The five planets are Jupiter (the Planet of the Year), Mars (the Wandering Planet), Saturn (the Planet of Filling), Venus (the Shining Planet), and Mercury (the Star of the Morning). The speed of movement of these five planets varies, sometimes fast and sometimes slow, sometimes advancing and sometimes retreating. Since the dawn of time, the concepts of yin and yang have been distinguished, and the sun, moon, and five planets have all gathered in the Celestial Records (one of the twenty-eight lunar constellations). Starting from the Celestial Records, they all move together in the sky, influencing one another with their varying speeds and movements. When the planets and the sun appear simultaneously in a constellation, it is called a "conjunction." The interval between one "conjunction" and the next is referred to as "one end." By comparing the days of "one end" to the days of a year, we calculate their ratio, which is referred to as the "conjunction year number" and "conjunction ratio." Once these two ratios are determined, other calculations can be derived. Multiplying the number of days in a year by the "conjunction ratio" gives the "conjunction month method"; multiplying the "conjunction month method" by the "conjunction ratio" gives the "daily method." Multiplying the number of days in a year by the "conjunction year number" gives the "conjunction month fraction"; if the "conjunction month fraction" equals the "conjunction month method," it is called the "conjunction month number," and the remainder is the "month remainder." Multiplying the "conjunction year number" by the "conjunction month number," then dividing by the "daily method," gives the "major remainder." Subtracting the "major remainder" from 60 leaves the "star conjunction new moon major remainder," and the leftover from the "major remainder" is the "new moon small remainder." Multiplying the "conjunction year number" by the "month remainder," and adding the product of the "conjunction month method" and the "new moon small remainder," then dividing by the product of the "daily method" and the "conjunction month method," gives the date of the star conjunction new moon, with the leftover being the "entry month day remainder." Subtracting the "new moon small remainder" from the "daily method" leaves the "new moon void fraction." Multiplying the degrees of the Dipper constellation in the calendar by the "conjunction ratio" provides the degree measurements of the stars in the Dipper. For Jupiter, Mars, and Saturn, subtracting the "conjunction year number" from the "conjunction ratio," multiplying the result by the degrees of the zodiac, and dividing by the "daily method" provides the degree measurements of the planets, with the leftover being the degree remainder. For Venus and Mercury, multiplying the degrees of the zodiac by the "conjunction year number" and dividing by the "daily method" provides the degree measurements of the planets, with the leftover being the degree remainder.
Jupiter: Total years, one thousand two hundred and fifty-five; total conjunctions, one thousand one hundred and forty-nine; conjunctions with the moon, twenty-one thousand eight hundred and thirty-one; daily calculation value, two hundred and eleven thousand seven hundred and six; total moons, thirteen; moon remainder, eleven thousand one hundred and twenty-two; major lunar remainder, twenty-three; minor lunar remainder, four thousand nine hundred and ninety-three; moon entry date, the fifteenth; moon balance, one hundred and ninety-nine thousand five hundred and sixty-four; major lunar remainder, four hundred and sixty-six; lunar mansion division, fifty-two thousand two hundred and ninety-five; planetary degree, thirty-three; degree remainder, one hundred and forty-seven thousand two hundred and eighty-nine.
Mars: Total years, five thousand one hundred and five; total conjunctions, two thousand three hundred and eighty-eight; conjunctions with the moon, forty-five thousand three hundred and seventy-two; daily calculation value, four hundred and forty-one thousand eighty-four; total moons, twenty-six; moon remainder, twenty-three; major lunar remainder, forty-seven.
Wow, these numbers are making my head spin! Is this some sort of astronomical calendar calculation? Let's break it down step by step.
Firstly, "minor lunar remainder, three thousand six hundred and twenty-seven" means that the minor lunar remainder is three thousand six hundred and twenty-seven. Next, "moon entry date, the thirteenth" means that the moon enters on the thirteenth day of the month. "Daily remainder, three million five hundred and eighty-five thousand two hundred and thirty" is a really large daily remainder, at three million five hundred and eighty-five thousand two hundred and thirty. The following "major lunar remainder, nine hundred and thirty-two," "lunar mansion division, one hundred and eighty-six thousand five hundred and forty," "planetary degree, thirty-three," and "degree remainder, one hundred and forty-seven thousand two hundred and eighty-nine" are all terms and values used in astronomical calculations, and I’m not quite sure what they specifically refer to.
Next, "Saturn: Total age, three thousand nine hundred and forty-three," this is the calculation result about Saturn; the total age is three thousand nine hundred and forty-three. "Total conjunction, three thousand eight hundred and ninety," "Conjunction moon method, seventy-two thousand three hundred and seventy-one," "Sun method, seven million one hundred and ninety-eight thousand," "Total moon number, twelve," "Moon remainder, fifty-eight thousand one hundred and fifty-three," "New Moon big remainder, fifty-four," "New Moon small remainder, one thousand six hundred and seventy-four," "Day of the month, twenty-four," "Day remainder, six hundred seventy-five thousand three hundred sixty-four," "New Moon virtual division, two thousand eight hundred and eighty-five," "Dou division, one hundred seventy-three thousand three hundred ninety-five," "Planetary degree, twelve," "Degree remainder, five hundred ninety-six thousand two hundred and fifty-six." These are all kinds of calculations about Saturn, one number after another, making people dizzy.
Then comes the calculation result of Venus, "Venus: Total age, one thousand nine hundred and seven," "Total conjunction, two thousand three hundred and eighty-five," "Conjunction moon method, forty-five thousand three hundred and fifteen," "Sun method, four hundred thirty-nine thousand five hundred and fifty-five," "Total moon number, nine," "Moon remainder, forty-three thousand one hundred and ten," "New Moon big remainder, twenty-five," "New Moon small remainder, three thousand five hundred and thirty-five," "Day of the month, twenty-seven," "Day remainder, nineteen thousand four hundred and ninety," "New Moon virtual division, one thousand two hundred and twenty-four," "Dou division, one hundred eighty-five thousand one hundred and seventy-five," "Planetary degree, two hundred ninety-two," "Degree remainder, nineteen thousand four hundred and ninety." These figures are just as dazzling.
Finally, the calculation of Mercury: "Water: total age number, 1,870", "Total conjunction number, 11,789", "Combined lunar method, 223,991", "Daily method, 21,727,127", "Combined lunar number, 1", "Monthly remainder, 215,459", "Major new moon remainder, 29", "Minor new moon remainder, 2,419", "Enter month day, 28", "Daily remainder, 23,404,261", "Shuo virtual division, 2,140", "Dou division, 5,363,995", "Planetary degree, 57", "Degree remainder, 23,404,361." The calculation results of Mercury are also quite complex. In short, it's just a jumble of astronomical figures! If you are not a professional, it is difficult to understand the meaning behind these numbers.
This passage talks about ancient astronomical calculation methods, which can feel pretty overwhelming. Let's take it one step at a time. The first paragraph discusses calculating the conjunction cycle of the five stars. It first asks you to calculate a "cumulative sum," which is to add up all the years from the year of Renchen (a specific year) to the year you want to calculate, then multiply by a certain "total combined number" (the exact number isn't provided). If the result is exactly a multiple of the "total combined age" (another unspecified number), it is called "cumulative sum"; otherwise, the remainder is called "combined remainder." Then subtract the "total combined number" from the "combined remainder." If the result is 1, it means the five stars will conjoin in that year; if it's 2, they conjoin the year before; if the result is 0, then they will conjoin in that year. Finally, divide the "combined remainder" by the "total combined number" to get a quotient, which is the "degree division." If the "cumulative sum" of Venus and Mercury is even, they will conjoin in the morning; if it is odd, they will conjoin in the evening.
The second paragraph discusses the calculation of the conjunction of the five stars and the moon. By multiplying the number of lunar months and the remainder of the moon by "product sum," you obtain a "product month" and a "moon remainder." Then divide a number called "calendar month" by "product month"; the quotient is "calendar month," and the remainder is "record entry." Next, multiply "chapter intercalation" (likely referring to an intercalary month) by "record entry." If the result is a whole number multiple of "chapter month" (possibly referring to the length of a month), it indicates an intercalary month. Then subtract "record entry" from it, and the remainder is "record year." Starting from Tianzheng (possibly referring to a specific point in time), the calculated time indicates the conjunction of the five stars and the new moon. If it is at the juncture of the intercalary month, use the new moon day (the first day of each month in the lunar calendar) to adjust.
The third paragraph calculates the conjunction time of the new moon day. Multiply a "total number" by the "record entry"; if the result is a whole number multiple of the number of days, it is called "product day," and the remainder is referred to as "small remainder." Then subtract 60 from "product day"; the remainder is "big remainder." Starting from "record entry," the calculated time is the conjunction time of the stars and the new moon day.
The fourth paragraph discusses the calculation of the moon's entry day. Multiply the "total number" by the "moon remainder," then multiply the "moon conjunction method" by the "new moon small remainder," add these two results together, then divide by the "total method." If the result is an integer multiple of the number of days, it is the moon's entry day; if not, a remainder remains. Starting from the new moon day, the calculated time is the entry day of the moon.
The fifth paragraph calculates the degree of conjunction for celestial bodies. Multiply the circumference (360 degrees) by "degree fraction"; if the result is a whole number multiple of the day degree method, the quotient is the degree, and the remainder is the remainder. Starting from the first five degrees of the zodiac (possibly referring to a point in the ecliptic coordinate system), the calculated degree is the degree of conjunction for celestial bodies.
The sixth paragraph calculates the time of the next conjunction of stars and the moon. Add the number of months to "record year," add the remainder to "moon remainder"; if the result is an integer multiple of the "moon conjunction method," you get the month of the next conjunction. If this month falls within the current year, it is counted as that year; if it exceeds one year, subtract one year from the total. If there is an intercalary month, consider it, and the remaining is the year of the next conjunction; if it exceeds another year, it is counted in the next two years. For Venus and Mercury, a morning conjunction shifts to an evening conjunction, while an evening conjunction shifts to a morning conjunction.
The last two paragraphs calculate the time of the next conjunction of the new moon and the moon's visibility, as well as the degree of the next conjunction of celestial bodies. The method is similar to the previous one, using the previous remainders plus new remainders, and then adjusting according to certain rules. In summary, this is a very complex calendar calculation method that requires a strong mathematical foundation to understand.
In the morning when the sun rises, Jupiter and the sun coincide, and then Jupiter "hides" from view. Its speed varies; it roughly takes sixteen days to cover ninety-nine thousand seven hundred and thirty-two minutes, resulting in a positional change of two degrees and one hundred seventy-nine thousand five hundred thirty-eight minutes. After that, Jupiter can be seen in the east in the morning, behind the sun. Sometimes it moves quickly, covering eleven-fifty-sevenths of a degree each day, totaling eleven degrees over fifty-seven days; sometimes it moves slowly, covering nine minutes each day and nine degrees over fifty-seven days before it remains stationary for twenty-seven days before it begins to move again. If it goes retrograde, it moves one-seventh of a degree per day, retreats twelve degrees over eighty-four days, and then stops. After twenty-seven days, it slows down, covering nine minutes each day and nine degrees over fifty-seven days before resuming direct motion. When it moves quickly, it covers eleven minutes each day and eleven degrees over fifty-seven days, at which point it is positioned in front of the sun and can be seen "hiding" in the west in the evening. Then, sixteen days later, it coincides with the sun again, completing one cycle. The entire cycle lasts three hundred ninety-eight days and one hundred ninety-nine thousand five hundred sixty-four minutes, resulting in a positional change of thirty-three degrees and one hundred forty-seven thousand two hundred eighty-nine minutes.
When the sun rises in the morning, Mars also conjuncts with the sun, becomes invisible and cannot be seen. It takes about seventy-two days to traverse 1,792,615 minutes, with a change in planetary position of 56 degrees and 124,945 minutes. Afterwards, Mars can be seen rising in the east behind the sun. Its speed varies; sometimes it moves quickly, covering 14/23 of a degree per day, and in a hundred and eighty-four days it covers 112 degrees. Sometimes it moves slowly, covering 12 minutes per day, 48 degrees in 92 days before stopping. It remains stationary for eleven days before resuming rotation. If it is in retrograde motion, it covers 17/62 of a degree per day, retreats 17 degrees after 62 days before stopping again. Eleven days later, it resumes direct motion. When moving slowly, it covers 48 degrees in 92 days before accelerating. When moving quickly, it covers 112 degrees in 184 days, at which point it can be seen setting in the west at dusk. Then, seventy-two days later, it conjuncts with the sun again, completing one cycle. The entire cycle lasts 780 days and 358,523 minutes, with a change in planetary position of 415 degrees and 249,890 minutes.
In the morning, Saturn and the sun are in conjunction, and then Saturn "vanishes." Nineteen days, which is 3,847,675 minutes, Saturn moves 2 degrees and 649,121 minutes, and then in the morning it can be seen in the east; at this point, it is positioned behind the sun. When moving forward, it travels 13/172 of a degree; over eighty-six days, it moves 6.5 degrees and then stops. After a pause of 32.5 days, it starts moving again. When moving backward, it covers 1/17 of a degree each day; after one hundred and two days, it retreats by 6 degrees and then stops again. After a pause of 32.5 days, it starts moving forward again, covering 1/13 of a degree each day and moving 6.5 degrees over eighty-six days; at this point, it is located in front of the sun, and in the evening, it "vanishes" in the west. Moving forward for nineteen days, which is 3,847,675 minutes, Saturn moves 2 degrees and 649,121 minutes, and then it is in conjunction with the sun again. At the end of one cycle, it totals 378 days and 67,364 minutes, and Saturn has moved 12 degrees and 596,256 minutes.
In the morning, Venus and the sun are in conjunction, and then Venus "vanishes." In six days, it moves 4 degrees backward, and then in the morning it can be seen in the east; at this point, it is behind the sun, continuing to move backward. When moving slowly, it covers 3/5 of a degree each day; after ten days, it retreats by 6 degrees. After stopping for 7 days, it starts moving again. When moving forward, when moving slowly, it covers 33/45 of a degree each day, traveling 33 degrees over 45 days. When moving fast, it covers 14/91 of a degree each day, moving 105 degrees over 91 days. When moving even faster, it covers 21/91 of a degree each day, moving 112 degrees over 91 days; at this point, it is behind the sun, and in the morning, it "vanishes" in the east. Moving forward for 42 days, totaling 19,490 minutes, Venus travels 52 degrees over 19,490 minutes, and then it is in conjunction with the sun again. During one conjunction, it totals 292 days and 19,490 minutes, and Venus has moved this many degrees.
In the evening, Venus and the sun align, then Venus "vanishes." Following this, for forty-two days, equivalent to 194,990 minutes, Venus travels 52 degrees and 19 minutes, then in the evening it can be seen in the western sky, appearing in front of the sun. Continuing, at a faster pace, it moves 1 degree and 31 minutes per day, covering a total of 112 degrees in 91 days, then continues. At a slower pace, it moves 1 degree and 14 minutes per day, covering a total of 150 degrees in 91 days. At an even slower pace, it moves 45 minutes and 33 seconds daily, covering 33 degrees in 45 days and then resumes its motion. After stopping for seven days, it starts moving again. When it goes retrograde, it moves 3 minutes and 5 seconds per day, retreating 6 degrees after ten days; then it is positioned in front of the sun, "vanishing" in the western sky by evening. Retrograding, it retreats 4 degrees after six days, then aligns with the sun again. Two alignments mark the completion of one cycle, totaling 584 days, 389,980 minutes, and Venus travels this distance. In the morning, Mercury aligns with the sun, and Mercury vanishes, retreating 7 degrees after eleven days, then is behind the sun, reappearing in the east at dawn. If Mercury goes retrograde, if it moves quickly, it retreats 1 degree after one day and then stops; after a day of no movement, it resumes direct motion. If Mercury moves direct, at a slower pace, it moves 7 degrees in eight days, then stops. If it moves quickly, covering 22 degrees in 18 days, it remains behind the sun, reappearing in the east at dawn. Moving direct, it covers 36,361 minutes in 18 days, then aligns with the sun. Each alignment lasts a total of 57 days and 36,361 minutes, and this is how Mercury travels.
In the evening, Mercury will conjoin with the Sun, hiding, moving three thirty-six thousand three hundred sixty-one degrees over eighteen days, then appearing in the west before the Sun at dusk. If it moves forward, it travels quickly, covering four-fifths of a degree each day, totaling twenty-two degrees over eighteen days, and then continues in direct motion. If it moves slowly, it covers seven-sevenths of a degree in a day, stops after seven degrees in eight days, and if it remains stationary for a day, it will reverse its course. If it moves retrograde, it will retreat one degree in a day, hiding in the west before the Sun at dusk. Continuing in retrograde motion, it will retreat seven degrees over eleven days, then conjoin with the Sun. After two conjunctions, a total of 115 days and 18,961,395 minutes will have elapsed; this describes the motion of Mercury.
The calculation method is as follows: add the remainder of the degrees during Mercury's hiding period (the time it is concealed) to the remainder of the degrees when Mercury and the Sun conjoin. If the remainder reaches a full day (according to the solar degree system), a complete cycle is achieved. Following the previous rules, one can calculate the time and degree remainder of Mercury's appearance. Multiply the denominator of Mercury's motion by the degree of its appearance; if the result equals the solar degree system, the calculation is correct; if it does not divide evenly but exceeds half, it is also considered correct. Then add the calculation result to the degrees of Mercury's motion; if the total degrees exceed the denominator, increase the count by one degree. The denominators for retrograde and direct motion differ; the denominator in use must be multiplied by the previous degrees. If the result equals the previous denominator, that is the current degree of motion. When Mercury stops moving, the previous degrees must be referenced; for retrograde motion, subtract the previous degrees. If the degrees during Mercury's hiding are insufficient, use doufen (a unit of angle) for calculations, using the denominator of the motion as a ratio. Degrees may increase or decrease, and must mutually constrain each other.
The courtier Liu Zhi of Emperor Wu of Han modified the calendar using the斗历 (douli, a type of calendar), calculating the "Four Division Method," which had a three-hundred-year error of one day, using one hundred fifty as the degree rule and thirty-seven as doufen. He calculated that the Jiazi year (the starting point of a specific cycle in the Chinese calendar) was the beginning of the calendar, arriving at the tenth year of Taishi, the year of Jiawu, totaling ninety-seven thousand four hundred eleven years. The beginning of the Jiazi lunar month coincided with the winter solstice, marking the start of the solar and lunar cycles with the five planets, establishing the point of the calendar. He embellished it with grandiose claims and named it "Zhengli."
Dongyang Marquis Du Yu wrote the "Spring and Autumn Chronicle," in which it is stated:
The sun travels one degree each day, while the moon moves thirteen degrees and nineteen minutes, plus a bit more each day. Officials in charge of astronomical calendars must calculate the new moon (the first day of the lunar month) and the full moon (the last day) of each month, as well as whether or not to insert a leap month, based on the speed of the sun and moon. The leap month lacks solar terms (the two times each month when the sun reaches 15° of the yellow path), and the direction of the Big Dipper also differs from other months, so the leap month is different from the others. Accurate calculations ensure that the four seasons and eight solar terms align properly, allowing for the establishment of an accurate calendar system, the intricacies of which are truly profound! If one can grasp the subtleties within, then one can align with the laws of nature, allowing things to proceed smoothly without error. The "Spring and Autumn Annals" states: "The leap month is for time correction, and time is for arranging events."
However, the laws of yin and yang's movement will experience slight deviations over time. These accumulated deviations will result in differences between the actual calendar and the calculated one. Therefore, Confucius and Zuo Qiuming often wrote about the issues of the new moon and leap month to address these deviations and clarify calendar calculation principles.
Liu Zijun developed the "Three Correct Calendars" to study the "Spring and Autumn" chronicle. The book records thirty-four solar eclipses, but according to the "Three Correct Calendars," there was only one recorded solar eclipse, which had the largest discrepancy when compared to other calendars. Furthermore, he believed that adding a day every six thousand years would accumulate a significant discrepancy between the number of years and the actual number of days, making this method impractical.
From ancient times to the present, many scholars studying the "Spring and Autumn" chronicle have made errors, some creating their methods, others using various calendars since the time of the Yellow Emperor to calculate the new moons recorded in the texts, resulting in discrepancies. Solar eclipses occurring on new moons serve as astronomical verification, as recorded in the classics, verifying that the records are in accordance with the laws of nature. However, Confucian scholars such as Liu Xiang and Jia Kui believed that solar eclipses occurred two or three days after the new moon, contradicting the clear records left by the sages. Their mistake lies in clinging to a single theory and ignoring the changes in celestial phenomena.
I have studied the issues of time in the "Spring and Autumn" Annals before and also wrote an article on the "Calendar," explaining in detail the principles governing the calendar. The main idea is that celestial bodies never stop moving; the sun, moon, and stars each move in their own orbits, all of which are moving objects. Moving objects cannot be completely identical; although the degree of movement can be roughly calculated, over time, year after year, there will always be slight differences, which is a natural phenomenon. Therefore, during the "Spring and Autumn" period, some years experienced frequent solar eclipses, while in some years there were no solar eclipses for several years. The patterns were inconsistent, but the calculated values remained consistent, so there would always be differences in the calendars. The initial error may be negligible and imperceptible, but over time, the error will accumulate and become larger, leading to deviations in the lunisolar calendar, which then necessitates modifying the calendar to accommodate it. The "Shang Shu" says, "Respect the vast sky, observe the movements of the sun, moon, and stars," and the "I Ching" states, "Manage the calendar and understand the times," meaning that the calendar should be established in accordance with celestial phenomena, rather than using the calendar to verify celestial phenomena. From this reasoning, during the Spring and Autumn period of over two hundred years, there were numerous changes to the calendar. Even if the ancient algorithms have been lost, we can still uncover clues within the "Classics" to roughly calculate the values and verify the deviations in the calendar at that time. Researchers should carefully examine the records of the moon dates and solar eclipses in the "Classics" to calculate the lunisolar calendar and verify its accuracy; yet, in reality, everyone did not do this, each relying on their own methods to calculate the era of the "Spring and Autumn" period, which is akin to measuring one's own foot to make shoes, yet trying to trim others' feet to fit those shoes.
After I completed "Theories on Calendars," in the Xianning era, two calculation-savvy scholars, Li Xiu and Bu Xian, created a new calendar drawing from my theories, called the "Qiandu Calendar," and presented it to the court. This calendar defined the sun's daily movement as a quarter of a degree, marginally increased the moon's orbital speed, revised the calendar every three centuries, calculated using a binary system, and made further adjustments after over seventy years based on the magnitude of discrepancies. The discrepancies were minimal, but sufficient to ensure the calendar's long-term effectiveness. At that time, officials and historians compared the "Qiandu Calendar" with the "Taishi Calendar" and found that the "Qiandu Calendar" showed forty-five more points of accuracy than the official calendar of that time. This calendar is still in use today. I also compared ten ancient and modern calendars against the "Spring and Autumn Annals" and found that the "Santong Calendar" had the largest error.
The "Spring and Autumn Annals" totaled seven hundred and seventy-nine days (three hundred and ninety-three days in the "Classic" and three hundred and eighty-six days in the "Commentary") and recorded thirty-seven solar eclipses, with three instances where the dates were not recorded.
The "Huangdi Calendar" calculated a total of four hundred and sixty-six days, with one solar eclipse recorded. The "Zhuanxu Calendar" resulted in five hundred and ninety days, recording eight solar eclipses. The "Xia Calendar" resulted in five hundred and thirty-six days, with fourteen solar eclipses. The "Zhenxia Calendar" calculated four hundred and sixty-six days, with one solar eclipse. The "Yin Calendar" resulted in five hundred and three days, with thirteen solar eclipses. The "Zhou Calendar" resulted in five hundred and sixty days, with thirteen solar eclipses. The "Zhenzhou Calendar" resulted in four hundred and eighty-five days, with one solar eclipse. The "Lu Calendar" resulted in five hundred and twenty-nine days, with thirteen solar eclipses. The "Santong Calendar" resulted in four hundred and eighty-four days, with one solar eclipse. The "Qianxiang Calendar" resulted in four hundred and ninety-five days, with seven solar eclipses. The "Taishi Calendar" resulted in five hundred and ten days, with nineteen solar eclipses. The "Qiandu Calendar" resulted in five hundred and thirty-eight days, with nineteen solar eclipses. The current "Changli Calendar" results in seven hundred and forty-six days, with thirty-three solar eclipses. However, there's a slight issue: the "Classic Commentary" contains errors, undercounting by thirty-three days; in reality, there were only four solar eclipses, and the dates for three of them are missing.
In the late Han Dynasty, there was a scholar named Song Zhongzi who collected seven types of calendars to study the historical chronicle "Spring and Autumn Annals." He discovered that the algorithms for the Xia calendar and the Zhou calendar differed from the calculations recorded in "Han Shu: Yiwen Zhi," so he renamed them the "True Xia Calendar" and the "True Zhou Calendar." In the eighth year of Emperor Mu's Yonghe reign, which corresponds to 352 AD, the court official Wang Shuo of Langya invented a new calendar called "Tongli." This calendar uses the Jiazi cycle as its starting point, with a cycle that lasts 97,000 years, using 4,883 years as its epoch, and the Doufen value is 125. He believed that this calendar's starting point marked the beginning of the universe.
During the time of Yao Xing in Later Qin, in the year 409 AD, Master Jiang Ji created a new calendar in Tianshui called the "Three Epochs Jiazi Yuan Calendar." The book roughly states: "To understand the calendar, one must first understand the principles governing the movements of the sun and moon, in order to calculate celestial phenomena and understand the changes on Earth. If this basic point is misunderstood, the four seasons will be thrown into disarray. Take a look at how Confucius compiled the 'Spring and Autumn Annals,' which records day by day, month by month, season by season, year by year, in sequence, because he knew that understanding celestial phenomena is the basis of human affairs; thus, calendars have always been valued by emperors. From the time of Fuxi to the Han and Wei periods, each dynasty has established its own calendar, striving for accuracy. The accuracy of a calendar is primarily assessed by its predictions of solar and lunar eclipses. However, in previous records, only the 'Spring and Autumn Annals' detailed the changes in solar eclipses, recording 36 solar eclipses over a period of 242 years, from Duke Yin to Duke Ai, but it remains unclear which specific calendar was utilized. Ban Gu believed that the 'Spring and Autumn Annals' used the "Lu calendar," but the "Lu calendar" itself is inaccurate, so the arrangement of intercalary months was quite chaotic. The state of Lu took the intercalary year as the beginning of the calendar, but the arrangement of intercalary months in the 'Spring and Autumn Annals' does not match this beginning. The 'Record of the Mandate of Heaven' states: in order to compile the 'Spring and Autumn Annals,' Confucius also re-examined the old calendar of the Shang Dynasty to pass down its algorithms. It seems that the 'Spring and Autumn Annals' should be corrected using the Shang calendar. However, upon examination, it is found that the conjunction of the sun and moon indeed occurred in that month, but it is not necessary to say it was because the intercalary month was missed again. Liu Xin's calendar and the solar eclipse recorded in the 'Spring and Autumn Annals' coincide only once, with the other times differing by one or two days. He also wrote the 'Five Elements Commentary,' proposing the theory of Taoyu and Ceni, meaning that during the Spring and Autumn period, the feudal lords were corrupt, so the movement of the moon was always slow. Liu Xin refused to acknowledge any flaws in the calendar itself and instead concocted this theory to justify it. The occurrence of a solar eclipse on the new moon serves as a verification of celestial phenomena, but Liu Xin used his own calendar to deny this celestial phenomenon, which is unfair to the celestial phenomenon and disappoints the calendar! Du Yu also believed that during the decline of the Zhou Dynasty and the chaos in the world, scholars were unable to grasp the true calendar, and the seven calendars currently in circulation may not have been the ones used by the royal court at that time. Now we use these seven calendars to verify the conjunction of the sun and moon in ancient and modern times and discover that all of them are inaccurate, all because of the different values of the Doufen. The Doufen in the 'Shang Calendar' is one-fourth, in the 'San Tong Calendar' it is 1539 to 385, in the 'Qianxiang Calendar' it is 589 to 145, and in the 'Jingchu Calendar' it is 1843 to 455; the Doufen values in these calendars vary, and their algorithms differ as well. The Doufen in the 'Shang Calendar' is too coarse, so it cannot be used now; the Doufen in the 'Qianxiang Calendar' is too fine, so it does not match with ancient times; the Doufen in the 'Jingchu Calendar' is relatively moderate, but the position of the sun is off by four degrees, and the results of solar and lunar eclipse calculations are also imprecise. For example, if a solar eclipse occurs when the sun is in Dongjing, and it is verified by the moon, it is actually in the position of Canxiu six degrees; with such a large deviation, how can it be relied upon to calculate celestial phenomena and human events? Now I have developed a new calendar that uses a ratio of 2451 to 605 for the Doufen, with the sun at 17 degrees in the Dou constellation, which is the starting point of Tianzheng; it can match with the 'Spring and Autumn Annals' and can also be used to verify modern times. Using this calendar to verify the 36 solar eclipses recorded in the 'Spring and Autumn Annals,' 25 new moons are accurate, 2 are off by one day, 2 are off by one day, and 5 have errors, resulting in a total of 34 solar eclipses; the remaining two eclipses are not specifically documented in the 'Spring and Autumn Annals,' so their accuracy cannot be verified. Various astronomical texts state, "After three hundred years, the Dou calendar undergoes changes."
If the new calendar we are currently using were applied to the Spring and Autumn Period, most solar eclipses would have occurred on the first day of the lunar calendar (New Moon). From the Spring and Autumn Period to the present, and it has been over a thousand years, the occurrence of solar eclipses has consistently varied around the three conjunctions of the New Moon and Full Moon. Therefore, this method can be used indefinitely, unlike the previous practice of changing the calendar every few hundred years, which was troublesome!
This method is reliable and can be used continuously. Unlike before, when the calendar was changed frequently, it was really troublesome! You know, it's been over a thousand years since the Spring and Autumn Period, and the pattern of solar eclipse occurrences has always changed around the three conjunctions of the New Moon and Full Moon. By using this new calendar, it will be accurate!
From 770 BC (Jiazi, Shangyuan) to 71 BC (Lu Yin Gong, Year of the Jiwei), a total of 82,736 years have passed. By 370 AD (Jin Xiaowu, Taiyuan, Year of the Jia-shen), a total of 83,841 years have passed; this is the grand total.
- Yuan calendar: 7,353 years
- Ji calendar: 2,451 years
- Total: 179,444 years
- Ri method: 662 years
- Month Cycle: 32,766 years
- Qi Fen: 12,860 years
- Yuan Yue: 99,445 years
- Ji Yue: 33,115 years
- Mei Fen: 44,761 years
- Mei Fa: 643 years
- Dou Fen: 650 years
- Zhou Tian: 895,220 years (also called Ji Ri)
- Zhang Yue: 235 years
- Zhang Sui: 19 years
- Zhang Run: 7 years
- Sui Zhong: 12 years
- Hui Shu: 47 (Day and Month: 893 years, a total of 47 meetings, divided evenly)
- Qi Zhong: 12 years
- Jiazi Ji Jiao Cha: 9,157 years
- Jiashen Ji Jiao Cha: 6,337 years
- Jiachen Ji Jiao Cha: 3,517 years
- Zhou Ban: 127 years
- New Moon and Full Moon combination: 941 years
- Hui Sui: 893 years
- Hui Yue: 11,415 years
- Xiao Fen: 2,196 years
- Zhang Shu: 129 years
- Xiao Fen: 2,183 years
- Zhou Run Da Fen: 76,269 years
- Li Zhou: 447,610 years (half Zhou Tian)
- Hui Fen: 38,134 years
- Cha Fen: 11,986 years
- Hui Lv: 1,882 years
- Xiao Fen Fa: 2,290 years
- Ru Jiao Xian: 11,140 years
- Xiao Zhou: 254 years
- Jiazi Ji Cha Lv: 49,178 years
- Jiashen Ji Cha Lv: 58,231 years
- Jiachen Ji Cha Lv: 67,284 years
- Tong Zhou: 167,630 years
- Zhou Ri Ri Yu: 3,362 years
- Zhou Xu: 2,710 years.
This text lists a series of astronomical calendar calculation results, involving various calendrical methods, cycles, and numerical values, which require specialized astronomical knowledge for interpretation. These numbers reflect the complexity and precision of ancient astronomical calendar systems, as well as the exploration and understanding of the laws of the universe by people at that time. Although it may be difficult for us to fully grasp the calculation methods behind them, these numbers themselves demonstrate the achievements of ancient China in the field of astronomical calendars.
The method of calculating the positions of the five stars is determined based on actual observation results, rather than rigidly adhering to previous calculation results. In other words, the starting point of the calculations should go back to their origins, while simplified methods should be applied to current use, carefully exploring the mysteries involved. Both methods have their own merits, so the author simultaneously proposes two methods.
Ji (pronounced jí) used lunar eclipses to verify the sun's degree of movement and position, which later became a standard for calendar researchers. He also wrote "The Treatise on the Celestial Sphere," using precise calculations to determine the sun's trajectory on the ecliptic, refuting some misconceptions held by previous Confucian scholars, and ultimately finding the correct answers.
