Yang Wei wrote in his book: "I have consulted a variety of classics and carefully studied the calendar. The calendar records agricultural production by years and various events by months. This method has ancient origins. Starting from the time of Shaohao, the Xuan Niao bird was responsible for signaling the seasons; during the periods of Zhuanxu and Di Ku, Chongli oversaw astronomical matters; by the time of Tang Yao and Yu Shun, Xihe was in charge of managing the calendar. The Xia, Shang, and Zhou dynasties adhered to this system, with officials in charge of the calendar in each generation. Once the calendar officials had established it, it was distributed to various vassal states, which then disseminated it to all parts of the country. During the Xia Dynasty, Xihe became preoccupied with wine and pleasure, neglecting the calendar, resulting in date confusion, as noted in the 'Yin Zheng' chapter of the 'Book of Documents'. From these examples, it can be seen that each dynasty placed great importance on organizing affairs in accordance with the agricultural seasons.
As the Zhou Dynasty weakened and the states vied for supremacy during the Warring States period, sacrificial sheep were no longer provided, and the ceremonial rituals for sacrifices were disregarded. The sequence of leap months and regular months became muddled, the date of the first day of the first month was also lost, the constellation of the Great Fire was still moving westward, and people wondered why the hibernating insects had not hidden. During that time, the emperor failed to act in accordance with the seasons; the officials in charge of the calendar did not record the dates; the vassals neglected their duties and failed to determine the months in a timely manner, resulting in the neglect of agricultural production. Confucius chronicled these events in the 'Spring and Autumn Annals', correcting mistakes through praise and criticism. If there was an error regarding the leap month in the calendar, he would criticize and record it; if the vassals announced the months on stage, he would say it was in accordance with etiquette."
Since then, up until the Qin and Han dynasties, people returned to using Mengdong (November) as the start of the year, placing the leap month after September, resulting in a disorganized arrangement of solar terms and months, causing frequent errors in the timing of months. Furthermore, solar eclipses did not occur on the new moon (the first day of the lunar month), and these mistakes persisted for hundreds of years without correction. People only became aware of these errors in the seventh year of Emperor Wu of Han's Yuanfeng era, and improved the calendar by enlisting talented individuals to revise the Taichu Calendar, correcting the discrepancies regarding the new moon and leap month, accurately calculating the degrees of the constellations to determine their arrangement, and designating the Yin month (the first month) as the start of the year, with the Huangzhong month (November) set as the initial month of the calendar. However, the degrees of the Dipper in the Taichu Calendar (representing the twenty-eight constellations) were excessive, which later fell into increasing neglect. By the second year of the Tang Yuanhe era, the Sifen Calendar was adopted and has been in use ever since. Observations of solar eclipses showed that they consistently occurred at the end of the month, indicating that the degrees of the Dipper were excessive, resulting in a calendar that was initially too detailed and then became sparse, rendering it unusable.
Therefore, I utilized my leisure time to study astronomy and calendars, consulted ancient texts, and verified them using solar eclipses and new moons. After careful consideration, I established a new calendar that neither advances nor delays, making it the best option among both ancient and modern calendars. Just like in the ancient times of Emperor Yao, it can accurately grasp the calendar and seasons, enabling various skills to flourish and allowing the common people to lead better lives. I hope that the ceremonies and various systems of today's nation can conform to the standards of ancient times, being complete and perfect. Therefore, I aim to correct the new moon and revise the calendar, setting the start of the year in the Month of Great Lu (October) and the starting month of the calendar as Jianzi Month (November). I believe that in the era of Emperor Ku, the calendar was known as the "Zhuanxu Calendar"; during the time of the Yellow Emperor, it became known as the "Yellow Emperor Calendar"; during the reign of Emperor Wu of Han, the calendar was reformed and became known as the "Taichu Calendar." Now that we are renaming it to Jingchu, it should be referred to as the "Jingchu Calendar." The "Jingchu Calendar" I have established is simple in its methods and easy to use, is less demanding to manage, and is easy to understand for learning. Even experts in astronomy and calendars, like Yansang, Lishou, Chongli, and Xihe, who have used various methods to calculate astronomical calendars, cannot reach the precision of my system. Therefore, the calendars of past dynasties have generally been imprecise and lacking in detail, and since the time of the Yellow Emperor, they have been continuously reformed. From the founding of the Yuan Dynasty in the Renchen year to the Jingchu year in the Dingsi year, a total of 4,446 years have passed, as calculated. The calendar of the Yuan Dynasty started from the Tianzheng month of Jianzi (the first month), with the first year of the Yuan Dynasty being the first day of the Jiashen cycle, which coincides with the midnight of the winter solstice.
The Yuan Dynasty's calendar regulations stipulated that 10,000 equals 1,580. According to the era law, 1,000 years equals 1,843. There are 22,795 months in an era. A chapter spans 19 years and consists of 235 months, with 7 of those being leap months. In total, there are 134,630 days. One day is equivalent to 4,559 days, with a remainder of 9,670. There are 673,150 days in a week. An era contains 12 qi phases, each comprising 12. The total for month-weeks is 24,638, the communication law totals 47, and the communication figure is 791,110. The synodic month total is 67,315, the limit for conjunctions is 722,795, the communication week is 125,621, the remaining days in the week total 2,528, there are 231 empty days, and the star division totals 455.
Next is the first year of the Jiazi era, the first day is a new moon, with the moon aligned on the sun's orbit. The conjunction difference rate is 412,919, and the delay difference rate is 103,947. In the second year of the Jiaxu era, the first day is a new moon, with the moon aligned on the sun's orbit. The conjunction difference rate is 516,529, and the delay difference rate is 73,767. In the third year of the Jiashen era, the first day is a new moon, with the moon aligned on the sun's orbit. The conjunction difference rate is 621,139, and the delay difference rate is 43,587. In the fourth year of the Jiawu era, the first day is a new moon, with the moon aligned on the sun's orbit. The conjunction difference rate is 723,749, and the delay difference rate is 13,470. In the fifth year of the Jiachen era, the first day is a new moon, with the moon aligned on the sun's orbit. The conjunction difference rate is 37,249, and the delay difference rate is 108,848. In the sixth year of the Jiayin era, the first day is a new moon, with the moon aligned on the sun's orbit. The conjunction difference rate is 14,859, and the delay difference rate is 78,668.
This passage documents the results of ancient calendar calculations, filled with various astronomical figures. Understanding the specific meanings requires specialized knowledge of ancient calendars. In simpler terms, it outlines the calendar parameters for various years in different eras. This passage details a highly specialized method of astronomical calendar calculation from ancient times.
Paragraph 1: The method of calculating the "ji cha" is about if there is a difference of one hundred thirty-six thousand one hundred and ten days between two calendar epochs. In order to know how this difference is calculated, first calculate how many months are in an epoch, then multiply this number of months by a constant ("tong shu"), subtract a fixed value ("hui tong"), and the remainder is the "ji cha". Add this "ji cha" to the starting date of the previous epoch, and you will get the starting date of the next epoch. If the sum does not exceed "hui tong", then the first day of the first month of that year is exactly the new moon day (lunar New Year's Day); if it exceeds "hui tong", then the first day of the first month is after the new moon day. The part that exceeds "hui tong" means the first day of the first month is later than the new moon day; the part that does not exceed "hui tong" means the first day of the first month is earlier than the new moon day.
Paragraph 2: This part discusses another method of calculating the "ji cha", which this time amounts to thirty-one thousand eight hundred days. The calculation method is similar; first calculate how many months are in an epoch, then multiply this number of months by a constant ("tong shu"), subtract a cycle value ("tong zhou"), and then subtract this number from "tong zhou". The result is the "ji cha". Subtract this "ji cha" from the starting date of the previous epoch, and you will get the starting date of the next epoch. If the subtraction is not complete, then add "tong zhou". It also explains how to calculate the "ci yuan ji cha rate" and "ci ji"; the method is similar to the previous one, all obtained through subtraction.
Paragraph 3: This paragraph introduces how to calculate the "tui suo ji yue shu", which involves calculating how many months there are from the Renchen epoch (the starting year of an epoch) to a certain year. The method is: first calculate the total number of years from the Renchen epoch to the target year, then divide by a fixed value ("ji fa"). The quotient represents how many epochs have passed, and the remainder indicates which year within that epoch. Then multiply the number of years by a constant ("zhang yue"); the result is the total number of months, and the portion that is less than a month is referred to as the leap remainder. If the leap remainder is greater than or equal to twelve, then there is a leap month in that year. The leap month is determined by the absence of solar terms.
Paragraph 4: This paragraph explains how to calculate the "retrogression method," which is to determine the first day of the first month of a certain year. The method is: first use a constant multiplied by the total number of months calculated earlier (accumulated months) to get the accumulated points; then use another constant ("day method") to divide, obtaining the accumulated days, and the part less than one day is the small remainder. Divide the accumulated days by sixty, and the remainder is the big remainder. Using the big remainder and the epoch number, you can determine the date of the first day of the first month of a certain year.
Paragraph 5: This paragraph explains how to calculate the first day of the next month. The method is: add 29 to the big remainder, then add the small remainder (2419). If the small remainder exceeds the "day method," deduct it from the big remainder, and then determine the first day of the next month according to the previous method. If the small remainder exceeds 2140, this month is considered a major month.
Paragraph 6: This paragraph explains how to calculate the lunar phases (waxing moon, full moon, waning moon). The method is: add 7 to the big remainder of the first day of the month, add 1744 to the small remainder, and a small fraction. If a small fraction exceeds 2, deduct it from the small remainder; if the small remainder exceeds the "day method," deduct it from the big remainder; if the big remainder exceeds 60, subtract 60. The remaining value can be used to determine the date of the waxing moon. By extension, you can calculate the full moon, waning moon, and the first day of the next month. If a lunar eclipse occurs, the calculation method will be adjusted based on the position of the moon during the eclipse.
Paragraph 7: The last paragraph introduces how to calculate the twenty-four solar terms, using the winter solstice as an example. The method is: first calculate the number of years from a specific epoch to the target year, multiply a constant by the number of years; the part that exceeds the length of an epoch is the big remainder, and the part that is insufficient is the small remainder. If the big remainder exceeds 60, subtract 60; the remaining result and the epoch number can determine the date of the winter solstice.
First, let's calculate the "next solar term date." Add fifteen, subtract four hundred and two, then divide by eleven, and see what the remainder is. If this remainder is large enough, subtract it from the subtracted four hundred and two; if it is not large enough, subtract it from the added fifteen. After the calculation, note down the date; this is the "next solar term day."
Next is the method of calculating the leap month: subtract the chapter age from the leap year remainder, multiply the remaining number by the age. If the result is enough for a chapter, then leap a month; if only half is enough, still leap a month. Remember, we start counting from November, and the result is the leap month. The leap month sometimes comes early, sometimes late, mainly depending on whether there is a solar term to determine.
Heavy snow occurs in November, with a range from 1242 to 1248; the winter solstice also occurs in November, with a range from 1254 to 1245; Minor Cold occurs in December, with a range from 1235 to 1224; Major Cold also occurs in December, with a range from 1213 to 1192; Beginning of Spring occurs in January, with a range from 1172 to 1147; Rain Water occurs in January, with a range from 1122 to 993; Awakening of Insects occurs in February, with a range from 1065 to 1036; the spring equinox occurs in February, with a range from 1008 to 979; Clear and Bright occurs in March, with a range from 951 to 925; Grain Rain occurs in March, with a range from 900 to 879; Beginning of Summer occurs in April, with a range from 857 to 840; Grain Fullness occurs in April, with a range from 823 to 812; Grain in Ear occurs in May, with a range from 800 to 799; Summer Solstice occurs in May, with a range from 798 to 801; Minor Heat occurs in June, with a range from 805 to 815; Major Heat occurs in June, with a range from 825 to 842; Beginning of Autumn occurs in July, with a range from 859 to 883; End of Heat occurs in July, with a range from 907 to 935; White Dew occurs in August, with a range from 962 to 992; Autumn Equinox occurs in August, with a range from 1021 to 1051; Cold Dew occurs in September, with a range from 1080 to 1107; Frost occurs in September, with a range from 1133 to 1157; Beginning of Winter occurs in October, with a range from 1181 to 1198; Light Snow occurs in October, with a range from 1215 to 1229.
This passage describes the ancient calendar's method of calculating leap months and solar terms, as well as the range of each solar term. These numbers represent the results of some astronomical calculations used to determine the specific dates of solar terms and leap months. The range of values may be related to observational errors or calculation methods at the time. In short, this is a rather complex astronomical calendar calculation process.
First, let's talk about how to calculate the "mei ri," which is the number of days after the winter solstice in a year. On the day of the winter solstice, if there are any remaining days, add one day, then multiply it by a specific number (mei fen), and then divide it by another specific number (mei fa). The portion that divides evenly is the large remainder, and the portion that cannot be divided is the small remainder. If the large remainder exceeds sixty, subtract sixty from it, and the remaining portion is the large remainder used for counting days. This large remainder is the number of days after the winter solstice last year.
To calculate the next "mei ri," add 69 to the large remainder and 592 to the small remainder. If the small remainder exceeds mei fa, add one to it again, and the large remainder will change accordingly, using the same method as above. When the small remainder is exhausted, it signifies "extinguished," and a cycle has ended.
Next, let's see how to calculate the day when the five elements take charge, that is, which day of the week wood, fire, gold, and water take over. The four solar terms of the beginning of spring, summer, autumn, and winter are the days when wood, fire, gold, and water take charge. Subtract 18 from the big remainder of these four days, subtract 483 from the small remainder, and subtract 6 from the small fraction. The remaining number is used to record the day. After calculating, the day prior to these four days is when Earth takes charge. If the big remainder, small remainder, and small fraction are not enough to subtract, follow certain rules to add or subtract.
Next is to calculate the auspicious day for casting hexagrams. Use the big remainder of the winter solstice day, then multiply the small remainder by 6, which marks the beginning of the "Kan Gua." Next, add 191 to the small remainder; if this exceeds a certain threshold (Yuan method), subtract from the big remainder, and so on, which is the day when the "Zhong Fu" starts. To calculate the next hexagram, add 6 to the big remainder, add 967 to the small remainder, and so on. The four main hexagrams are all calculated based on the middle day, by multiplying the small remainder by 6.
Let's talk about how to calculate the solar position in the sky. Multiply a number (based on the calendar system) by the number of days since the new moon (lunar first day); if it exceeds a circle (week), subtract a circle, then divide the remaining by the calendar, the quotient is the degree, and the remainder is the minute. Start counting from the fifth star of the Ox constellation, divide by the order of the constellation; if it is less than one full constellation, it is the position of the sun at midnight on the new moon of the eleventh month.
To calculate the next day, add one degree while keeping the minutes the same. If the minutes are not enough, subtract one degree.
Next is the lunar position, which is the position of the moon in the sky. Multiply the number of days since the new moon by the length of a lunar cycle; subtract a circle if it exceeds, then divide the remaining by the calendar, the quotient is the degree, and the remainder is the minute. The method is the same as calculating the solar position, which allows you to determine the position of the moon at midnight on the new moon of the eleventh month.
To calculate the next month, add 22 degrees and 860 minutes for a short month, and add 1 day, 13 degrees, and 679 minutes for a large month; if the minutes exceed the limit, add one degree. In the latter part of winter, the moon is located near the Zhang and Xin constellations.
Finally, let's talk about how to calculate the conjunction degree, which is the degree when the sun and moon are in the same position. Multiply a number (zhang sui) by the remainder of the new moon day; if it exceeds a specific number (standard method), subtract it, and the remainder is divided into large units and small units. Subtract the sun's degree and minute at midnight from the new moon day; if the minute exceeds the rule, subtract it from the degree, and so on, you can calculate the degree when the sun and moon are together in the 11th month of Tianzheng.
To calculate the next month, add 29 degrees, 977 large units, and 42 small units. If the small unit exceeds the standard method, subtract it from the large unit; if the large unit exceeds the rule, subtract it from the degree. Finally, adjust the values using the constellations, and you can calculate the degree when the sun and moon are together in the next month.
First, let's see how to calculate the position of the sun and moon. To calculate the position of the first quarter, add the new moon day's degree (the first day of the lunar month) plus seven conjunction degrees, seven hundred and five large units, ten minutes, and one minute. If the minute exceeds two, subtract two from the ten minutes; if the ten minutes are full, subtract it from the large units; if the large units are full, subtract it from the degree. By following this method, the obtained degree is the position of the first quarter. Continue to add, and you can calculate the position of the full moon (15th day of the lunar month), last quarter, and the position of the new moon next month.
To calculate the position of the first quarter, add ninety-eight conjunction degrees, one thousand two hundred and seventy-nine large units, and thirty-four minutes to the new moon day's degree. Then, continue to calculate according to the method just now, and you can get the position of the first quarter. Similarly, continue to add, and you can calculate the position of the full moon, last quarter, and the position of the new moon next month.
Next, let's see how to calculate the brightness and darkness of the sun and moon (length of day and night). To calculate the brightness and darkness of the sun and moon, use the rule to calculate the sun, use the lunar cycle to calculate the moon, then multiply by the number of night hours of the nearest solar term, and then divide by two hundred to get the brightness (length of the day). Subtract the number of days from the rule, subtract the number of months from the lunar cycle, and the remaining is the darkness (length of the night). Add the brightness and darkness to midnight (the Zi hour) accordingly, and calculate the degree using the same method.
Next, let's explore how to calculate the timings for lunar conjunction, solar conjunction, and lunar eclipse.
To calculate the calculations of new moons, conjunctions, and lunar eclipses, first record the integral of the new moon days for the calculated year, then add the value of the conjunction difference rate for that year, and subtract the number of synodic months (one cycle). The remainder is the degree and minute distance from the node when the new moon occurs in November of that year. Continuously adding and subtracting the synodic months will allow you to calculate the degree and minute distance from the node when the new moon occurs each month in the future. Add the total number of new moon cycles and the degree and minute distance from the node when the new moon occurs on the 15th day of the lunar calendar each month, and subtract the synodic months; the remaining value is the degree and minute distance from the node when the full moon occurs. If the degree and minute distance from the node when the new moon occurs is below the total number of new moon cycles and above the limit of conjunction, then a solar eclipse will occur on the day of the new moon, and a lunar eclipse will occur on the day of the full moon.
Finally, let's explore how to determine the moon's position in its ecliptic path. To calculate the moon's position in the ecliptic path, first record the integral of the new moon days for the calculated year, then add the value of the conjunction difference rate for that year, and subtract twice the number of synodic months. If the remaining number is less than the number of synodic months, then if the record is in the ecliptic path, the first month of the lunar calendar will also be in that path; if the record is in orbit, the first month of the lunar calendar will be in the orbit. If the remaining number is greater than or equal to the number of synodic months, after subtracting the number of synodic months, if the path is full, it will be in the orbit; if the orbit is full, it will be in the path.
To calculate the situation for the next month, add the number of synodic months and subtract the number of synodic months; if it was previously in orbit, it will now be in the ecliptic path; conversely, if it was in the ecliptic path, it will now be in orbit. If a conjunction occurs before a lunar eclipse, the new moon will be in the ecliptic path, while the full moon will also be in the ecliptic path; if the new moon is in the orbit, the full moon will be in the orbit. If a lunar eclipse occurs before a conjunction, then the new moon of the eclipse will be in the orbit, and the full moon will be in the ecliptic path; if the new moon is in the ecliptic path, the full moon will be in the orbit. If the timing of the conjunction occurring first is closer to the limit, observations should be conducted earlier; if the lunar eclipse occurs first and is closer to the limit, observations should be postponed.
To calculate the distance to the intersection point, if the intersection occurs before the meeting, divide the distance to the intersection point by the daily degree; the result will be the distance to the intersection point. If the meeting happens before the intersection, subtract the distance to the intersection point from the meeting distance, then divide by the daily degree; the result will be the distance before the intersection point. All other measurements are in degrees. If the intersection distance exceeds fifteen degrees, even if the meeting occurs, a solar eclipse will not happen; if it is below ten degrees, a solar eclipse will occur; if it is above ten degrees, the solar eclipse will be minimal, causing only a slight dimming of light. The size of the solar eclipse is determined by using fifteen degrees as the divisor.
To calculate the direction of the start of the solar eclipse, if the moon is in the outer path, if the intersection happens before the meeting, the solar eclipse starts from the southwest corner; if the meeting occurs before the intersection, the solar eclipse starts from the southeast corner. If the moon is in the inner path, if the intersection happens before the meeting, the solar eclipse starts from the northwest corner; if the meeting occurs before the intersection, the solar eclipse starts from the northeast corner. The size of the solar eclipse is determined by using fifteen degrees as the divisor. If it happens to be at the intersection point, it is a total solar eclipse. A lunar eclipse occurs when the moon is in opposition to the sun; the direction of the deficit angle is opposite to what was mentioned above.
Below are the speeds of the moon's movement, the waxing and waning rates, the waxing and waning points total, and the daily degree of the moon's movement:
On the first day, the moon moves fourteen degrees (fourteen minutes), the waxing and waning rate increases by twenty-six, and the total waxing and waning points is two hundred eighty.
On the second day, the moon moves fourteen degrees (eleven minutes), the waxing and waning rate increases by twenty-three, and the total waxing and waning points is one hundred eighty-five thousand three hundred thirty-four, and the daily degree of movement is two hundred seventy-seven.
On the third day, the moon moves fourteen degrees (eight minutes), the waxing and waning rate increases by twenty, and the total waxing and waning points is two hundred twenty-three thousand three hundred ninety-one, and the daily degree of movement is two hundred seventy-four.
On the fourth day, the moon moves fourteen degrees (five minutes), the waxing and waning rate increases by seventeen, and the total waxing and waning points is three hundred fourteen thousand five hundred seventy-one, and the daily degree of movement is two hundred seventy-one.
On the fifth day, the moon moves fourteen degrees (one minute), the waxing and waning rate increases by thirteen, and the total waxing and waning points is three hundred ninety-two thousand seven hundred fourteen, and the daily degree of movement is two hundred sixty-seven.
On the sixth day, the moon moves thirteen degrees (fourteen minutes), the waxing and waning rate increases by seven, and the total waxing and waning points is four hundred fifty-one thousand three hundred forty-one, and the daily degree of movement is two hundred sixty-one.
On the seventh day, the moon traveled thirteen degrees (seven minutes), the gain-loss ratio decreased, and the gain-loss points were four hundred eighty-three thousand two hundred fifty-four, with a daily movement of two hundred fifty-four degrees.
On the eighth day, the moon traveled thirteen degrees (one minute), the gain-loss ratio decreased by six, and the gain-loss points were four hundred eighty-three thousand two hundred fifty-four, with a daily movement of two hundred forty-eight degrees.
On the ninth day, the moon traveled twelve degrees (sixteen minutes), the gain-loss ratio decreased by ten, and the gain-loss points were four hundred fifty-five thousand nine hundred, with a daily movement of two hundred forty-four degrees.
On the tenth day, the moon traveled twelve degrees (thirteen minutes), the gain-loss ratio decreased by thirteen, and the gain-loss points were four hundred one thousand three hundred ten, with a daily movement of two hundred forty-one degrees.
On the eleventh day, the moon traveled twelve degrees (eleven minutes), the gain-loss ratio decreased by fifteen, and the gain-loss points were three hundred fifty-one thousand four hundred thirteen, with a daily movement of two hundred thirty-nine degrees.
On the twelfth day, the moon traveled twelve degrees (eight minutes), the gain-loss ratio decreased by eighteen, and the gain-loss points were two hundred eighty-two thousand six hundred fifty-eight, with a daily movement of two hundred thirty-six degrees.
On the thirteenth day, the moon traveled twelve degrees (five minutes), decreased by twenty-one points, and the gain points were two hundred thousand five hundred ninety-six, totaling two hundred thirty-three.
On the fourteenth day, it traveled twelve degrees (three minutes), decreased by twenty-three points, and the gain points were one hundred four thousand eight hundred fifty-seven, totaling two hundred thirty-one.
On the fifteenth day, it traveled twelve degrees (five minutes), increased by twenty-one, and the total gain-loss points were two hundred thirty-three.
On the sixteenth day, it traveled twelve degrees (seven minutes), increased by nineteen, and the total gain-loss points were ninety-five thousand seven hundred thirty-nine, totaling two hundred thirty-five.
On the seventeenth day, it traveled twelve degrees (nine minutes), increased by seventeen, and the total gain-loss points were one hundred eighty-two thousand three hundred thirty-six, totaling two hundred thirty-seven.
On the eighteenth day, it traveled twelve degrees (twelve minutes), increased by fourteen, and the total gain-loss points were two hundred fifty-nine thousand eight hundred sixty-three, totaling two hundred forty.
On the nineteenth day, it traveled twelve degrees (fifteen minutes), increased by eleven, and the total gain-loss points were three hundred twenty-three thousand six hundred eighty-nine, totaling two hundred forty-three.
On the twentieth day, it traveled twelve degrees (eighteen minutes), increased by eight, and the total gain-loss points were three hundred seventy-three thousand eight hundred thirty-eight, totaling two hundred forty-six.
On the 21st, at 13 degrees and 3 minutes, there was an increase of four; the reduced integral value is 413,311, totaling 250. On the 22nd, at 13 degrees and 7 minutes, there was a decrease; the reduced integral value is 428,546, totaling 254. On the 23rd, at 13 degrees and 12 minutes, there was a decrease of five; the reduced integral value is 428,546, totaling 259. On the 24th, at 13 degrees and 18 minutes, there was a decrease of eleven; the reduced integral value is 405,751, totaling 265. On the 25th, at 14 degrees and 5 minutes, there was a decrease of seventeen; the reduced integral value is 355,602, totaling 271.
On the 26th, at 14 degrees and 11 minutes, there was a decrease of twenty-three; the reduced integral value is 278,099, totaling 277. On the 27th, at 14 degrees and 12 minutes, there was a decrease of twenty-four; the reduced integral value is 173,242, totaling 278. On Sunday, at 14 degrees and 13 minutes, plus an additional 626 minutes, there was a decrease of twenty-five (including 626 decimals); the reduced integral value is 63,826, totaling 279 (including 626 decimals).
This text describes the ancient astronomical calendar calculation method, which is very professional. Let's explain it sentence by sentence in modern spoken language.
First paragraph:
"The method for calculating the timing of the new moon and lunar eclipses in the ancient astronomical calendar is as follows: start with the integral for the new moon, add the rate of the delay and speed difference below the integral, divide by 360 degrees of the week; if the remainder equals a full day, count it as one day; if not, record the remainder, which is referred to as the 'remaining day.' This calculation will yield the date when the new moon of the eleventh month enters the calendar for that year."
"To calculate the next month, add one day based on the previous month, resulting in a remainder of 4450; to calculate the full moon (lunar 15th), add fourteen days to the previous month's calculation, resulting in a remainder of 3489. If the remainder is sufficient for one day, add one day; if the remainder exceeds 27 days, subtract 27 days; if the remainder is not enough, subtract one day and add a 'weekly void' (this concept requires an understanding of the historical calendar system).
In the calculation of the time of the conjunction of the sun and moon and the occurrence of lunar eclipses, additional detailed factors must be taken into account, such as the slight variations in the moon's orbit. Determine the 'size remainder' through complex calculations and methodologies, and adjust the time of the conjunction and lunar eclipse based on the calculation results. The specific calculation process is very complex and cannot be easily explained here; a profound understanding of ancient calendrical systems is necessary for comprehension."
Timing: Multiply twelve by the remainder after dividing by the full day's law, and one chen is obtained, starting from the rat; the chen corresponding to the new moon is also determined. If there is a non-exhausted remainder, divide it by four, with one counting as slight, two as half, and three as too much. There is also a remainder of three, with one counting as strong, arranged above half a law, and discarded if not a full half a law. Strong and weak are combined accordingly, with half as half strong and too much as too strong. Two strong are considered as weak, combined with a slight as half weak, with half as too weak, and combined with too much as one chen weak. By assigning it to the chen, each receives its slight, too much, half, and strong, as well as weak. If the lunar eclipse occurs within four days before or after the midsection, it is considered a limit; if it occurs five days before or after the midsection, it is considered an interval. The remainder is considered a day if it falls below the interval or limit. This passage calculates the precise time of the new moon and lunar eclipse in chen units, considering finer time adjustments and using terms such as "slight, half, too much, strong, weak" to indicate slight time differences. The lunar eclipse occurs a few days before and after the fifteenth day of the lunar month, and different calculation methods are used depending on the situation.
The final two paragraphs enumerate constellations and their corresponding degrees, which serve as auxiliary data for astronomical observations. Here is the literal translation without any further explanation:
Dipper twenty-six (expressed in four hundred fifty-five divisions) Ox eight Girl twelve Void ten Peril seventeen Chamber sixteen Wall nine
Ninety-eight degrees to the north (expressed in four hundred fifty-five divisions)
Kui sixteen Lou twelve Stomach fourteen Ma eleven Bi sixteen Zhi two Can nine
Eighty degrees to the west
The following numbers, Well thirty-three, Ghost four, Willow fifteen, Star seven, Zhang eighteen, Wing eighteen, Zhen seventeen, refer to the degrees of the twenty-eight constellations in the south, totaling one hundred twelve degrees. Then there are Jiao twelve, Kang nine, Di fifteen, Fang five, Xin five, Wei eighteen, Ji eleven, which are the degrees corresponding to seventy-five degrees east. (The following degree table is omitted).
The following is the method of calculating the twenty-four solar terms. The book states that by using a specific calculation method, one can determine that the winter solstice occurs in November. From this result, the months of the other solar terms can be calculated in order. As for the positions of the stars, they are based on the position of the sun. First, calculate the small remainder for each of the twenty-four solar terms each year, then multiply by four; if the result is an integer, it indicates "less"; if not, multiplying by three indicates "strong." This result allows for the determination of the dawn and dusk times for each solar term, as well as the positions of the constellations.
The five stars refer to: Jupiter is called the "year star," Mars is called the "wandering star," Saturn is called the "filling star," Venus is called the "bright star," and Mercury is called the "morning star." The speeds of these five stars vary; they may move quickly at times, slowly at others, and can either advance or retreat. Since the universe was formed and yin and yang were separated, the sun, moon, and five stars have come together in the celestial records. Starting from the celestial records, they move together in the sky, affecting each other's speeds and directions. If a star and the sun appear simultaneously in the same constellation, it is called a "conjunction." The period between one "conjunction" and the next is referred to as the "termination." Then, using the date of the "termination" and the date of the year, the "conjunction termination year number" and the "conjunction termination number" can be calculated. Once these two numbers are established, additional calculation methods can be derived. Multiply the "chapter year" by the "conjunction number" to get the "conjunction month method"; multiply the "chronicle method" by the "conjunction number" to obtain the "day degree method"; multiply the "chapter month" by the "year number" to get the "conjunction month fraction"; divide the "conjunction month method" by the "conjunction month fraction" to obtain the "conjunction month number," with the remainder being the "month remainder." Multiply the "total number" by the "conjunction month number," then divide by the "day degree method" to obtain the "large remainder." Subtract the "large remainder" from sixty; the remainder is the "large remainder of the star conjunction new moon," and the remainder of the "large remainder" is the "small remainder of the new moon." Multiply the "total number" by the "month remainder," add the product of the "conjunction month method" and the "small remainder of the new moon," then divide by the product of the "day degree method" and "conjunction month method" to obtain the "star conjunction entry month day number." Simplify the remainder using the "total method" to get the "entry month day remainder." Subtract the "small remainder of the new moon" from the "day degree method"; the remainder is the "new moon virtual fraction." Multiply the "calendar斗 fraction" by the "conjunction number" to obtain the "star degree斗 fraction." For Jupiter, Mars, and Saturn, subtract the "conjunction number" from the "year number," multiply the remainder by the celestial circle, then divide by the "day degree method" to obtain the planet's degree, with the remainder being the degree remainder. For Venus and Mercury, multiply the celestial circle by the "year number," then divide by the "day degree method" to obtain the planet's degree, with the remainder being the degree remainder.
The "combined final year number" for Jupiter is 1,255, the "combined final number" is 1,149, the "combined month method" is 21,831, the "solar degree method" is 2,117,607, the "combined month number" is 13, the "month remainder" is 11,122, the "new moon major remainder" is 23, the "new moon minor remainder" is 4,093, and the "entry month day" is 15.
After a little over a month, the total calculation results in 1,995,664.
The new moon virtual fraction is 466.
The Dou fraction (斗分) is 522,795.
The planet's degree is 33.
The remaining degrees are 1,472,869.
Next is the calculation for Mars: the combined final year number is 5,105; the combined final number is 2,388; the combined month method is 45,372; the solar degree method is 441,184; the combined month number is 26; the month remainder is 23,000; the new moon major remainder is 47; the new moon minor remainder is 3,627; the entry month day is 13; the day remainder is 3,585,230; the new moon virtual fraction is 932; the Dou fraction (斗分) is 186,540; the planet's degree is 50; the remaining degrees are 1,412,150.
The results of the Mars calculations are as follows: the combined final year number is 3,943; the combined final number is 3,890; the combined month method is 72,371; the solar degree method is 719,987; the combined month number is 12; the month remainder is 58,153; the new moon major remainder is 54; the new moon minor remainder is 1,674; the entry month day is 24; the day remainder is 675,364; the new moon virtual fraction is 2,885; the Dou fraction (斗分) is 173,395; the planet's degree is 12; the remaining degrees are 5,962,256.
Next is the calculation for Venus: the combined final year number is 1,970; the combined final number is 2,385; the combined month method is 45,315; the solar degree method is 4,395,555; the combined month number is 9; the month remainder is 43,310; the new moon major remainder is 25; the new moon minor remainder is 3,535; the entry month day is 27; the day remainder is 194,990; the new moon virtual fraction is 1,224; the Dou fraction (斗分) is 185,175; the planet's degree is 292; the remaining degrees are 194,990.
Finally, the calculation for Mercury: the combined final year number is 1,870; the combined final number is 11,789; the combined month method is 223,991.
In a specific AD year, let's begin by calculating the days. The solar degree method is 21,727,127, the combined month number is 1, the month remainder is 215,459, the new moon major remainder is 29, the new moon minor remainder is 2,419, the entry month day is 28, the day remainder is 23,442,261, the new moon virtual fraction is 2,140, the Dou fraction (斗分) is 5,363,995, the planet's degree is 57, and the remaining degree is 23,443,361.
Next, calculate the movements of the five planets. The method is to start from the year of Renchen and multiply the year you want to calculate by a number called "Hezhong Number." If the result is exactly an integer multiple of the Hezhong year, it is called "Ji He"; if it is not an integer multiple, the remainder is called "He Yu." Subtract "He Yu" from "Hezhong Number," and if the result is 1, it indicates that the celestial body is in conjunction in the calculated year; if it is 2, it indicates that the celestial body is in conjunction in the second year being calculated; if the result is neither 1 nor 2, it indicates that the celestial body is in conjunction in that year. The remaining value, when subtracted from "Hezhong Number," gives you "Du Fen." If the "Ji He" of Venus and Mercury is an even number, it is a morning conjunction; if it is an odd number, it is an evening conjunction.
Then calculate the conjunction of the five planets with the moon. The method is to multiply the month number and the month remainder by "Ji He." If the result is an integer multiple of the lunar conjunction method, it is called "Ji Yue"; if not, the remainder is the "Yue Yu." Divide the month by "Ji Yue," the quotient is the entering year, and the remainder is the entering month. Then multiply the Zhang Run by the entering month; if the result is an integer multiple of the Zhang month, it indicates a leap month, and then subtract it from the entering month, leaving the entering year and month. Starting from the year of Tianzheng, the result is the timing of the planetary conjunction with the moon. If it coincides with the leap month transition, use the new moon to adjust.
Next, calculate the conjunction of the moon and the new moon. The method is to multiply the common number by the entering month; if the result is an integer multiple of the day method, it is called "Ji Ri"; if not, the remainder is called "Xiao Yu." Divide 60 by "Ji Ri," the remainder is "Da Yu," and starting from the entering year, the result obtained is the day of the planetary conjunction with the new moon.
Then calculate the entering month and day. The method is to multiply the common number by the month remainder, then multiply the lunar conjunction method by the small remainder from the new moon, add these two results together, and then simplify using the common method. If the result is an integer multiple of the day degree method, that is the star conjunction entering month day; if not, the remainder is the day remainder. Starting from the new moon, the result obtained is the entering month day.
Finally, calculate the star conjunction degree. The method is to multiply the Zhou Tian value by Du Fen; if the result is an integer multiple of the day degree method, that is the degree; if not, the remainder is the remainder. Starting from the five degrees before Niu, the resulting value is the degree of the planetary conjunction.
The final calculation pertains to the next conjunction month. The method is: add the number of months to the current year, and add the remainder to the month remainder. If the result is a multiple of the conjunction month, then a month is obtained. If this month does not complete a year, it remains within that year; if it is complete, subtract one year; if there is a leap month, add it; the remainder indicates the following year; if it is complete again, it moves into the following two years. For Venus and Mercury, when they are in conjunction in the morning and adding a year, they become in conjunction in the evening; when they are in conjunction in the evening and adding a year, they become in conjunction in the morning.
The final calculation pertains to the next conjunction new moon. The method is: add the large and small remainders of the current new moon, respectively, to the large and small remainders of the conjunction new moon. If the remainder exceeds one month, then add an additional large remainder of 29 days and a small remainder of 24 or 19 days. If the small remainder is a multiple of the day, subtract it from the large remainder, using the same calculation method as previously described.
This passage talks about ancient astronomical calculations; let's go through it sentence by sentence. The first paragraph explains how to calculate the next new moon (the first day of the lunar month). First, calculate how many days have already passed in this month, then add the remaining days, and then apply the "remainder full day method" to get a result. If this result is slightly less than the new moon, subtract one day; if it is much more than the new moon (exceeding 24 or 19 days), subtract 29 days; if it is not much more, subtract 30 days. The remaining days are the number of days until the next new moon from now, and then the next new moon can be determined. The method for calculating the next conjunction degree is similar to the method for calculating the new moon.
The second paragraph explains the movement pattern of Jupiter. Jupiter appears in the morning with the sun, then goes into hiding. When it moves forward (counterclockwise), it travels 997,832 minutes in 16 days, which equals 2° 17' 52.38''. Then Jupiter appears in the east, after the sun. The speed of direct movement is sometimes fast and sometimes slow. When it is fast, it moves 11 degrees in 57 days; when it is slow, it moves 9 degrees in 57 days and then stops. It stops for 27 days before continuing to move. When it moves in the retrograde direction (clockwise), it moves 1/7 degree per day, retreats 12 degrees in 84 days, and then stops. Twenty-seven days later, it moves slowly in the direct direction, moves 9 degrees in 57 days, then accelerates, moving 11 degrees in 57 days; at this point, it is positioned ahead of the sun, appears in the west in the evening, and then meets the sun again after 16 days. In one complete cycle, it totals 398 days and 1,995,664 minutes, with Jupiter moving 33 degrees and 1,472,869 minutes.
The third paragraph talks about the movement pattern of Mars. Mars appears together with the sun in the morning, then becomes less visible. It moves forward for 72 days, covering 1,792,615 minutes, equivalent to 56 degrees and 1,249,345 minutes. Then it rises in the east, following the sun. The speed of its movement varies; when moving quickly, it travels 112 degrees in 184 days; when moving slowly, it travels 48 degrees in 92 days before stopping. It pauses for 11 days before resuming its movement. When in retrograde, it moves back 17 degrees over 62 days before stopping. After 11 days, it resumes slow forward movement, covering 48 degrees in 92 days, then accelerates, covering 112 degrees in 184 days. At this point, it is positioned ahead of the sun, becoming visible in the west at night, and then it reunites with the sun after 72 days. In one cycle, the total duration is 780 days and 3,585,230 minutes; Mars has covered 415 degrees and 2,498,690 minutes.
In the morning, Saturn appears together with the sun, then Saturn becomes less visible. For nineteen days, which equals 3,847,675.5 minutes, Saturn moves 2 degrees and 6,491,121.5 minutes, then it can be seen in the east in the morning, at this point, it is behind the sun. When moving forward, it covers 13/172 of its path, traveling 6.5 degrees in 86 days and then stops. After a pause of 32.5 days, it begins moving again. When moving backward, it retreats 6 degrees after 102 days, moving 1/17 of its path each day, and then stops again. After a pause of 32.5 days, it resumes forward motion, covering 6.5 degrees in 86 days; at this point, it is in front of the sun, becoming less visible in the west at night. After moving forward for 19 days, equivalent to 3,847,675.5 minutes, Saturn moves 2 degrees and 6,491,121.5 minutes before it reunites with the sun. At the end of one cycle, the total duration is 378 days and 675,364 minutes; Saturn has moved 12 degrees and 5,962,256 minutes.
In the morning, Venus is together with the Sun, and then Venus goes into hiding. It retreats four degrees over six days, and then in the morning, it can be seen in the east, at which point it is behind the Sun and in retrograde motion. When it moves slowly, it travels 0.6 degrees each day, retreating six degrees in ten days. After a pause of seven days, it resumes its motion. When in direct motion and moving slowly, it travels 0.6 degrees each day, covering thirty-three degrees in forty-five days, and then continues in direct motion. When moving quickly, it travels one degree and four-fifths each day, covering one hundred five degrees in ninety-one days, and then continues in direct motion. When moving even faster, it travels one degree and twenty-one-fifths each day, covering one hundred twelve degrees in ninety-one days, at which point it is behind the Sun, "hiding" in the east in the morning. After forty-two days in direct motion, Venus travels fifty-two degrees and then is again together with the Sun. One conjunction totals two hundred ninety-two days, and the total degrees traveled by Venus is the same.
In the evening, Venus is together with the Sun, and then Venus goes into hiding. After forty-two days in direct motion, Venus travels fifty-two degrees and then can be seen in the west in the evening, at which point it is in front of the Sun. When in direct motion and moving quickly, it travels one degree and twenty-one-fifths each day, covering one hundred twelve degrees in ninety-one days, and then continues in direct motion. When moving slowly, it travels one degree and one-fourteenth each day, covering one hundred five degrees in ninety-one days, and then continues in direct motion. When moving even slower, it travels 0.6 degrees each day, covering thirty-three degrees in forty-five days, and then stops. After stopping for seven days, it begins to move again. In retrograde motion, it travels 0.6 degrees each day, retreating six degrees in ten days, at which point it is in front of the Sun, "hiding" in the west in the evening. After six days in retrograde, it retreats four degrees, and then it is again together with the Sun. Two conjunctions complete one cycle, totaling five hundred eighty-four days, and the total degrees traveled by Venus is the same.
In the morning, Mercury and the Sun will appear in the same direction. Mercury will initially become invisible, then after eleven days, it will shift seven degrees to the east and appear behind the Sun. If Mercury appears to be in retrograde, it moves quickly, moving back one degree each day, then stops. After one day, it will turn again. If Mercury is in direct motion, it moves slowly, covering seven-sevenths of a degree in a day, seven degrees in eight days, and then stops. If it moves quickly, it covers two-tenths of a degree in a day, twenty-two degrees in eighteen days, and appears behind the Sun, rising in the east in the morning. If in direct motion, it covers 36,361 degrees in eighteen days, and then appears in the same direction as the Sun. Mercury and the Sun will align every fifty-seven days and 36,361 minutes; this is how Mercury operates.
In the evening, Mercury and the Sun will also appear in the same direction. Mercury will initially become invisible, then, after eighteen days, it will cover 36,361 degrees and appear in the west in the evening, in front of the Sun. If in direct motion, it moves quickly, covering two-tenths of a degree in a day, twenty-two degrees in eighteen days, and then continues in direct motion. If it moves slowly, it covers seven-sevenths of a degree in a day, seven degrees in eight days, and then stops. After one day, it will turn again. If it's in retrograde, it moves back one degree each day, appears in front of the Sun, and then becomes invisible in the west at night. If it's in retrograde, it moves back seven degrees over eleven days, and then appears in the same direction as the Sun. Mercury and the Sun will meet twice, taking a total of one hundred fifteen days and one million eight hundred ninety-six thousand three hundred fifty minutes; this is how Mercury operates.
When calculating the conjunction of Mercury and the Sun, you should sum the remainders of the degrees of Mercury and the Sun. If the remainder exceeds the solar day calculation, you need to start over and calculate as before to obtain the time and degree remainders of Mercury's appearance. Multiply the denominator of Mercury's movement by the visible degree fraction; if it equals the solar day calculation, you will obtain a whole number; if it does not divide evenly, any remainder greater than half is counted as a whole number. Then add the movement fraction; when the fraction reaches the denominator, you get one degree. The denominators for retrograde and direct motion differ; you need to multiply the current movement denominator by the original fraction. If it equals the original denominator, it is the current movement fraction. In the event of a pause, refer back to the earlier calculations, subtracting for retrograde. If the hidden degrees are insufficient, divide by the Dou fraction, using the movement denominator as a proportion. The fractions will increase or decrease, mutually constraining each other.
The Grand Preceptor of Emperor Han Wu, Liu Zhi of Pingyuan, modified the calendar with the Dou calendar, calculating the "Four-Fold Method," which would result in a loss of one day every three hundred years, using one hundred and fifty as the rule for degrees and thirty-seven as the Dou fraction. Calculating the Jiazi as the first year, reaching the tenth year of Taishi, with the year in Jiawu, after 97,411 years, the era commenced at midnight on the winter solstice of Jiazi, with the sun, moon, and five stars aligning with the Xingji, establishing the starting point of the era. He used some exaggerated statements to embellish it, naming it the "Correct Calendar."
Duke Yang Yu of Du Yu wrote the "Chunqiu Changli" (Long Calendar of Spring and Autumn), which notes: "The sun travels one degree each day, while the moon travels approximately thirteen degrees and nineteen minutes each day." Officials in charge of astronomy and calendars need to determine the first day of each month and the end of the month, as well as whether to add a leap month, based on the speed of the sun and moon. The leap month does not have a solar term (one of the twenty-four solar terms), and the orientation of the Big Dipper varies from that of other months. By considering these factors comprehensively, the four seasons and eight solar terms can be accurately determined, ultimately leading to the establishment of an accurate calendar, the intricacies of which are indeed profound! As long as one understands the essence, they will align with the laws of celestial mechanics, and everything will proceed in an orderly manner without error. As stated in the "Spring and Autumn Annals": "The leap month is for correcting time; time is for arranging events."
However, the principles governing Yin and Yang are constantly changing, and there are always slight deviations. These deviations accumulate continuously and will eventually conflict with the calendar. Therefore, Confucius and Qiuming often discussed the matters concerning the first day of each month and intercalary months, as they sought to correct errors in the calendar and clarify its rules.
Liu Zijun compiled the "Three Correct Calendars" to examine the "Spring and Autumn Annals." The book records two types of solar eclipses, a total of thirty-four times, but the "Three Correct Calendars" only calculated one solar eclipse, which had the greatest discrepancies compared to other calendars. Furthermore, it also stipulates adding a day every six thousand years. This means that the longer the years pass, the greater the discrepancies become; this method is fundamentally unworkable.
From ancient times to the present, many scholars studying the "Spring and Autumn Annals" have made errors. Some have invented their own methods, while others have used various calendars since the time of the Yellow Emperor to calculate the new moons (first day of each month) recorded in the texts, but their findings do not align. Solar eclipses occur on new moons; this serves as a verification of celestial phenomena. The "Classic of History" also records these solar eclipses occurring on new moons, which shows that the records of the "Classic of History" are consistent with celestial phenomena. However, Confucian scholars such as Liu Xiang and Jia Kui believe that solar eclipses occur on the second or third day of each month, which goes against the clear records of the sages. Their error lies in their stubborn adherence to a particular theory and refusal to adjust the calendar based on celestial phenomena.
I once wrote an article titled "On Calendars," carefully studying the principles of calendars. The main idea is that celestial bodies are in constant motion; the sun, moon, and stars each follow their own orbits, all are moving objects. Objects in motion cannot be completely identical, although the general degree of movement can be determined and limited. Accumulating days form months, and accumulating months form years; during the transition between the new and the old, there will inevitably be slight differences; this is a natural phenomenon. Therefore, during the Spring and Autumn period, some years had frequent solar eclipses, while in other years there were no solar eclipses for several years. The patterns were inconsistent, but the calculations relied on fixed values, 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 sighting of the new moon, and then it would be necessary to modify the calendar to adapt to it. The Book of Documents states, "Respect the vast heavens and observe the sun, moon, and stars," and the Book of Changes says, "Manage the calendar and understand the times," which means that calendars should be created based on celestial phenomena, rather than being established to verify celestial phenomena. By inference, during the more than two hundred years of the Spring and Autumn period, there were numerous revisions to the calendar. Even if the ancient algorithms were lost, we can still trace back to the classics and deduce approximate values, and the discrepancies noted in the classics have been confirmed. Scholars should carefully study the dates and solar eclipses recorded in the classics to verify the sighting of the new moon and calculate time. But in reality, this is not the case; everyone holds their own opinions, speculating about the circumstances of the Spring and Autumn period, and it's like measuring your own foot and then trying to fit someone else's shoe.
After I finished writing "On Calendars," in the Xianning era, Li Xiu and Bu Xian, who were proficient in calculations, established a calendar called the "Qiandu Calendar" based on my theories and submitted it to the court. This calendar slightly increases the sun's daily movement by a quarter of a degree to adjust the moon's movement. The calendar is revised every three hundred years using binary calculations. After more than seventy years, adjustments are made using a method based on the succession of strengths and weaknesses. The difference in strengths and weaknesses is very small, but enough to ensure the accuracy of long-term calculations. At that time, the officials compared the "Qiandu Calendar" with the "Taishi Calendar" against ancient and modern records. The "Qiandu Calendar" was 45 times more accurate than the official calendar of the time. This algorithm is still in use today. I also validated the Spring and Autumn period calendar using ten ancient and modern calendars and found that the "San Tong Calendar" had the largest error. The "Spring and Autumn" period comprised a total of seven hundred seventy-nine days (393 days in the "Chronicles" and 386 days in the "Annals"), recording thirty-seven solar eclipses, three of which were not classified as A or B. The "Huangdi Calendar" totaled four hundred and sixty-six days with one solar eclipse. The "Zhuanxu Calendar" totaled five hundred and ninety days with eight solar eclipses. The "Xia Calendar" totaled five hundred and thirty-six days with fourteen solar eclipses. The "Zhenxia Calendar" totaled four hundred and sixty-six days with one solar eclipse. The "Yin Calendar" totaled five hundred and three days with thirteen solar eclipses. The "Zhou Calendar" totaled five hundred and six days with thirteen solar eclipses. The "Zhenzhou Calendar" calculated to be 485 days with one solar eclipse. The "Lu Calendar" calculated to be 529 days with thirteen solar eclipses. The "San Tong Calendar" calculated to be 484 days with one solar eclipse. The "Qianxiang Calendar" calculated to be 495 days with seven solar eclipses. The "Taishi Calendar" calculated to be 510 days with nineteen solar eclipses. The "Qiandu Calendar" calculated to be 538 days with nineteen solar eclipses. The current "Chang Calendar" calculated to be 746 days with thirty-three solar eclipses. However, there is a small discrepancy here; the "Chronicles" and "Annals" contain an error, missing thirty-three days. In fact, there should have been four solar eclipses, with three solar eclipses not classified as A or B, possibly indicating incomplete records. At the end of the Han Dynasty, there was a scholar named Song Zhongzi who collected seven different calendars to investigate the chronology of the "Spring and Autumn" period. He discovered that the algorithms of the Xia Calendar and Zhou Calendar differed from those recorded in the "Book of Han: Yiwenzhi," and subsequently renamed them the "Zhenxia Calendar" and "Zhenzhou Calendar."
In the eighth year of Emperor Mu of the Jin Dynasty, there was an author named Wang Shuo who created a new calendar called the "Tongli Calendar". This calendar started with the year of Jiazi, with a cycle lasting 97,000 years, using 4,883 years as a major cycle, dividing the heavens into 1,225 degrees. He believed that this starting point was the beginning of the universe.
During the time of Yao Xing in Later Qin, in the year 414 AD, a man named Jiang Ji from Tianshui wrote the "Sanji Jiazi Yuanli". The book states: to establish a calendar, one must first understand the rules of the sun and moon's movements in order to calculate celestial phenomena and understand the changes on earth. If the fundamentals are incorrect, the four seasons will be disrupted. Therefore, Confucius wrote the "Spring and Autumn Annals", recording each day, each month, each season, and each year in order, because he understood that celestial changes are fundamental to human affairs, which is why calendars have always been highly regarded by rulers. From the time of Fuxi to the Han and Wei dynasties, each dynasty created its own calendar, aiming for accuracy. To test the accuracy of a calendar, one must see if its calculations of solar and lunar eclipses match up. However, ancient records only detailed solar eclipses in the "Spring and Autumn Annals", covering a period of 242 years from Duke Yin to Duke Ai, recording 36 solar eclipses, but the specific calendar used remains unclear.
Ban Gu believed that the "Spring and Autumn Annals" adopted the calendar of the state of Lu, which was inaccurate, leading to a chaotic arrangement of leap months. Lu took the surplus year as the beginning of the era, but when examining the leap month arrangement in the "Spring and Autumn Annals", it does not align with this era. The "Mingli Xu" states: to write the "Spring and Autumn Annals", Confucius re-examined the ancient calendar of the Shang Dynasty to ensure its methods were passed down through generations. Therefore, it seems that the "Spring and Autumn Annals" should be corrected using the Shang Dynasty calendar. However, when using the Shang Dynasty calendar to verify the records in the "Spring and Autumn Annals", we find that many new moons (the first day of each month) do not align, some are a day early, some are a day late. The records of new moons in the "Gongyang Commentary" and the "Spring and Autumn Classic" also vary, which can be understood, but the "Spring and Autumn Classic" has solar eclipse verifications, while the "Commentary" does not match.
Fu Qian explained, "The Gongyang Commentary uses the Era of Tai Chu Shang Yuan, which is the era of Liu Xin's 'San Tong Li.' What's the connection to the 'Spring and Autumn Annals'? Is it really appropriate to interpret the 'Spring and Autumn Annals' using the Han Dynasty calendar? The Gongyang Commentary has many errors, not just this one. In the twenty-seventh year of Duke Xiang, on the day of Yihai in the eleventh month, a solar eclipse occurred. The Gongyang Commentary said, 'Chen and Su are in Shen, the calendar officials were negligent, and both attempts to arrange a leap month were wrong.' However, according to calculations, a solar eclipse occurring in this month is normal, and there was no situation where two attempts to arrange a leap month were wrong. The calendar of Liu Xin and the solar eclipse recorded in the 'Spring and Autumn Annals' differ by only one day in some cases and by two days in most cases. He also associated the 'Spring and Autumn Five Elements Commentary,' suggesting that solar eclipses are caused by political turmoil. Liu Xin did not admit any errors in the calendar but instead fabricated reasons, which is truly unfair to celestial phenomena and misguides future generations!
Du Yu also believed that during the decline of the Zhou Dynasty and the widespread chaos, scholars were unable to understand the true calendar. The seven calendars currently in circulation may not necessarily have been used by the royal family at the time. When we use these seven calendars to check the ancient and modern conjunctions of the sun and moon, we find that they are all inaccurate, all due to the different values of the divisions. The divisions of the Yin calendar are one-fourth, the San Tong calendar is 1539 parts out of 385, the Qianxiang calendar is 589 parts out of 145, and the Jingchu calendar is 1843 parts out of 455. The units in these calendars vary in size, and their algorithms are also different. The divisions of the Yin calendar are too coarse, so they cannot be used now; the divisions of the Qianxiang calendar are too fine, so they cannot be used in ancient times; although the divisions of the Jingchu calendar are relatively moderate, the position of the sun is off by four degrees, and the predictions for solar and lunar eclipses are inaccurate. For example, if a solar eclipse occurs in Dongjing, but the position of the moon is calculated at six degrees in Can Su, given such a significant margin of error, how can it be relied upon for predicting celestial events and human activities?"
Now I have established a new calendar, using 605 out of 2451 as the unit of measurement, with the sun positioned at 17 degrees in the Dou constellation as the starting point of the celestial phenomenon. This can help correct the records in the "Spring and Autumn Annals" and verify modern celestial events. When using this calendar to verify the 36 recorded solar eclipses in the "Spring and Autumn Annals," 25 were accurate, 2 were off by just one day, and 5 were inaccurate. In total, there were 34 occurrences, with the remaining two lacking recorded dates for solar eclipse events, making it impossible to confirm their accuracy. Various historical texts state that "the Dou calendar was revised every three hundred years."
If we apply our current new calendar to the Spring and Autumn period, most solar eclipses would occur on the first day of the lunar month. From the Spring and Autumn period to the present, over a thousand years have passed, and the occurrence of solar eclipses has consistently fluctuated around the new moon and full moon. Therefore, this method can be used continuously, without the need to change calendars every few hundred years like before.
This method is reliable! Just think about it: over a thousand years have passed since the Spring and Autumn period, and the pattern of solar eclipses has remained unchanged, always occurring around the new moon and full moon. Our new calendar can fully adapt to this pattern and can be utilized for hundreds or even thousands of years without issues. Unlike the previous calendars that were constantly changing, which was quite a hassle! Changing every three hundred years, isn't that exhausting?
From the start of the Jiazi cycle to the first year of Duke Yin of Lu in the Jiwei year, a total of 82,736 years have elapsed. Continuing on to the ninth year of the Taiyuan era under Emperor Xiaowu of Jin in the Jiashen year, a total of 83,841 years have passed.
According to the Yuan law, it is 73,553; according to the Ji law, it is 2,451; the total comes to 179,044. The daily calculation yields 6,620; the monthly cycle yields 32,766; the qi division is 12,860; the Yuan month is 99,045; the Ji month is 33,115; the Mei division is 44,761; the Dou division is 65; the week day is 895,220 (also known as Ji day); the Zhang month is 235; the Zhang year is 19; the leap Zhang is 7; the middle year is 12; there are 47 meetings (893 years in total, divided into 47 meetings, just enough); the qi is 12; the Jiayin record discrepancy is 9,157; the Jiashen record discrepancy is 6,337; the Jiachen record discrepancy is 3,517; a week and a half is 127; the total for the new moon is 941; the meeting year totals 893; the meeting month totals 11,045; the small division totals 2,196; the Zhang number is 129; the small division totals 2,183; the leap large division is 76,269; the historical week totals 447,610 (representing half a week); the meeting division totals 38,134; the difference division totals 11,986; the meeting rate totals 1,882; the small division method is 2,209; the entry limit totals 11,104; the small week totals 254; the Jiayin record discrepancy rate is 49,178; the Jiashen record discrepancy rate is 58,231; the Jiachen record discrepancy rate is 67,284; the total week is 167,063; the surplus of week days is 3,362; the virtual week is 2,701.
Let's first discuss the calculation methods of these five stars. The book states that the result derived from the "Five Star Approximation Method" should be considered the standard answer, without needing to align with the original data. Think about it, the initial calculation method must differ from the current method; each has its own advantages, thus, the author developed two methods.
Mr. Ji still used lunar eclipses as a means to test the degree of the sun's movement. This serves as a benchmark for researchers of calendars! He also wrote a book titled "On the Celestial Sphere" to calculate the sun's path along the ecliptic, disputing several mistaken beliefs held by earlier Confucian scholars, thus providing a thorough clarification of the matter.
