daily movement standard, it is recorded as one. If the denominator cannot be divided completely, and if it exceeds half, it is also recorded as one. Then add the movement fraction; if the fraction is full, it is recorded as one degree. The denominators of retrograde and direct movement are different; multiply the current denominator of movement by the original fraction. If it is the same as the original denominator, it is recorded as one; this represents the current movement fraction. The remaining parts inherit the previous ones; subtract for retrograde. If retrograde has not completed the specified degrees, use the Big Dipper to divide by the fraction, using the denominator of movement as a ratio. The fraction will increase or decrease, mutually restricting each other. Any concepts described as full or filled, similar to profit and loss, refer to the division of real numbers; "go" and "divide" are all division by taking the remainder.
Jupiter appears in the morning alongside the sun and then goes retrograde and then goes direct. Over sixteen days, it spans 1,742,323 minutes, and the planet itself also travels 2 degrees and 323,467 minutes. It then appears in the eastern sky in the morning, behind the sun. During direct motion, it moves at a rate of 11 out of 58 per day, covering 11 degrees over 58 days. Then, during direct motion, it slows down, moving at a rate of 9 out of 58 per day, covering 9 degrees over 58 days. It pauses, remaining stationary for 25 days before starting to rotate again. In retrograde, it moves at a rate of 1/7 per day, retreating 12 degrees over 84 days. After another pause of 25 days, it goes direct again, moving at a rate of 9 out of 58 per day, covering 9 degrees over 58 days. Direct motion resumes, with a fast speed, moving at a rate of 11 minutes per day, covering 11 degrees over 58 days. At this point, it is positioned in front of the sun and disappears in the western sky by evening. Over sixteen days, it spans 1,742,323 minutes, and the planet itself also travels 2 degrees and 323,467 minutes before appearing with the sun again. In this manner, one cycle is completed, totaling 398 days and spanning 3,484,646 minutes, with the planet traveling a total of 43 degrees and 250,956 minutes.
Sun: It appears alongside the sun in the morning, then goes into hiding. Next is direct motion, a total of 71 days, spanning 1,489,868 minutes, which indicates that the planet travels 55 degrees and 242,860.5 minutes. Then, it can be seen in the eastern sky in the morning, behind the sun. During direct motion, it moves at a rate of 14 out of 23 per day, covering 112 degrees over 184 days. Direct motion resumes, but the speed decreases, moving at a rate of 12 out of 23 per day, covering 48 degrees over 92 days. It then stops, remaining stationary for 11 days. Then it goes retrograde, moving at a rate of 17 out of 62 per day, retreating 17 degrees over 62 days. After another 11-day pause, it goes direct again, moving at a rate of 12 minutes per day, covering 48 degrees over 92 days. Direct motion resumes, with the speed increasing, moving at a rate of 14 minutes per day, covering 112 degrees over 184 days. At this point, it is positioned in front of the sun and disappears in the western sky by evening. Over seventy-one days, it spans 1,489,868 minutes, and the planet travels 55 degrees and 242,860.5 minutes before appearing again with the sun. In this manner, one cycle is completed, totaling 779 days and spanning 973,113 minutes, with the planet traveling a total of 414 degrees and 478,998 minutes.
Saturn: It appears in the morning with the sun, then it disappears. Next, it goes direct for a total of 16 days, traveling a distance equivalent to 1,122,426.5 minutes, while the planet moves 1 degree and 1,995,864.5 minutes. Then it can be seen in the east in the morning, behind the sun. During its direct motion, it moves 3/35 of a degree each day, covering 7.5 degrees in 87.5 days. After that, it stops and stays still for 34 days. Then it goes retrograde, moving backward 6 degrees over 102 days, while moving 1/17 of a degree each day. After another 34 days, it goes direct again, moving 3 minutes each day, covering 7.5 degrees in 87 days. At this point, it is in front of the sun and hides in the west in the evening. In 16 days, it travels 1,122,426.5 minutes, while the planet moves 1 degree and 1,995,864.5 minutes, and then it appears with the sun again. In total, this cycle takes 378 days and 166,272 minutes, and the planet moves 12 degrees and 1,733,148 minutes.
As for Venus, when it is in conjunction with the sun in the morning, it first hides away, then goes retrograde, moving back 4 degrees in 5 days, after which it can be seen in the east, behind the sun. During its retrograde motion, it moves 3/5 of a degree each day, retreating 6 degrees in 10 days. Then it stops for 8 days. After that, it turns direct and slows down, moving 33/46 of a degree each day, covering 33 degrees in 46 days, and begins moving direct. Its speed increases, moving 15/91 of a degree each day, covering 160 degrees in 91 days. Then it moves even faster in direct motion, covering 113 degrees in 91 days at a rate of 22/91 of a degree each day, at which point it is behind the sun, appearing in the east in the morning. During the direct motion, it traces out 1/56,954 of a full circle in 41 days, while the planet also travels 50 degrees and 1/56,954 of a full circle, and then it conjunctions with the sun again. In total, the conjunction takes 292 days and 1/56,954 of a full circle, and the distance traveled by the planet matches this.
When Venus meets the sun at night, it first hides, then moves along, traversing one fifty-six-thousand-ninth of a circle for forty-one days. The planet moves fifty degrees, which is one fifty-six-thousand-ninth of a circle, then it can be seen in the west, positioned in front of the sun. It moves along, speeding up, moving one degree and ninety-one twenty-second of a degree per day, totaling one hundred thirteen degrees in ninety-one days. Then the speed decreases to fifteen minutes of movement per day, covering one hundred sixty degrees in ninety-one days. After that, the speed slows down, moving forty-six minutes and thirty-three degrees per day, totaling thirty-three degrees in forty-six days. Then it comes to a halt for eight days. Next, it turns around and moves backward, retreating five minutes and three degrees per day, retreating six degrees in ten days; at this point, it is located in front of the sun and can be seen in the west at night. The backward movement accelerates, retreating four degrees in five days, and then it meets the sun again. Two conjunctions are counted as one cycle, a total of five hundred eighty-four days and one hundred thirty-nine thousand eight hundred one-thousandth of a circle, with the planet covering the same distance.
When Mercury meets the sun in the morning, it first hides, then moves backward, retreating seven degrees in nine days, and then it can be seen in the east in the morning, behind the sun. The backward movement accelerates, retreating one degree per day. Then it stops, not moving for two days. After that, it shifts to forward motion, slowing down, moving eight-ninths of a degree each day, totaling eight degrees in nine days, before starting to move forward. Speeding up, it moves one and a quarter degrees per day, covering twenty-five degrees in twenty days; at this point, it is located behind the sun and can be seen in the east in the morning. While moving forward, it covers six hundred forty-one million nine hundred sixty-seven-thousandths of a circle in sixteen days, and the planet also moves thirty-two degrees six hundred forty-one million nine hundred sixty-seven-thousandths of a circle, before meeting the sun again. One conjunction totals fifty-seven days and six hundred forty-one million nine hundred sixty-seven-thousandths of a circle, with the planet covering the same distance.
When the sunset and the sun appear at the same time, Mercury will hide, following its orbital pattern. At 16 days, 6 hours, and 41 minutes, 967 thousand, and 667 seconds, it will move to the position of 32 degrees, 6 minutes, and 41 million, 967 thousand, and 667 seconds on the ecliptic, then appear in the west in the evening, appearing before the sun. If Mercury moves quickly, it travels one and a quarter degrees in a day, which means it covers twenty-five degrees in twenty days. If it moves slowly, it travels 0.875 degrees in a day, covering eight degrees over nine days. If Mercury stops, it remains still for two days. If Mercury is moving slowly in retrograde, it retreats one degree in a day, appearing before the sun, and then hides in the west at dusk. If it's moving slowly in retrograde, it retreats seven degrees over nine days, before eventually conjoining with the sun. From one conjunction to the next, it takes a total of 115 days, 6 hours, and 1 million, 225 thousand, and 505 seconds, and the same is true for other planets.
Jupiter is called the Planet of the Year, Mars is called the Star of Fire, Saturn is called the Star of Filling, Venus is called the Star of Purity, and Mercury is called the Morning Star. The movements of these five planets vary in speed and can be direct or retrograde. From the beginning of the heavens and the earth, when yin and yang separated, the sun, moon, and five planets gathered in the twenty-eight mansions of the zodiac. Starting from the constellations, they move together in the sky at different speeds, chasing each other. When stars meet the sun in the same constellation on the same day, it is called a conjunction. The time between one conjunction and the next is called a cycle. By simplifying the number of days in each cycle and the days in a year, the ratio of the cycle's days to the days in a year can be calculated, which is the conjunction cycle number, and the number of conjunctions occurring in a year is the conjunction count. Once these two ratios are determined, various data can be calculated. Multiplying the number of days in a year by the conjunction count gives the formula for calculating the conjunction month; multiplying the number of days in an era by the conjunction count gives the formula for the daily number; multiplying the number of months in a year by the number of cycles gives the fraction of the conjunction month; if the formula for the conjunction month is a whole number, the remainder is the month remainder. Multiplying the total number of days by the month remainder, then multiplying by the formula for the conjunction month, adding these two numbers, and then dividing by the formula for the daily number multiplied by the formula for the conjunction month gives the number of days for the star conjunction in the month. The remainder is then reduced by the total number of days to get the remainder of the day in the month. Subtracting the month remainder from the daily number formula gives the fractional month remainder. Multiplying the numerical value of the constellation in the calendar by the number of conjunctions gives the star degree constellation. For Jupiter, Mars, and Saturn, subtracting the total days in a year from the number of conjunctions, multiplying the remaining number by the number of degrees in a week, and then dividing by the daily number formula gives the degree of the planet, with the remaining value representing the degree remainder. For Venus and Mercury, multiplying the number of degrees in a week by the number of days in a year, and then dividing by the daily number formula, gives the degree of the planet, with the remaining value representing the degree remainder.
For Jupiter: Conjunction end age, 1255.
Conjunction end number, 1149.
Conjunction month formula, 21831.
Daily number formula, 2117607.
Conjunction count, 13.
Month remainder, 11122.
Shuo Dayu: twenty-three.
Shuo Xiaoyu: four thousand ninety-three.
The first day is the fifteenth of the month.
There is an additional 199,564 (unit unclear) this month.
Shuo Xu Fen: four hundred sixty-six.
Dou Fen: five hundred twenty-two thousand seven hundred ninety-five.
The planetary degree is thirty-three.
An additional 1,472,869 degrees have been recorded.
Mars: The total age is 5,105, the total combination is 2,388, the monthly combination is 45,372, the daily degree method is 4,001,084, the combined month number is twenty-six, and there is an additional twenty-three thousand in a month.
Shuo Dayu is forty-seven, Shuo Xiaoyu is three thousand six hundred twenty-seven.
The second day is the thirteenth of the month.
An additional 358,530 (unit unclear) is recorded on this day.
Shuo Xu Fen: nine hundred thirty-two.
Dou Fen: one hundred eighty-six thousand five hundred forty.
The planetary degree is fifty.
An additional 1,412,150 degrees have been recorded.
Saturn: The total age is three thousand nine hundred forty-three, the total combination is three thousand eight hundred nine, the monthly combination is seventy-two thousand three hundred seventy-one, the daily degree method is seven million nineteen thousand eight hundred eighty-seven, the combined month number is twelve, and there is an additional fifty-eight thousand one hundred fifty-three in a month.
Shuo Dayu is fifty-four, Shuo Xiaoyu is one thousand six hundred seventy-four.
The third day is the twenty-fourth of the month.
An additional 67,536 (unit unclear) is recorded on this day.
Shuo Xu Fen: two thousand eight hundred eighty-five.
Dou Fen: one hundred seventy-three thousand three hundred ninety-five.
The planetary degree is twelve.
An additional 596,256 degrees have been recorded.
Venus: The total age is 1,907, the total combination is 2,385, the monthly combination is 45,315, the daily degree method is 439,555, the combined month number is nine, and there is an additional forty-three thousand ten in a month.
Shuo Dayu is twenty-five, Shuo Xiaoyu is three thousand five hundred thirty-five.
The final day is the twenty-seventh of the month.
On this day, an additional 194,990 units (exact unit unspecified) were recorded. The Shuo division is 1,024. The Dou fraction is 1,851,175. The degree of the planet Mercury is 292. An additional 194,990 degrees were added. For Mercury: the total age is 1,870, the total combination is 11,789, and the total lunar method is 223,991.
In a certain year AD, the calculation method is as follows: there are 21,727,127 days in a year. A month is counted as 1 day. There are 215,459 additional days beyond the standard calendar in a year. The remaining days after the new moon are 29 days, and the remaining days after the new moon cycle total 2,419 days. Each month is comprised of 28 days. There are 23,444,261 additional days in a year. The Shuo division is 2,140. The Dou fraction is 5,363,995. The planetary degree is 57. The remaining degrees are 2,344,361 days.
The method for calculating the positions of the five celestial bodies is as follows: starting from the year of Renchen, multiply the year you want to calculate by a number called the "total combination number." If the result is exactly a multiple of the total age, it is called "integrated combination"; if it is not a multiple, the leftover value is referred to as "the combined remainder." Subtract the "combined remainder" from the "total combination number." If the result is 1, it means that the celestial body is in conjunction in the year being calculated; if it is 2, it means that the conjunction is in the second year of the calculation; if the result is 0, it means the conjunction is in the current year. The remaining part is used to calculate the degree division. For the conjunction of Venus and Mercury, even numbers indicate morning, while odd numbers indicate evening.
The method for calculating the conjunction of the five celestial bodies with the moon is as follows: multiply the number of months and the remaining months by the integrated combination; if the result is a multiple of the lunar method, it is called integrated month, and the remaining part is the lunar remainder. Divide the recorded month by the integrated month; the quotient is the recorded month, and the remainder is the entry month. Then multiply the chapter intercalation by the entry month; if the result is a multiple of the chapter month, it means it is an intercalary month; then subtract the entry month, and the remaining part represents the entry year. Starting from Tianzheng, the quotient is the star conjunction month. If it is at the time of the leap month transition, use the new moon day to adjust.
The method for calculating the conjunction of the moon's conjunction with the new moon is as follows: multiply the common number by the entry month; if the result is a multiple of the daily method, it is called the integrated day, and the leftover value is referred to as the small remainder. Divide the integrated day by 60; the remaining part is the large remainder. Starting from the entry month, the quotient is the star conjunction new moon day.
The method for calculating the entry month and day is outlined below: multiply the common number by the remaining month, then multiply the lunar method by the new moon small remainder, add these two results together, then reduce using common methods. If the result is a multiple of the daily method, it means the celestial body is in conjunction on the entry month and day; if it is not a multiple, the remaining part is the daily remainder. Starting from the new moon day, the quotient is the entry month and day.
To calculate the degree of conjunction, the method is: multiply the sidereal day by the degrees and minutes; if the result is a whole number multiple of the solar degree method, it is called a degree, and the remainder is the leftover part. Starting from the five degrees before Taurus, the quotient is the degree of the celestial body conjunction.
To calculate the next new moon, the method is: add the lunar month to the current year, and add the remainder to the lunar remainder. If the result is a whole number multiple of the new moon method, it indicates a month. If the result is less than a year, it is within that year; if the result is greater than or equal to a year, subtract one year, and if there is a leap month, take that into account; the remaining part indicates the year of the next new moon; if the result is greater than two years, continue in this manner. For Venus and Mercury, if they are conjunct in the morning, the next conjunction will be in the evening; if they are conjunct in the evening, the next conjunction will be in the morning.
To calculate the next conjunction of the new moon, the method is: add the large remainder and small remainder of the new moon respectively to the large remainder and small remainder of the new moon month. If the lunar remainder exceeds one month, add 29 days (large remainder) and 2419 days for the small remainder. If the small remainder is a whole number multiple of the solar method, start calculating from the large remainder, using the same method as before.
This passage outlines methods used in ancient astronomical calculations; let’s break it down sentence by sentence.
Firstly, the first paragraph discusses how to calculate the "next new moon date," which is the date of the new moon (the first day of the lunar month) next month. "To find the next new moon date: add the new moon date, the solar remainder to the new moon date and the remainder, and if the remainder exceeds the standard (solar degree method, the specific standard is not clearly stated), subtract one." This means: to calculate the first day of next month, first add the date of this month's first day, the remaining days of this month, and the remaining days of last month. If the total number of days exceeds a certain standard (solar degree method, which is not specifically stated), subtract one. "If the previous new moon's small remainder fills its virtual division, subtract one day; if the later small remainder exceeds 2419, subtract 29 days; if not, subtract 30 days; the remaining days will be the new moon date of next month, referred to as the new moon." If the remaining days calculated from last month's first day are insufficient, subtract one day; if this month's remaining days exceed 2419, subtract 29 days; if not, subtract 30 days. The remaining days will be the date of next month's first day, referred to as the new moon. "To find the next conjunction degree, use the degree and minutes, as previously calculated for the star positions." Next, calculate the degree for the first day of next month using the same method as before for determining the positions of the stars.
Next, the second paragraph talks about the orbit of Jupiter. "Jupiter: in the morning, it merges with the sun, hides and follows. On the sixteenth day, it travels 99,783.2 minutes and 2 degrees, and 179,523.8 minutes, and is seen in the east after the sun." This sentence describes the trajectory of Jupiter after it merges with the sun. "Following at a fast speed, it travels 11/57 degrees in a day and 11 degrees in 57 days. Following at a slow speed, it travels 9/57 degrees in a day and 9 degrees in 57 days, and then it stops." When Jupiter moves forward at a fast speed, it travels 11/57 degrees in a day and 11 degrees in 57 days; when it moves slowly, it travels 9/57 degrees in a day and 9 degrees in 57 days, and then it stops. "It remains stationary for 27 days before resuming movement." After stopping for 27 days, it starts moving again. "In reverse, it travels one-seventh of a degree per day, retreats 12 degrees after 84 days." When it moves in reverse, it travels one-seventh of a degree per day and retreats 12 degrees after 84 days. "After 27 days, it resumes slow movement, traveling 9/57 degrees in a day and 9 degrees in 57 days, and then moves forward again." After 27 days, it moves forward slowly again, traveling 9 degrees in 57 days. "At a fast speed, it travels 11/57 degrees in a day and 11 degrees in 57 days, moves ahead of the sun, and sets in the west at dusk." Then the speed increases, traveling 11/57 degrees in a day and 11 degrees in 57 days, moving ahead of the sun and setting in the west at dusk. "Following, on the sixteenth day, it travels 99,783.2 minutes and 2 degrees, and 179,523.8 minutes, and merges with the sun." Then it follows again, and after sixteen days, it merges with the sun again. "Overall, after completing one cycle, it takes 398 days, totaling 995,664 minutes, covering a distance of 33 degrees." Jupiter completes its orbit in 398 days, covering a distance of 33 degrees.
The last paragraph discusses the movement pattern of Mars, which is similar to the description of Jupiter. "Mars: When the morning and the sun are in conjunction, hidden, 72 days later, 1,790,215 minutes, 56 degrees, 124,945 minutes of the planet, and appears in the east, trailing behind the sun." After Mars and the sun meet, it appears in the east 72 days later, trailing behind the sun. "In the direction, the sun moves 14 minutes out of 23 minutes of arc each day, 184 days to 112 degrees. More smoothly, late, the sun moves 12 minutes, 92 days to 48 degrees before coming to a halt." When Mars is moving smoothly and quickly, it travels 14 minutes of arc each day, and 112 degrees in 184 days; when it is slow, it travels 48 minutes of arc over 92 days, before coming to a halt. "It does not move for 11 days and then rotates." After stopping for 11 days, it starts moving again. "When in retrograde, it travels 17 minutes of arc each day for 62 days, retreats 17 degrees, and then stays again." When in retrograde, it travels 17 minutes of arc each day for 62 days, retreats 17 degrees, and then stops. "After 11 days, it resumes direct motion, traveling 48 minutes of arc over 92 days, and then speeds up." After 11 days, it resumes direct motion, traveling 48 minutes of arc over 92 days, and then speeds up. "It moves 14 minutes a day, 112 degrees in 184 days, moving ahead of the sun, and sets in the west at dusk." It moves 14 minutes a day, 112 degrees in 184 days, moving ahead of the sun, and sets in the west in the evening. "In the direction, 1,790,215 minutes, 56 degrees, 124,945 minutes of the planet, and in conjunction with the sun." Then it moves smoothly again; 72 days later, it meets the sun again. "In total, over one cycle of 780 days, Mars has traveled 415 degrees, which is equivalent to 3,585,230 minutes of arc." In one cycle, a total of 780 days, Mars has traveled 415 degrees.
In the morning, Saturn and the sun are in conjunction, and then Saturn becomes obscured. For nineteen days, three hundred eighty-four thousand seven hundred sixty-five and a half minutes, Saturn moved two degrees six hundred forty-nine thousand one hundred twenty-one and a half minutes. Then, in the morning, it can be seen in the east, at this time it is behind the sun. During direct motion, it moves thirteen parts for every one hundred seventy-two. After eighty-six days, it moves six and a half degrees and then stops. After stopping for thirty-two and a half days, it then resumes its rotation. In retrograde motion, it moves one seventeenth each day; after one hundred and two days, it retreats six degrees and then stops again. After stopping for thirty-two and a half days, it resumes direct motion, moving one thirteenth each day, and moves six and a half degrees in eighty-six days. At this time, it is in front of the sun and obscured in the west at night. During direct motion, for nineteen days, three hundred eighty-four thousand seven hundred sixty-five and a half minutes, Saturn moved two degrees six hundred forty-nine thousand one hundred twenty-one and a half minutes, and then it is in conjunction with the sun again. At the end of one cycle, a total of three hundred seventy-eight days, sixty-seven thousand five hundred thirty-six minutes, Saturn moved twelve degrees five hundred ninety-six thousand two hundred fifty-six minutes.
In the morning, Venus and the sun are in conjunction, and then Venus becomes obscured. In six days, it moves backward four degrees, then in the morning it can be seen in the east, at this time it is behind the sun and moving retrograde. During slow motion, it moves three parts for every five each day, moving backward six degrees in ten days. After stopping, it starts rotating again after seven days of stillness. During direct motion, when moving slowly, it moves thirty-three parts for every forty-five each day, moving thirty-three degrees in forty-five days and then continues forward. During rapid motion, it moves fourteen parts for every ninety-one each day, moving one hundred five degrees in ninety-one days and then continues forward. During even faster motion, it moves twenty-one parts for every ninety-one each day, moving one hundred twelve degrees in ninety-one days; at this time, it is behind the sun and obscured in the east in the morning. During direct motion, for forty-two days, nineteen thousand four hundred ninety minutes, Venus moved fifty-two degrees nineteen thousand four hundred ninety minutes, and then it is in conjunction with the sun again. At one conjunction, a total of two hundred ninety-two days nineteen thousand four hundred ninety minutes, Venus also moved this many degrees.
In the evening, Venus aligns with the Sun, and then Venus "disappears." In direct motion, for forty-two days, 194,990 minutes, Venus travels 52 degrees and 194,990 minutes, and then it can be seen in the evening in the west; at this time, it is in front of the Sun. In direct motion, at a fast pace, it moves 0.21 degrees each day, traveling 112 degrees in ninety-one days, and then continues in direct motion. At a slow pace, it moves 1/14 of a degree each day, traveling 105 degrees in ninety-one days, then continues in direct motion. At an even slower pace, it moves 33/45 of a degree each day, traveling 33 degrees in forty-five days, and then stops. After stopping for seven days, it resumes its motion. In retrograde motion, it moves 0.6 degrees each day, moving 6 degrees in ten days; at this time, it is in front of the Sun, "disappearing" in the western sky in the evening. In retrograde motion, it moves 4 degrees in six days, and then aligns with the Sun again. The cycle ends with two alignments, totaling 584 days and 389,980 minutes, and Venus travels the same number of degrees.
In the morning, Mercury aligns with the Sun, and then Mercury "lies low," meaning it becomes invisible to observers. About eleven days later, Mercury retreats 7 degrees each day, and then reappears in the east; at this time, it is behind the Sun. If Mercury is moving in retrograde at a fast pace, retreating 1 degree each day, it will stop and then start moving in the opposite direction again after one day. If Mercury is in direct motion at a slow speed, moving 1 degree each day, it will travel 7 degrees in eight days, and then stop. If moving quickly, it travels 4/18 of a degree each day, and it will travel 22 degrees in eighteen days; at this time, it is behind the Sun, appearing in the east in the morning. If the direct motion speed is slow, it will travel 36 degrees and 23,446,261 minutes in eighteen days, and then align with the Sun again. The total cycle for one alignment is 57 days and 23,446,261 minutes.
At night, Mercury and the sun meet, then "retreat," walking thirty-six degrees and two million three hundred and forty-four thousand two hundred and sixty-one minutes over the course of eighteen days, and then appearing in the west. At this point, it is positioned in front of the sun. If Mercury's speed in direct motion is fast, it moves one degree every eighteen minutes, totaling twenty-two degrees in eighteen days, and then continues moving forward. If the speed is slow, it moves seven degrees in eight days, then stops. One day later, it starts moving in reverse. If Mercury is in retrograde, it retreats one degree per day; at this time, it is in front of the sun and appears in the west at night. If the retrograde speed is fast, it retreats seven degrees after eleven days and then meets the sun again. The complete cycle of the two conjunctions lasts one hundred and fifteen days and includes one million eight hundred and ninety-six thousand one hundred and ninety-five minutes, and that's how Mercury moves.
The method of calculating Mercury's movement is: add up the extra degrees when Mercury and the sun meet. If it exceeds the number of degrees in a year, subtract the number of degrees in a year; the remaining amount is the time and number of degrees when Mercury reappears. Multiply the denominator of Mercury's movement by the degrees seen; if the result equals the number of degrees in a year, the calculation is considered complete. If it does not divide evenly and exceeds half, it is considered the number of degrees in a year. Then add the number of degrees of Mercury's movement; if the number of degrees exceeds the denominator, add one degree. The denominators of direct motion and retrograde motion are different; use the current denominator of movement multiplied by the previous degrees, divided by the previous denominator, to get the current degrees of movement. If Mercury stops, use the previous degrees; if in retrograde, subtract the previous degrees. If the time of "retreat" is not sufficient to complete one degree, use dipper units to calculate, using the denominator of movement as a proportion. The number of degrees will increase or decrease, affecting each other before and after.
Liu Zhi, the attendant of Emperor Wu of Han, modified the calendar with the Dipper calendar, calculated the "Four-Part Method," and stated that one day would be lost every three hundred years, using one hundred and fifty as the rule for degrees and thirty-seven as the dipper units. It was calculated that Jiazi is the upper element, and by the tenth year of Taishi, it is Jiawu year, totaling ninety-seven thousand four hundred and eleven years. During the midnight of the winter solstice, at the start of the Jiazi cycle, the sun, moon, and five stars align according to the celestial records, which led to the establishment of the calendar's origins. He also incorporated some fictional elements, naming this calendar "Zhengli."
The Duke of Dongyang, Du Yu, wrote the "Spring and Autumn Calendar," which says: ... The sun moves one degree each day, and the moon moves thirteen degrees, nineteen minutes, and seven seconds each day. The officials in charge of astronomy and calendar calculations must determine the new moon day (first day of the lunar month) and the last day of the month based on the speed of the sun and moon, in order to decide whether to add an intercalary month. The intercalary month does not have a solar term (the second solar term of each month). The direction of the Big Dipper is also different during this month. These are characteristics of an intercalary month. By cross-checking these methods, the eight solar terms of the four seasons can be accurately determined, and only then can an accurate calendar be established; the intricacies of this are truly profound! If one can accurately grasp these subtle points, they can align with the laws of nature and avoid mistakes in their actions. The "Spring and Autumn Annals" says: "The establishment of an intercalary month is to correct the time. With accurate time, things can be done well."
However, the laws of the operation of yin and yang will produce slight deviations over time; these deviations accumulate, leading to discrepancies with the actual calendar. Therefore, Confucius and Qiu Ming (also known as Zuo Qiu Ming) often write articles when discussing new moon days and intercalary months, with the aim of correcting errors in the calendar and clarifying its laws. Liu Zijun's "Three Correct Calendars" was compiled to check the "Spring and Autumn Annals," recording a total of 34 solar eclipses in two distinct records, but when calculated using the "Three Correct Calendars," only one solar eclipse was recorded, resulting in the largest discrepancy compared to other calendars. Moreover, it stipulates adding one day every six thousand years, so the more years, the greater the error; this is simply unworkable.
From ancient times to the present, many people who interpret the "Spring and Autumn Annals" have made mistakes. Some invent their own methods, while others use various calendars since the time of the Yellow Emperor to calculate the new moon days recorded in the classics, but the results do not align. Solar eclipses occurring on new moon days are a verification of celestial phenomena; the "Spring and Autumn Annals" also record these solar eclipses occurring on new moon days, which indicates that the records in the "Spring and Autumn Annals" are consistent with celestial phenomena. However, scholars such as Liu Xiang and Jia Kui believed that solar eclipses occur one or two days after the new moon, which contradicts the clear records of the sages. Their mistake lies in rigidly adhering to a particular theory and not adjusting the calendar based on changes in celestial phenomena.
I previously wrote an article on calendars titled "On Calendars" that details the principles of calendar systems. Essentially, it states that celestial bodies are in constant motion; the sun, moon, and stars each follow their own orbits, as they are all moving objects. Although their movement can be roughly calculated, over time, year after year, with the cycles of change, there will always be slight variations, which is a natural law. Therefore, during the Spring and Autumn Period, some years experienced frequent solar eclipses, while others went several years without any. The patterns were inconsistent, yet the calculations needed to adhere to a constant value, so there would always be differences in calendars. The initial error was very small, almost imperceptible, but with accumulation, it would lead to deviations in the timing of new moons, and then it was necessary to modify the calendar to adapt to it. The "Book of Documents" states, "Respectfully observe the heavens and the sun, moon, and stars," and the "Book of Changes" says, "Managing the calendar and understanding the time," which means calendars should be established in accordance with celestial phenomena, rather than creating calendars to verify celestial phenomena. Based on this inference, during the Spring and Autumn Period of over two hundred years, there were many revisions and changes to the calendar. Although the ancient methods of calculation have been lost, we can still find traces in the classics and generally understand the situation of the calendars at that time. The discrepancies noted in the classics can also be validated. Scholars should carefully study the records of months, days, and solar eclipses in the classics to confirm the timing of new moons and calculate the dates from that period; however, they did not do so. Each scholar maintained their own views and speculated on the circumstances of the Spring and Autumn Period, which is akin to measuring your own feet to make shoes while disregarding others' sizes.
After I finished writing "Chronicles of Calendars," during the Xian Ning era, two scholars skilled in calculations, Li Xiu and Bu Xian, created a new calendar drawing from my theories, called the "Qiandu Calendar," and submitted it to the court. This calendar employs the method of slightly increasing the moon's speed by a quarter of a degree daily in accordance with the sun's movement, modifying the calendar every three hundred years using binary calculations. About every seventy years, adjustments are made based on gains and losses, with minimal discrepancies between gains and losses, sufficient to guarantee long-term accuracy in calculations. At that time, the officials and historians compared the "Qiandu Calendar" with the "Taishi Calendar" and determined that the "Qiandu Calendar" significantly outperformed the "Taishi Calendar," showing 45 points of greater accuracy than the official calendar of the time. This calendar remains in use today. I also compared ten ancient and modern calendars against the "Spring and Autumn Annals" and found that the "Santong Calendar" exhibited the greatest discrepancy.
The "Spring and Autumn Annals" totals 779 days (393 days in the text and 386 days in the commentary) and records 37 solar eclipses, without any record of solar eclipses during the years of Jia and Yi. The "Huangdi Calendar" calculates as 466 days with one solar eclipse. The "Zhuanxu Calendar" calculates as 590 days with eight solar eclipses. The "Xia Calendar" calculates as 536 days with fourteen solar eclipses. The "Zhenxia Calendar" calculates as 466 days with one solar eclipse. The "Yin Calendar" calculates as 503 days with thirteen solar eclipses. The "Zhou Calendar" calculates as 506 days with thirteen solar eclipses. The "Zhenzhou Calendar" calculates as 485 days with one solar eclipse. The "Lu Calendar" calculates as 529 days with thirteen solar eclipses. The "Santong Calendar" calculates as 484 days with one solar eclipse. The "Qianxiang Calendar" calculates as 495 days with seven solar eclipses. The "Taishi Calendar" calculates as 510 days with nineteen solar eclipses. The "Qiandu Calendar" calculates as 538 days with nineteen solar eclipses. The current "Changli Calendar" calculates as 746 days with 33 solar eclipses. However, there is a minor issue with this, since the records in the "Jingzhuan" are inaccurate and 33 days were omitted, resulting in a total of 4 solar eclipses, with 3 of these not recorded in the Jiazi year.
In the late Han Dynasty, there was a scholar named Song Zhongzi who collected seven types of calendars to study the book "Spring and Autumn." He discovered that the calculations of the Xia and Zhou calendars differed from those recorded in the "Book of Han: Treatise on Arts and Literature," so he rebranded them as the "True Xia Calendar" and the "True Zhou Calendar."
In the eighth year of Emperor Mu's Yonghe reign, a writer and the Prince of Langya named Wang Shuo created a new calendar called "Tongli." This calendar used the Jiazi year as its starting point, with a cycle of 97,000 years, with 4,883 years constituting a major cycle and dividing the celestial sphere into 1,225 degrees. He believed this starting point represented the dawn of heaven and earth.
During the time of Yao Xing of Later Qin, around the ninth year of Emperor Xiaowu's Taiyuan reign, which was the Jia Shen year, Jiang Ji, a scholar from Tianshui, compiled a book called "Sanji Jiazi Yuanli." The book asserts that to establish a calendar, one must first grasp the principles governing the movements of the sun and moon in order to predict celestial changes and understand climate changes on earth. If this fundamental principle is incorrect, the seasons will be thrown out of balance. Therefore, Confucius compiled "Spring and Autumn" in accordance with the order of the sun, moon, seasons, and years, highlighting that understanding celestial changes is crucial for governing human affairs, which is why rulers throughout history have attached great importance to it. From the time of Fuxi to the Han and Wei periods, various dynasties established their own calendars in pursuit of the most accurate calendar. To assess a calendar's accuracy, one primarily checks its ability to predict solar and lunar eclipses. However, among ancient texts, only "Spring and Autumn" provides detailed records of solar eclipses; spanning 242 years, from Duke Yin to Duke Ai, it recorded 36 solar eclipses, but the specific calendar used remains unknown.
Ban Gu believed that the "Spring and Autumn Annals" were compiled based on the calendar of the State of Lu, and since the Lu calendar was inaccurate, the scheduling of leap months was chaotic. The arrangement of leap years in the "Spring and Autumn Annals" did not align with the rules of the Lu calendar. The "Preface to the Book of Documents" mentions that Confucius, in order to compile the "Spring and Autumn Annals," reexamined the old calendar of the Yin and Shang dynasties to preserve the methods of calendar calculation. Therefore, it seems that the "Spring and Autumn Annals" should be corrected using the calendar of the Yin and Shang dynasties. However, further investigation revealed that the conjunction of the sun and moon in the "Spring and Autumn Annals" did not match the calendar of the Yin and Shang dynasties. Calculating the new moon day (first day of the month) in the "Spring and Autumn Annals" using the calendar of the Yin and Shang dynasties often resulted in discrepancies of one day, with the "Spring and Autumn Annals" texts frequently recording an extra day and the transmitted texts often missing a day. However, the records of the new moon day in the "Gongyang Commentary" and the "Spring and Autumn Annals" texts and transmitted texts were different, which is reasonable, as the records of solar eclipses in the "Spring and Autumn Annals" can be verified, whereas the transmitted texts of the "Spring and Autumn Annals" were incorrect in this regard. Fu Qian explained that the transmitted texts of the "Spring and Autumn Annals" used the Taiji Shangyuan calendar, which was the era of the "San Tong Calendar" established by Liu Xin. What relevance does this have to the "Spring and Autumn Annals"? Is it somewhat far-fetched to interpret the "Spring and Autumn Annals" through the lens of the Han Dynasty's calendar? There were many errors in the transmitted texts of the "Spring and Autumn Annals," not just this particular case. In the twenty-seventh year of Duke Xiang, on the new moon day of the eleventh month of the Yi Hai year, a solar eclipse occurred. The transmitted texts of the "Spring and Autumn Annals" recorded: "The morning star was in the position of Shen; the officials in charge of the calendar made mistakes in arranging leap months twice." However, according to calculations, the conjunction of the sun and moon did indeed take place in that month, and there was no case of leap months being incorrectly arranged twice. Liu Xin's calculations of the solar eclipses recorded in the "Spring and Autumn Annals" were accurate for only one eclipse date, while the rest were off by one or two days. Based on the records of solar and lunar eclipses in the "Five Elements Commentary of the Spring and Autumn Period," he proposed that during the Spring and Autumn Period, due to political turmoil among the states, the moon's movement was consistently slightly slower. Liu Xin did not admit any errors in the calendar calculations but rather sought alternative explanations. Solar and lunar eclipses are authentic reflections of celestial phenomena, yet Liu Xin used his own calendar to deny these phenomena, which was not only unjust but also undermined the integrity of the calendar itself!
Du Yu also believed that during the decline of the Zhou Dynasty and the chaos that engulfed the world, scholars were unable to grasp the true calendar. The seven calendars currently in circulation may not necessarily be the ones used by the emperors at that time. When using these seven calendars to calculate solar and lunar conjunctions in both ancient and modern contexts, it was found that they were all inaccurate, all due to the different values of Du Fen. The Du Fen of the "Yin Calendar" is one quarter, the Du Fen of the "San Tong Calendar" is 1539/385, the Du Fen of the "Qianxiang Calendar" is 589/145, and the Du Fen of the "Jingchu Calendar" is 1843/455. The values of Du Fen in these calendars are different, and the calculation methods are also different. The Du Fen of the "Yin Calendar" is too rough to be used now; the Du Fen of the "Qianxiang Calendar" is too fine to be used to calculate ancient celestial phenomena; the Du Fen of the "Jingchu Calendar" is relatively moderate, but the calculated position of the sun is off by four degrees, and the phenomena of solar and lunar eclipses cannot be calculated accurately. For example, if a solar eclipse occurs near the star "Dongjing," when calculated using the "Jingchu Calendar," the position of the moon will be near "Cansu," with such a significant discrepancy, how can it be used to calculate celestial phenomena and human affairs?
I have now developed a new calendar, with a Du Fen of 2451/605, and the sun is positioned at 17 degrees in the Dou constellation, which is the starting point of celestial motion. It can be used to analyze solar eclipses in the "Spring and Autumn Annals" and modern celestial phenomena. Using this new calendar to analyze the 36 solar eclipses recorded in the "Spring and Autumn Annals," 25 dates are accurate, 2 are off by one day, 2 are off by one day (on the last day of the month), and 5 have errors, totaling 34 solar eclipses. The remaining two solar eclipses are not documented in the "Spring and Autumn Annals," so their accuracy cannot be verified. Various astronomical texts note that "the Du calendar has been revised every three hundred years."
If we were to use the new calendar system currently in use during the Spring and Autumn period, solar eclipses would mainly occur on the new moon (the first day of the lunar month). From the Spring and Autumn period to now, over a thousand years have elapsed, and solar eclipses have consistently occurred around the new moon and full moon (the fifteenth day of the lunar month), varying between the three lunar phases. Therefore, this method can be used continuously without needing to revise the calendar every few hundred years as was done previously.
The meaning of this passage is that the author believes that their proposed new calendar system is superior to the old one because it can be used for a long time without frequent modifications. They use the Spring and Autumn period as an example to illustrate that this calendar is suitable for long-term use, unlike the old system which required constant adjustments. The author expresses skepticism towards the old calendar while affirming the new one in a rhetorical tone.
From the first month of the first year of the Jiazi cycle to the Jiwei year during Duke Yin of Lu's reign, a total of 82,736 years have passed. When calculated to the ninth year of the Taiyuan era in the reign of Emperor Xiaowu of Jin, a total of 83,841 years have elapsed.
How are these numbers calculated? The Yuan method totals 7,353, the Ji method totals 2,451, which adds up to a total of 179,044. The day method totals 6,620, the month week totals 32,766, the Qi division totals 12,860, the Yuan month totals 99,945, the Ji month totals 33,315, the Mei division totals 44,761, the Mei method totals 643, the Dou division totals 605, and the total for the week day is 895,220 (also referred to as Ji day). The chapter month totals 235, the chapter year totals 19, the chapter leap totals 7, the year consists of 12, and the total number of meetings is 47 (the total of days and months equals 893 years, exactly 47 meetings, perfectly divided). The Qi within totals 12.
The interval of the Jiazi era is 9157, the interval of the Jiashen era is 6337, and the interval of the Jiachen era is 3517. The Zhou half is 127, the total number of new moons is 941, the total number of leap years is 893, the total number of months is 11,405, the small interval is 2196, the total number of chapters is 129, the small interval is 2183, the total number of leap months is 76,269, and the total number of calendar weeks is 447,610 (this is half a week). The total number of meetings is 38,134, the difference value is 11,986, the total number of rates is 1882, the small interval method is 2209, the limit of intervals is 11,104, and the small interval week is 254.
The difference rate values of the Jiazi era are 49,178, the difference rate values of the Jiashen era are 58,231, and the difference rate values of the Jiachen era are 67,284. The total number of interchanges is 167,063, the remainder of the week is 3362, and the week is 2701.
These numbers are quite daunting, like ancient astronomers doing complex calculations. In any case, this records a significant period of time, as well as the results of various astronomical calendar calculations during this period.
Let’s begin by discussing the calculation method for these five celestial bodies. The book says that the results obtained from this method are accurate, and there's no need to adhere strictly to the original raw data. You see, the initial calculation method and the current method each have their own advantages, so the author simultaneously proposed two methods.
Regarding Ji, he used lunar eclipses to verify the degree of the sun's movement, which serves as an authoritative basis for calendar researchers. He also authored "Hun Tian Lun" to calculate the sun's trajectory along the ecliptic, refuting several misconceptions held by earlier Confucian scholars and uncovering the underlying principles.
