During the time of Emperor Xiaojing, the "Renzi Calendar" was used, but its new moon dates (the first day of the lunar month) were always somewhat off, the position of Mars was also misaligned, and the movements of four other planets did not conform to the calendar. In summary, there were fundamental issues with the calendar itself. In the tenth month of the first year of Xinghe, King Xian of Qi arrived in Ye City and ordered Li Yexing to revise the calendar, resulting in the establishment of the new "Jiazi Yuan Calendar." After this was completed, Left Minister Sima Ziruo, Right Minister Longzhi, and others submitted a memorial stating:
Since ancient times, the operation of heaven and earth, the alternation of day and night, the interplay of yin and yang, and the cycles of cold and heat all follow certain规律. Wise rulers in ancient times formulated calendars based on these规律, arranged various sacrificial activities, and aligned with the will of heaven. However, since the reign of Emperor Maojin (referring to a certain emperor), the calendar has been changed repeatedly, and the officials responsible for formulating the calendar have been replaced time and again, resulting in conflicting calendars without a unified standard. This is not due to the uneven levels of understanding among people, but because the way of heaven is inherently unpredictable, making it difficult to achieve complete stability.
The previously used "Zhengguang Calendar" had discrepancies in calculating the new moon dates, despite being in use. Measurements using shadow gauges or charcoal for timing revealed errors; observations of weather in a heavy chamber and using ashes to judge time also showed slight differences. We earnestly beseech Your Majesty, with your wisdom and martial prowess, to seize this opportunity, to be as fierce as a tiger and soar like a flying dragon, to unify the realm and bring peace to the world. Great Chancellor, King of Bohai, you are naturally wise and exceptionally talented, capable of governing the world, solving various problems, rectifying chaos, and ensuring national stability and prosperity. You have made great contributions to the country, and now that the nation is stable and rituals and music have been renewed, the issue of the calendar remains unresolved. Therefore, Your Majesty has ordered the Assistant Palace Attendant Li Yexing, the Eastern Pavilion Ritual Scholar Wang Chun of the Great Chancellor's Office, and the Household Minister Cao Canjun and others to be responsible for revising the calendar.
However, the calculation of the calendar can vary in speed and frequency. It cannot be assessed from just one perspective, nor can it be measured using only a single method. Therefore, His Majesty has also commanded Grand Chancellor Sun Qian, Chief Clerk, General Ye, Commander of the Cavalry, Ji Jing, the former Minister of the Yellow Gate, Cui Xian, the Prince of Bohai, along with Zishu, a student of Li Yexing, and others to participate collectively, engage in discussions, and ultimately establish the calendar's accuracy.
Although we officials are deeply concerned about this matter and are doing our best, we still fear that we may not be doing enough. I believe that getting this job done is like building a house; it requires everyone's joint effort and cannot be accomplished by one person alone. Therefore, we must bring together the expertise of various specialists to jointly revise and improve it. The experts involved in the revision of the calendar include: Lu Daoyue, the Left Minister; Li Xie, the Minister of Agriculture and Marquis of Pengcheng; Pei Xianbo, the Left Minister and Grand Zhongzheng of Dongyongzhou; Wen Zisheng, the Cavalry Regular Attendant and Grand Zhongzheng of Xiyanzhou; Lu Cao, the Chief Historian of the Grand Commandant's Office; Lu Yuanming, the Right Deputy Minister of the Ministry of Personnel and Duke of Chengyang; Li Tonggui, the Deputy Minister of the Central Secretariat; Xing Ziming, the Former Deputy Minister of the Central Secretariat; Yu Wenzhong, the Deputy Minister of the Central Secretariat; Yuan Zhongquan, the Former Chief Historian of the Grand Commandant's Office and Marquis of Jiankang; Du Bi, the Chief Minister of the Law Department; Li Puji, the Left Deputy Minister of the Ministry of War and Duke of Dingyang; Xin Shu, the Deputy Minister of the Ministry of Rites; Yuan Changhe, the Deputy Minister of the Ministry of Sacrifices; Hu Shirong, the Former Sima of Qingzhou Cavalry Command and Viscount of Anding; Zhao Hongqing, the Chief Historian of Luxiang County; Hu Fatong, the Chief Historian; and, responding to the emperor's summons, Zhang Zhe, the Left Minister; Cao Wei Zu, the External Sima Supervisor; Guo Qing, the Chief Historian; and Hu Zhonghe, the Chief Historian Doctor, among others. They are all capable and experienced individuals who can explain the calendar clearly, carefully comparing and analyzing, and meticulously verifying ancient and modern texts, ultimately completing this new calendar together. The new calendar uses the Jiazi as the era, with the Zi hour marking the start of the day, which is in line with heavenly principles, so we should begin calculations from this era. We are currently in the Xinghe years, and we use the era name for the calendar year, which isn't unique to the Han Dynasty's "Taichu Calendar" or the Wei Dynasty's "Jingchu Calendar." We have completed the compilation of the new calendar and humbly ask the emperor to distribute it to the relevant departments for implementation according to the methods within it.
The emperor showed the new calendar to Qi Xianwu Wang Tian Cao, the military advisor Xindu Fang. Xindu Fang was an expert in astronomy, and after careful study, he submitted a report saying: "On December 20 of this year, the new calendar states that the planet Jupiter is at 12 degrees in the Mansion constellation, moving very quickly; however, the actual observed position of Jupiter is at 11 degrees in the Mansion constellation. On the same day, the new calendar indicates that the planet Saturn is at 11 degrees in the Horn constellation, stationary; yet the actual observed position of Saturn is at 4 degrees in the Well constellation, also stationary. On the same day, the new calendar states that Venus is at 25 degrees in the Dipper, appearing in the early morning and moving in retrograde; while the actual observed position of Venus is at 21 degrees in the Dipper, also in retrograde. There are discrepancies between the new calendar and the actual observations."
Xindu Fang completed his report.
As for Jupiter, I have been observing it for nearly eight or nine years, and its speed of movement always fails to match the speed predicted by the calendar. Now the new calendar has added two degrees; what does that mean? It sets in the evening and rises in the morning, perfectly on time! Today, when I looked up, I felt that its speed of movement indeed seems to have increased by two degrees as stated in the new calendar; when it rises and sets, it should also match the new calendar's predictions.
As for Saturn, since I formulated the calendar based on the Renzi year's calendar, its speed of movement has always fallen short of the predicted speed, so I added seven degrees to the Renzi year's calendar. However, I know this is still not enough; it probably needs about five more degrees. I considered adding these five degrees too, but I worry that doing so would mean adjusting it every ten days, which would create too many inconsistencies. Therefore, the speed of Saturn's movement can be a bit faster at times and a bit slower at others, with the ultimate goal being to align it with the results predicted by the calendar.
Around the 20th of last month, I saw Saturn in the east in the morning. I calculated its position using both the new and old methods and found almost no difference between the two. After three days of planetary movement, I adjusted it by four degrees. Situations like this occur every year, and when Saturn is hidden or appears, its position still needs to be calculated according to the calendar.
Oh, Rufang, you only feel something is off when you fixate on the star signs for December 20th. In fact, I've been studying and working on calendars for over thirty years, tracing back a thousand years, consulting numerous historical texts, and referencing the achievements of He Chengtian, Liu Jun, and Zu Chongzhi from the Zhao Dynasty and Liu Yilong's era in Liangzhou. I finally completed my "Jiazi Yuanli." This calendar is twice as precise as the other three!
I have been living and studying in Luoyang for forty years. During this time, I closely observed the trajectories of Venus, Jupiter, Mars, Saturn, and Mercury. My calendar predicts the appearance and disappearance times of these planets. Compared to the actual situation, the maximum error is just one or two days or one or two degrees. Mars has an unpredictable trajectory, sometimes not matching the calculated results. Zu Chongzhi's calendar overestimates by ten days and six degrees, while He Chengtian's calendar underestimates by twenty-nine days and twenty-nine degrees; whereas my calendar aligns closely with the results of the "Renzhi Calendar," there is not much difference. Mercury appears less often, but whenever it does, my calendar prediction is completely accurate, which matches the "Renzhi Calendar" exactly. Only the sunrise and sunset times of Venus and Mercury are slightly different in my calendar compared to other calendars.
I understand that the laws of celestial motion are complex and unpredictable, and observation and calculation are very difficult. The paths of the five planets are even harder to pin down. It is impossible to be completely accurate with the naked eye. Therefore, my calendar primarily focuses on the general rules of planetary motion, overlooking minor discrepancies to keep it practical. If we only focus on the phenomena we see in front of us and do not consider the rules of planetary appearance and disappearance, then the calendar can't be created.
You know, creating a calendar is not an easy task. It has to span thousands of years with precise calculations of solar terms and lunar phases, accurate to the tiniest fraction of a day. It must achieve a proper balance, ensure precise time intervals, and even the decimal points for each day must be precise. The results of the lunar and solar calendars must be exact, and the timing of solar and lunar eclipses needs to be validated against the calculations of the calendar. It involves complex calculations and parameters required to ensure accuracy, similar to the challenge of determining the Jiazi year. Even if the year of Jiazi is identified, there may still be slight errors, and various situations make it difficult to balance. My calendar, starting from the year of Jiazi, follows the same rules in seven calculation methods, like perfect pearls strung together without any errors, with clear and reliable methods.
As for what you said, there is indeed some truth. However, the inaccuracies in those astrological predictions will eventually return to the correct track. For example, according to my calculations, the error noted in Saturn on December 20th last year was five degrees, but today it is only three degrees; the error for Venus was four degrees last year, but today it has been corrected completely. Thus, those apparent errors will ultimately be confirmed, so my calendar can be reliably used over the long term without significant errors.
Fang also noted that according to the new calendar calculated last December, the Zhen star was positioned at 11 degrees in the Jiao Constellation, while its actual position in the sky was at 4 degrees in the Kang Constellation, leading to a five-degree discrepancy between the new calendar and the actual celestial positions; there were discrepancies in Venus and Saturn as well. Compared to the old calendar of "Ren Zi," the Zhen star was also off by five degrees, and there were discrepancies in Venus and Saturn as well, with the discrepancies in the old calendar being larger than those in the new one.
In summary, calendar makers must spend many years observing, applying methods to calculate celestial phenomena, and understanding the varying degrees of celestial movements in order to determine which calendar is more accurate and then use that to establish the calendar. It is impossible to determine the correctness of a calendar within a month or two. For example, Mars orbits every 779 days and has seven distinct states: delayed, fast, stationary, retrograde, direct, hidden, and visible, going through a complete cycle of these seven states; Venus has an orbital period of 583 days, with periodic changes in morning and evening visibility, completing a full cycle of these seven states; Jupiter has an orbital period of 398 days, completing a complete cycle of these seven states; Saturn has an orbital period of 378 days, completing a full cycle of these seven states; Mercury has an orbital period of 115 days, with periodic changes in morning and evening visibility, completing a complete cycle of these seven states. Those who formulate calendars must calculate these seven states to develop an accurate calendar. Calendars that successfully calculate these seven states are relatively accurate, while those that fail to do so have significant errors, and these cannot be clarified in just a day or two.
From the Five Emperors and Three Dynasties to the Qin, Han, Wei, and Jin, calendar makers underwent years of observation and calculation, and their calendars are worth referencing. Those hastily formulated calendars may have been relatively accurate at the time, but over time they become unreliable. If a calendar is formulated within three or four years, even if it initially aligns closely with celestial phenomena, deviations will emerge over time. The current "Jiazi" calendar was established after many years of dedicated research; while it does have some minor discrepancies, it is closer to celestial phenomena compared to the "Renzi Yuanli." If validated over the long term, after ten or twenty years, the discrepancies in the movements of the three celestial bodies will be less significant compared to the "Renzi Yuanli."
Fang again reported these situations to King Wu, who ordered the nationwide implementation.
From the first year of the Jiazi calendar to the first year of Duke Yin of Lu during the Spring and Autumn period (722 BC), the total is 292,736 years; this is an astronomical figure.
From the Jiazi year to the end of the Jiaxu era, the total is 124,136 years; this is an astronomical figure.
From the first year of the Jiazi calendar to the second year of Wei Xinghe (238 AD), the total is 293,997 years; this is an astronomical figure.
From the Jiazi year to the Jiaxu year to the Gengshen year, the total is 125,397 years; this is an astronomical figure.
Original method: 111,160 (representing the three systems)
Unified method: 337,200 (representing the two eras)
Wow, these are all astronomical figures! The first paragraph talks about the calendar, which is a total of 168,600 days. I'm not sure how that was calculated, but the book says "a thousand as a calendar, days up to ten," and I don't know what that means either. Next are the calendar method and degree method, both of which are 16,860, which is said to be derived from multiplying thirty by the chapter year. Then there is the day method, 208,530, said to be calculated by thirty multiplied by the chapter month. The qi time method is 1,405, obtained by the twelve-minute degree method. The chapter year is 562 years, the chapter leap is 207 months, the chapter months total 6,951, and the chapter months total 6,744; all of these are calculated based on how many days are in a year, how many leap months there are, and so on.
The numbers for Zhou Tian, total number, and no division are all the same: 615,817. The book explains the calculations in a very complicated way, something like "degree method through degrees, internal division numbers," "day method through twenty-nine days, internal month surplus numbers," "surplus numbers through internal sixty-nine, divided into five thousand seven hundred forty-four to get this number." Anyway, I just can't make sense of it.
The surplus number is 88,417, the 'no division' number is also 88,417, the calculated division is 4,117, and the virtual division is 97,883. These numbers are all tied to the calculations involving the Zhou Tian, the sun, and the moon, and are obtained through various complex algorithms. The small division method is 24, the year is twelve, the number is 173, the remainder is 67,117, the total is 36,142,807, and the virtual is 141,413. These numbers are all related to the twenty-four solar terms, the trajectory of the moon, and so on. Finally, Zhou Ri is 27, and Zhou Yu is 115,631; these are also calculated based on the laws of Zhou Tian and the sun and moon.
In short, these numbers are all related to ancient astronomical calendars. Looking at them makes my head hurt; as someone from the modern era, I really struggle to grasp these complex algorithms. However, I find these poems and couplets quite intriguing, but unfortunately, I’m not able to translate them.
Total number, 5,745,941; Zhou Xu, 92,899; Xiao Zhou, 7,513; Yue Zhou, 225,390; the total number of new and full moons, 14; the remaining value, 159,588.5; the number of intercalary months, 158 degrees; the remainder, 116,058.5. These numbers represent calculations in astronomical calendars, and I won't explain their specific meanings, as they are quite complex.
To calculate the moon's movement, one must first calculate the accumulated months. Take the number of years from a certain epoch to the year you want to calculate, subtract one from the total years, and then multiply by the number of days in each lunar month (counting the lunar year as equivalent to one year), which gives you the accumulated months. If the accumulated months are not an integer, the remainder is the leap remainder. If the leap remainder exceeds 355, that year is a leap year. If the leap remainder exceeds 515, the intercalary month will be around November, specifically determined by the winter solstice.
Next, calculate the accumulated days. Multiply the total by the number of accumulated months and then divide by the number of days in a day to get the accumulated days. If the accumulated days do not divide evenly, the remainder is the small remainder. Then divide the accumulated days by 60; the remainder is the large remainder. Record the large and small remainders, and once calculated, you can determine which day in November the new moon occurs for the year you are calculating.
To calculate the new moon of the next month, add 29 to the large remainder and 11,647 to the small remainder, then calculate according to the method above to get the new moon of the next month. If the small remainder exceeds 97,883, that month is a long month; if the small remainder is less than this number, it is a short month. To calculate the first quarter, full moon, and last quarter, add 7 to the large remainder of the new moon, 79,794 to the small remainder, and 1 to the small fraction. If the small fraction reaches 4, subtract it from the small remainder; if the large remainder reaches the number of days in the month, subtract it from the large remainder; if the large remainder reaches 60, subtract 60, and then you can calculate the first quarter day. Continue adding to calculate the full moon, last quarter, and the new moon of the next month.
To calculate the twenty-four solar terms of the traditional Chinese calendar, first calculate the accumulated periods. Take the number of years from a certain epoch to the year you want to calculate, subtract one year, and then multiply by a certain number (the exact value of which is unclear to me) to get the accumulated periods. The part that does not divide evenly is the small remainder; divide the accumulated periods by 60, and the remainder is the large remainder. Record the large and small remainders, and once calculated, you can determine the date of the winter solstice in November for the year you are calculating.
To calculate the next solar term, add 15 to the big remainder, 3684 to the small remainder, and 1 to the small fraction. When the small fraction reaches 24, subtract from the small remainder; when the small remainder reaches a certain number (which I also don't understand), subtract 1 from the big remainder. By following this method, you can determine the date of the next solar term.
First, let's discuss how to calculate a leap month. One method is: first subtract the leap remainder from the chapter year, then multiply the remaining value by 12 (since there are 12 months in a year). If the result is 2700 or more, it means there will be a leap month; if it exceeds half of 2700, it is also considered a leap month. Starting from the eleventh month of the Tianzheng calendar, the calculated result will identify the leap month. The month that lacks a middle solar term is the leap month.
Another method: multiply the yearly value by the leap remainder and add the chapter leap; if the result is 1 or more, it indicates that there will be a leap month. If it exceeds 6744, starting from the Winter Solstice, the month in which the middle solar term ends is considered the leap month. If the middle solar term occurs on the new moon day or within two days after, the previous month is designated as the leap month.
Next is the solar term table: Winter Solstice is in November, Minor Cold is in December; Major Cold is in December, Beginning of Spring is in January; Rain Water is in January, Awakening of Insects is in February; Spring Equinox is in February, Qingming is in March; Grain Rain is in March, Beginning of Summer is in April; Minor Fullness is in April, Grain in Ear is in May; Summer Solstice is in May, Minor Heat is in June; Major Heat is in June, Beginning of Autumn is in July; End of Heat is in July, White Dew is in August; Autumn Equinox is in August, Cold Dew is in September; Frost's Descent is in September, Beginning of Winter is in October; Minor Snow is in October, Major Snow is in November.
Finally, let's discuss how to calculate the distance to the new moon (the first day of the lunar calendar) and the intersection point (when the solar and lunar longitudes differ by 180 degrees). First, calculate the total number of new moons from a certain epoch to the present, then add the difference in intersection points from that epoch to the present (for example, the intersection point difference for the Jiaxu epoch is 26522649). Then subtract this total from a constant (called "Huitong"), and the remainder will represent the accumulated intersection. Divide by the solar method to get the degrees, and the remaining remainder will be the degree remainder. This degree and degree remainder represent the distance and remainder from the new moon of the eleventh month of the Tianzheng calendar to the intersection point.
Below are some data on several epochs for easier calculations:
Jiazi epoch: (Epoch start coincides with the new moon, solar and lunar align, intersection in the middle.)
Jiaxu Era: (The beginning of the era coincides with the new moon, and the moon is on the solar path.) The angle difference at the intersection is one hundred twenty-seven degrees, with a remainder of 39,349 degrees.
Jiashen Era: (The beginning of the era coincides with the new moon, and the moon is within the solar path.) The angle difference at the intersection is eighty-one degrees, with a remainder of 11,561 degrees.
Jiamu Era: (The beginning of the era coincides with the new moon, and the moon is within the solar path.) The angle difference at the intersection is thirty-four degrees, with a remainder of 192,313 degrees.
Jiachen Era: (The beginning of the era coincides with the new moon, and the moon is on the solar path.) The angle difference at the intersection is one hundred sixty-two degrees, with a remainder of 23,122 degrees.
Jiayin Era: (The beginning of the era coincides with the new moon, and the moon is on the solar path.) The angle difference at the intersection is one hundred fifteen degrees, with a remainder of 203,874 degrees.
This passage describes the ancient astronomical calendar calculation methods, explained in modern spoken Chinese as follows:
To calculate how many degrees until the new moon (the first day of the lunar month) next month reaches the intersection (where the solar and lunar longitudes differ by 180 degrees), the calculation method is as follows: first add 29 degrees, then add 11,647 degrees (this is the remaining degrees). If the total exceeds one day's degrees (solar calculation method), divide by one day's degrees, and the remainder will be how many degrees until next month's first day reaches the intersection, along with the remaining degrees.
To calculate how many degrees until the full moon (the fifteenth day of the lunar month) reaches the intersection, the method is similar: add 14 degrees, then add 159,588.5 degrees (this is the remaining degree), and then use the above method to divide by the solar calculation method; the remainder will indicate how many degrees until the full moon reaches the intersection, along with the remaining degrees.
To calculate the moon's position on the ecliptic, the method is: first sum the degrees from the last epoch (a longer cycle) to the new moon (first day), then add the intersection position difference within this epoch, and then divide by a value called "Huitong" (a cyclical value used in these calculations) multiplied by 2. The remaining degrees are divided by the "Huitong" value; if it divides evenly, the moon is on the outer ecliptic; if not, the moon is on the inner ecliptic.
To determine the moon's position next month, add next month's degrees and the remaining degrees to the full cycle and remaining cycle numbers of the outer ecliptic; if the result indicates the inner degrees, it will be in the inner ecliptic; if the result indicates the outer degrees, it will be in the outer ecliptic.
Calculate the day on which the conjunction point (when the solar and lunar longitudes differ by 180 degrees) occurs. The method is as follows: use the degrees from the new moon of November to the conjunction point and the remaining degrees, subtract the conjunction counts and any remaining counts. If the remaining counts are insufficient to subtract, then subtract one degree from the total degrees, and use the solar calendar method to remove it; the remainder represents the remaining degrees. Then add the small remainder from the new moon of November; if it exceeds the solar calendar method, use the solar calendar method to remove it, and what remains is how many degrees are left to the conjunction point before the new moon of November, as well as the remaining degrees. Based on the length of the lunar months, start counting from the 11th month of the celestial calendar (which may refer to November of a certain year); if it is not a full month, count it into the next month, thus calculating the day on which the conjunction point occurs. Next, multiply the remaining degrees of the year by the remaining degrees of the month; use the solar calendar method to remove it, and the result indicates the Zi hour (midnight), allowing one to calculate which hour the conjunction point occurs. If the conjunction point occurs before the full moon (the 15th), then the new moon (the 1st) is identified as the conjunction point, and a lunar eclipse occurs on the full moon (the 15th); if the conjunction point is after the full moon, then a lunar eclipse occurs on the full moon, and the new moon of the next month is the conjunction point; if the conjunction point is exactly on the full moon, then a lunar eclipse occurs on the full moon, and the new moons before and after are also conjunction points; if the conjunction point is exactly on the new moon, then a solar eclipse occurs on the new moon, and both the preceding and following full moons experience a lunar eclipse.
To calculate the month and date of the next conjunction point, the method is as follows: add the conjunction counts and any remaining counts to the month and date of the last conjunction point; if it exceeds the solar calendar method, use the solar calendar method to remove it, and what remains is the month and date of the next conjunction point. Continue this process.
To calculate the direction of the conjunction point, the method is as follows: if the moon is outside the zodiac, if the conjunction (when the solar and lunar longitudes are the same) occurs first followed by the conjunction point, then the loss, which refers to either a solar or lunar eclipse, begins from the southeast; if the conjunction point occurs first followed by the conjunction, then the loss begins from the southwest. If the moon is inside the zodiac, if the conjunction occurs first followed by the conjunction point, then the loss begins from the northwest; if the conjunction and conjunction point occur simultaneously, then the solar or lunar eclipse reaches its peak. If a lunar eclipse occurs during the opposition of a solar eclipse (when the solar and lunar longitudes differ by 180 degrees), the direction of the loss is the same.
First, let's discuss how to calculate solar eclipses. If the degree of the new moon (on the first and fifteenth days of the lunar calendar) relative to the nodal point (the intersection of the orbits of the sun and moon), plus the remainder of the degree, exceeds one hundred fifty-eight degrees and eleven thousand six hundred fifty-eight point five, then subtract this number from the conjunction number (the time of the new moon) and the remainder of the conjunction number; the remaining degree indicates where a solar eclipse will not occur. If the degree of the new moon from the nodal point, plus the remainder, is less than fourteen degrees and fifteen thousand nine hundred eighty-eight point five, then a solar eclipse will not occur. Subtract fifteen from all of these, and the remaining value is the degree of the solar eclipse. If the new moon is exactly at the nodal point, then it is the point of a solar eclipse.
Next is the algorithm for calculating the delay of the new moon (the first day of the lunar calendar) entering the calendar. First, calculate the cumulative total of new moons from a certain epoch to the present, then add in the delay difference since this epoch. For example, the delay difference for the Jiaxu epoch is two million three hundred fifty-two thousand one hundred ninety-one. Then divide this total by the cycle length to find the remaining days; if there is a remainder, divide by the degree of a day to get the days, and the remaining value is the remainder of the days. The calculated days represent the date of the new moon entering the calendar in the first month of the year you want to calculate.
To calculate the entry date of the next month, add one day to the base of the previous month, then add twenty-three thousand five hundred forty-six to the remainder. If the number of days is complete, divide by the degree of a day, remove the full weeks of days and the remainder, and calculate as above; the result is the entry date of the next month.
To calculate the entry of the full moon (the fifteenth day of the lunar calendar), add fourteen days to the date of the new moon, then add the remainder of fifteen thousand nine hundred eighty-eight point five. Treat the full weeks of days and the remainder as above, and the calculated date is the entry date of the full moon.
The following table shows the moon's movement speed (in degrees and minutes), gain rate, waxing and waning rate, and waxing integral:
| Moon's Delay Degree (and minutes) | Gain Rate | Waxing and Waning Rate | Waxing Integral |
|---|---|---|---|
| Fourteen degrees per day (four hundred two minutes) | Gain seven hundred fifty | Waxing | Twenty-one thousand one hundred eleven |
| Day 2, 14 degrees of arc (334 minutes) | Gain 689 | Surplus 757 | 41,135 |
| Day 3, 14 degrees of arc (261 minutes) | Gain 616 | Surplus 1,446 | 57,232 |
| Day 4, 14 degrees of arc (190 minutes) | Gain 545 | Surplus 2,672 | 72,360 |
| Day 5, 14 degrees of arc (111 minutes) | Gain 466 | Surplus 267 | 85,294 |
| Day 6, 13 degrees (522 minutes) | Gain 215 | Surplus 3,073 | 94,037 |
| Day 7, 13 degrees (29 minutes and 96 seconds) | Gain 89 | Surplus 3,388 | |
This text describes the calculation methods related to solar and lunar eclipses and lunar motion in ancient astronomical calendars, which are very specialized and complex. The table presents the calculated results of the moon's orbital speed and related astronomical parameters.
On Day 1, August 8, a total of 13 observations were made, with the sum of the values from each observation amounting to 68 minutes. The result was a loss of 139 minutes and a surplus of 3,477 minutes. The cumulative surplus total reached 96,507.
On Day 2, August 9, 12 observations were made, with the sum of the values from each observation amounting to 468 minutes. There was a loss of 283 minutes and a surplus of 3,338 minutes. The cumulative surplus total increased to 92,649.
From the third to the seventh day, the daily observation results are as follows: On the third day (the tenth observation), twelve observations totaling 379 points, losses of 390, gains of 3,055 points, and total surplus points of 84,794; on the fourth day, twelve observations totaling 267 points, losses of 52, gains of 2,665 points, and total surplus points of 73,969; on the fifth day, twelve observations totaling 151 points, losses of 618, gains of 2,163 points, and total surplus points of 63,036; on the sixth day, twelve observations totaling 40 points, losses of 729, gains of 1,545 points, and total surplus points of 42,883; on the seventh day, eleven observations totaling 515 points, losses of 816, gains of 816 points, and total surplus points of 22,649.
Starting on the fifteenth day, the situation began to change. On the fifteenth day, twelve observations totaling 38 points, an increase of 731 points, beginning to show a "shrinking" trend. On the sixteenth day, twelve observations totaling 123 points, an increase of 636 points, a decrease of 731. The "shrink" points are 22,290 points.
Over the next few days, the process of "contraction" continued: On the 17th, twelve observations were recorded, totaling 211 points, with an increase of 558 and a decrease of 1,377, resulting in a "contraction" score of 38,220; on the 18th, twelve observations were recorded, totaling 224 points, with an increase of 445 and a decrease of 1,935, resulting in a "contraction" score of 53,700; on the 19th, twelve observations were recorded, totaling 435 points, with an increase of 334 and a decrease of 2,380, resulting in a "contraction" score of 66,059; on the 20th, twelve observations were recorded, totaling 555 points, with an increase of 214 and a decrease of 2,714, resulting in a "contraction" score of 75,329; on the 21st, thirteen observations were recorded, totaling 128 points, with an increase of 79 and a decrease of 2,928, resulting in a "contraction" score of 81,269.
In the last two days: On the 22nd, twelve observations were recorded, totaling 270 points, with a loss of 63 and a decrease of 3,700, resulting in a "contraction" score of 83,463; on the 23rd, thirteen observations were recorded, totaling 432 points, with a loss of 225 and a decrease of 2,944. This concludes the record of observations.
On the first day of the new scoring system, the score was 81,713. On the 24th, there were fourteen observations, resulting in a total of 33 points, with a decrease of 388, resulting in a new score of 2,719.
On the second day, the score was 75,468. On the 25th, there were fourteen observations, resulting in a total of 194 points, with a decrease of 549, resulting in a new score of 2,331.
On the third day, the score was 64,699. On the 26th, there were fourteen observations, resulting in a total of 319 points, with a decrease of 674, resulting in a new score of 1,782.
On the fourth day, the score was 49,461. On the 27th, there were fourteen observations, resulting in a total of 346 points, with a decrease of 701, resulting in a new score of 1,108.
On the fifth day, the score stood at thirty-seven thousand seven hundred fifty-four. On Sunday, at fourteen degrees (three hundred seventy-nine minutes), it decreased by seven hundred thirty-four, reducing the score to four hundred twenty-seven.
On the sixth day, the score was eleven thousand two hundred ninety-seven. Next is the method for calculating the size of the new moon and lunar eclipse, as follows: To determine the size of the new moon and lunar eclipse, the method states: multiply the remaining days in the lunar calendar by the rate of gains and losses for the days entered, and divide by seven thousand five hundred thirteen; the resulting gains and losses will be the fixed points. If the score is high, reduce the original small remainder; if low, add to it. If the addition reaches a full day, the conjunction adds time on the following day; if reducing, if insufficient to reduce, subtract one day, and the addition method will then reduce it, with the conjunction adding time on the previous day. For the lunar eclipse, the fixed size of the eclipse will determine the additional time.
The method for calculating additional time is as follows: The technique for adding time states: multiply the remaining small size by the year, and divide by the daily method; the result is determined as follows. If the new moon adds excess that does not reach the end, divide by four; if the result is one, it is considered small, two is half, three is too much. If there is still a remainder in the half, divide by three; if the result is one, it is strong, and anything above half is considered one, while anything below half is discarded. Combine strong and small for small strong, combine half strong for half strong, and combine too much for too strong. If two strong are obtained, it is considered weak; combine it with small for half weak, with half for too weak, and with too much for one time weak. Name it according to the time of occurrence, indicating its relative strength and weakness. The sun's clash is a break, and the moon is eclipsed below the break.
Finally, the method for calculating the degrees of the sun and moon conjunction and opposition phases, the fifth part, the method for calculating the solar degree, is as follows: The technique for calculating solar degrees states: since the beginning of the calendar, accumulate the days of the new moon, multiply by sixteen thousand eight hundred sixty using the solar degree method, subtract the full week, and the remainder should be approximated using the solar degree method to determine the degree; the remainder indicates the position of the new moon on the twelfth degree before the ox, excluding the next star; if it does not reach the star, it is counted outside, which gives the desired year, the precise eleventh month, and the position of the new moon for half a day in degrees and minutes.
Let's first calculate the position of the sun. The method is as follows: there are a total of 365 degrees in a complete solar year, and there are 4,117 subdivisions. Subtract one from the number of days from the winter solstice to the new moon, then subtract this number from the total degrees, and finally subtract the remaining degrees from the subdivisions; if the subdivisions are insufficient, subtract one degree from the total and then use the solar method to subtract again. After calculating using the above method, you will end up with the position of the sun at midnight on the new moon day in November of that year, expressed in degrees and subdivisions.
Next, let's calculate the positions of the sun and moon. To calculate the position of the sun on a certain day, the method is: add 30 degrees for a large month, 29 degrees for a small month, and then add one degree for each day, and then divide by the lunar subdivisions, with the remaining subdivisions divided by the solar subdivisions.
To calculate the conjunction degrees of the sun and moon at the new moon, the method is as follows: multiply the zhang year (562) by the remainder of the new moon day, then divide by the zhang month (6,951); the result is the large subdivision, and the remainder is the small subdivision. Add the small subdivision to the degrees of the sun at midnight; if the subdivisions exceed the solar method, carry over from the total degrees. After completing the calculations using the above method, you will have the conjunction degrees of the sun and moon for the new moon in November of that year.
There is another method to calculate the conjunction degrees of the sun and moon at the new moon: add 29 degrees, with the large subdivision being 8,945 and the small subdivision being 6,919. If the small subdivision reaches a full zhang month, carry over from the large subdivision; if the large subdivision reaches the full solar method, carry over from the degrees, divide by the lunar subdivisions, and divide the remaining subdivisions by the solar subdivisions. After completing the calculations, you will have the conjunction degrees of the sun and moon at the new moon for the next month.
Now let's calculate the position of the moon. The method is: take the total number of days since the epoch to the new moon, multiply it by the total number of lunar days in a cycle (225,390), and then subtract the integer multiples of the lunar days. Simplify the remaining number using the solar method to obtain degrees and subdivisions, then starting from the twelve degrees preceding the Ox constellation, divide by the lunar subdivisions; if there isn't enough for one lunar subdivision, it can be ignored, and the remainder will be the position of the moon at midnight on the new moon day in November of that year.
There is another method to calculate the position of the moon: use the small lunar cycle multiplied by the remainder of the new moon day as the numerator, and the zhang year multiplied by the solar method as the denominator; divide the numerator by the denominator to get the degrees; if it doesn't divide evenly, divide the zhang month by the remainder to get the large and small subdivisions. Subtract the obtained result from the degrees and subdivisions at the new moon, and the result will be the position of the moon at midnight on the new moon day in November of that year.
To calculate the position of the moon next month, here's how to do it: add 22 degrees and 7373 minutes for the new moon; add 35 degrees and 13583 minutes for the full moon. If the minutes exceed the daily degree calculation, carry over from the degrees, then divide by the mansion degrees, and ignore any remainder less than one mansion. This gives you the position of the moon for next month.
Finally, to calculate the position of the moon on a specific day next month, you add 13 degrees and 6210 minutes. If the minutes exceed the daily degree calculation, carry over from the degrees, then divide by the method above. The result is the position of the moon on that day next month.
First, let's look at how to calculate the position of the waxing crescent moon. You add 7 degrees to the degree of the new moon (first day of the lunar month), then add 6451 minutes, 3461 seconds, and 2 milliseconds. If the milliseconds reach 4, subtract from the seconds; if the seconds exceed a full month, subtract from the minutes; if the minutes reach the daily degree, subtract from the degrees. By following this method, the remaining value gives you the position of the waxing crescent moon. The same method can be used to calculate the positions of the full moon, waning crescent moon, and the first day of the next month.
Next, to calculate the positions of the waxing crescent and full moons, you add 98 degrees, 11695 minutes, 5225 seconds, and 1 millisecond to the degree of the new moon. Subtract as per the method above when reaching full; the remaining value gives you the position of the waxing crescent and full moons. Continue adding to calculate the positions of the full moon, waning crescent moon, and the first day of the next month.
Then, the degrees of the twenty-eight mansions are: Dipper mansion 26 degrees, Ox mansion 8 degrees, Girl mansion 12 degrees, Void mansion 10 degrees; Danger mansion 17 degrees, Room mansion 16 degrees, Wall mansion 9 degrees. The seven mansions of the northern Xuanwu total 98 degrees (equivalent to 4117 minutes). Leg mansion 16 degrees, Lou mansion 12 degrees, Stomach mansion 14 degrees, Pleiades mansion 11 degrees; Big mansion 16 degrees, Beak mansion 2 degrees, Canopus mansion 9 degrees. The seven mansions of the western White Tiger sum to 80 degrees. Well mansion 33 degrees, Ghost mansion 4 degrees, Willow mansion 15 degrees, Star mansion 7 degrees; Net mansion 18 degrees, Wing mansion 18 degrees, Dipper mansion 17 degrees. The seven mansions of the southern Vermilion Bird total 112 degrees. Horn mansion 12 degrees, Neck mansion 9 degrees, Root mansion 15 degrees, Room mansion 5 degrees; Heart mansion 5 degrees, Tail mansion 18 degrees, Winnowing basket mansion 11 degrees. The seven mansions of the eastern Azure Dragon sum to 75 degrees.
The entire celestial sphere is 365 degrees, equivalent to 16860 minutes, where 4117 minutes correspond to the celestial sphere. Adding these up, we get 6158117, which represents the total degrees in the celestial sphere.
The following is the method for calculating the position of the Earth King Star (possibly referring to a specific celestial phenomenon or solar term) before the four solar terms: the Beginning of Spring, the Beginning of Summer, the Beginning of Autumn, and the Beginning of Winter. First, subtract 18, 4420, 18, and 2 from the major remainder, minor remainder, small fraction, and milliseconds of the four solar terms respectively. If the major remainder is insufficient, add 60 before subtracting; if the minor remainder is not enough, subtract one day, then add the corresponding value before subtracting; if the small fraction is not enough, subtract one minor remainder, add 24, and then subtract; if the milliseconds are not enough, subtract one millisecond, add 5, and then subtract. The remaining value after calculations will be the position of the Earth King Star before the four solar terms.
Another method is to calculate the position of the Earth King Star on the Winter Solstice. The method is: add 27 to the major remainder, 6631 to the minor remainder, 6 to the small fraction, and 3 to the milliseconds. If the milliseconds reach 5, subtract from the small fraction; if the small fraction reaches its limit, subtract from the minor remainder; if the minor remainder is full, subtract 1 from the major remainder. The remaining value after calculations will be the position of the Earth King Star on the Winter Solstice.
First, let's calculate when the next Earth King Day will be. Add 91 to the major remainder, 5244 to the minor remainder, and 6 to the small fraction. If the small fraction reaches its limit, subtract from the minor remainder; if the minor remainder is full, subtract from the major remainder; if the major remainder is full, subtract 60. Note the result; this will indicate the next Earth King Day.
Next, calculate how many days after the Winter Solstice is the "Mie" Day. If no small remainder has accumulated since the Winter Solstice, add 1, then multiply the small fraction by this number, and divide by 88417 to get the accumulated days; the remainder will be the small remainder. Subtract 6 xun (60 days) from the accumulated days, and the remainder will be the "Mie" Day. Record the result; this will be the "Mie" Day after the Winter Solstice.
Then calculate when the next "Mie" Day will be. Add 69 to the "Mie" Day and 57244 to the minor remainder. If the minor remainder is full, subtract from the "Mie" Day; if the "Mie" Day is full, subtract 60. Record the result; this will be the next "Mie" Day. If there is no remainder, then it is the "Mie" Day.
Next, calculate when the next "Mei" Day will be. Add 69 to the "Mei" Day, 1915 to the minor remainder, and 62285 to the small fraction. If the small fraction reaches its limit, subtract from the minor remainder; if the minor remainder is full, subtract from the "Mei" Day; if the "Mei" Day is full, subtract 60. Starting from the previous month, subtract the number of days in each month; if it is less than a month, the remaining amount will be the next "Mei" Day, along with the minor remainder and small fraction. Record the result; this will be the next "Mei" Day.
Now, let's explore the four hexagrams: the Winter Solstice corresponds to the "Kan" hexagram, the Spring Equinox to "Zhen," the Summer Solstice to "Li," and the Autumn Equinox to "Dui." The "Zhongfu" hexagram follows right after "Kan."
Calculate the next hexagram. Add 6 to the large surplus of the "Kan" hexagram, 1473 to the small surplus, 14 to the small remainder, and 4 to the micro remainder. When the micro remainder is full, subtract from the small remainder; when the small remainder is full, subtract from the small surplus; when the small surplus is full, subtract from the large surplus; when the large surplus is full, subtract 60. Note the result; this marks the beginning of the "Fu" hexagram.
November: "Wei Ji," "Jian," "Yi," "Zhong Fu," "Fu"
December: "Tun," "Qian," "Kui," "Sheng," "Lin"
January: "Xiao Guo," "Meng," "Yi," "Jian," "Tai"
February: "Xu," "Sui," "Jin," "Xie," "Da Zhuang"
March: "Yu," "Lun," "Gu," "Ge," "Jue"
April: "Lu," "Shi," "Bi," "Xiao Xu," "Qian"
May: "Da You," "Jia Ren," "Jing," "Xian," "Gou"
June: "Ding," "Feng," "Huan," "Lv," "Dun"
July: "Heng," "Jie," "Tong Ren," "Sun," "Pi"
In August, the corresponding hexagrams are "Xun," "Cui," "Da Xu," "Bi," "Guan." September is "Gui Mei," "Wu Wang," "Ming Yi," "Kun," "Bo." October is "Gen," "Ji Ji," "Shi Ke," "Da Guo," "Kun."
These hexagrams correspond to ancient official titles as follows: The four hexagrams for January correspond to Fang Bo (local officials in ancient times), "Zhong Fu" corresponds to San Gong (the highest-ranking officials of ancient times), "Fu" corresponds to the Son of Heaven, "Tun" corresponds to feudal lords, "Qian" corresponds to nobles, "Kui" corresponds to the nine ministers, and "Sheng" is also related to San Gong, repeating in a cycle.
The relationship between the third line and the top line: if the top line is a Yang line, it corresponds to "clear, slightly warm, and Yang wind"; if it is a Yin line, it corresponds to "descending red, decisive warmth, and Yin rain." The relationship between the sixth line and the top sixth line is the same: if the top sixth line is a Yang line, it corresponds to "sunshine, cold, and Yin rain"; if it is a Yin line, it corresponds to "curved dust, decisive cold, and Yang wind." In summary, if the hexagram above has a Yang line, it corresponds to Yang wind; if it has a Yin line, it corresponds to Yin rain.
The following is the method for calculating the seventy-two solar terms, which is a bit complex: First, based on the remainder of the day length on the winter solstice, known as the "big and small remainder," calculate the day when the tiger begins mating. Then, you need to add certain specific values to the big and small remainders and perform a series of calculations, such as adding five, or one thousand two hundred twenty-eight, or one for the infinitesimal, and so forth. This continues with the small remainder and the big remainder, calculating the date for each solar term.
Specifically for each solar term, the seventy-two solar terms are described as follows: The winter solstice is "the tiger begins mating, the herb begins to sprout, the lychee buds emerge"; the minor cold is "earthworms begin to coil, deer antlers shed, springs stir"; the major cold is "geese head north, magpies begin nesting, pheasants start to call"; the beginning of spring is "chickens begin to lay eggs, the east wind thaws, hibernating insects begin to stir"; the rainwater period is "fish do not break the ice, otters sacrifice fish, wild geese arrive"; the awakening of insects is "the beginning of rainwater, peach blossoms begin to bloom, the cuckoo sings"; the spring equinox is "eagles transform into doves, the black bird arrives, thunder begins to sound"; the Qingming festival is "lightning begins to appear, hibernating insects stir, hibernating insects open their doors"; the grain rain period is "paulownia begins to flower, field mice transform into quails, rainbows begin to appear"; the beginning of summer is "duckweed begins to grow, the crested ibis descends onto mulberry trees, crickets chirp"; the minor fullness is "earthworms emerge, king gourds grow, bitter herbs flourish"; the grain in ear is "the grass dies, the minor heat arrives, mantises are born"; the summer solstice is "the cuckoo begins to sing, the cuckoo’s call fades, deer antlers shed"; the minor heat is "cicadas begin to sing, half summer arrives, hibiscus flourishes"; the major heat is "warm winds arrive, crickets dwell in walls, eagles begin to learn"; the beginning of autumn is "decaying grass transforms into fireflies, the earth moistens in sultry summer, cool winds arrive"; the end of summer is "white dew falls, cold cicadas sing, eagles sacrifice birds"; and white dew is "heaven and earth begin to cool, storms arrive, wild geese come."
During the autumn equinox, swallows fly back to the south, birds begin to store food for the winter, and the sounds of thunder gradually fade away.
At the cold dew solar term, hibernating insects hide in cracks, the cold intensifies, and the yang energy diminishes with each passing day.
During the frost descent season, the water starts to dry up, and the wild geese fly in flocks. If sparrows fall into the water, they turn into toads. (This sentence is a bit exaggerated; it is an ancient description of natural phenomena.)
At the beginning of winter, yellow chrysanthemums bloom, wolves start to hunt their prey, and the river water begins to freeze.
In light snow, the ground starts to freeze, and if wild chickens fall into the water, they turn into mythical creatures, and rainbows are no longer visible.
In heavy snow, the ice layer becomes thicker, the ground starts to crack, and crows have stopped cawing.
Next is the method for calculating the Shangshuo date in traditional Chinese calendars: subtract one from the total number of years from the beginning of the era to the desired year, then multiply by six, and divide by six cycles of the Chinese calendar. The remainder is represented by the Jiazi, and after calculation, this gives you the Shangshuo date.
Next is the fifth method for calculating the trajectories of the five planets (Wood, Metal, Water, Fire, Earth): from the year of the Upper Yuan Jiazi (2697 BC) to the first year of Duke Yin of Lu in the Spring and Autumn period (770 BC), this amounts to a total of 292,736 calculation units. From the year of the Upper Yuan Jiazi to the second year of Wei Xinghe (222 AD), there are a total of 293,997 calculation units.
Jupiter (the Year Star) has a value of 6,723,888; Mars (the Red Planet) has a value of 13,149,083; Saturn (the Settling Star) has a value of 6,374,061; Venus (the Bright Star) has a value of 9,843,882; Mercury (the Star of the Dragon) has a value of 1,953,717, respectively.
The method for calculating the movement of the five planets is as follows: subtract one from the total number of years from the Shangyuan Festival to the target year, then multiply by the number of weeks to obtain the actual values of the five planets; then divide each planet's value accordingly, the quotient is the accumulated count, and the remainder represents the accumulated excess. Subtract the accumulated excess from the calculated value, and the remainder represents the entry into the annual degrees and minutes. Using the daily degree method simplifies the calculation to yield the accumulated degrees and remainder for the morning and evening after the winter solstice of the target year. For Venus and Mercury, use one accumulated day's number and subtract the accumulated excess to calculate the accumulated degrees and the remainder, indicating that one corresponds to the morning and the absence of one indicates the evening; if the remainder is insufficient to subtract, then subtract the accumulated degrees for one, add the daily degree method, and then subtract. Starting from the twelve degrees before the Ox constellation, calculate using the constellation sequence; if it does not complete one constellation, consider it as an external value, which gives the accumulated degrees and the remainder for the morning and evening after the winter solstice of the year in question.
This passage describes an ancient astronomical calculation method, which seems quite complex, so let's break it down sentence by sentence.
First paragraph: It states that to calculate the position of celestial bodies on the morning and evening of the winter solstice of a certain year, first subtract 1 from the number of days from the Shangyuan Festival (an ancient calendar epoch) to the year you want to calculate, and then calculate according to a certain method. If the calculation result exceeds the degrees of one day, subtract the degrees of one day and add 1. If the final result still exceeds the degrees of one day, subtract the degrees of one day again, then add a value called "Zhouxu." The final number obtained is the degrees and remainder of the celestial bodies' movement in the morning and evening after the winter solstice of that year. The calculation method for Venus and Mercury is the same.
Second paragraph: This section discusses how to calculate the time when the stars, moon, and sun align on the same day. First, calculate the number of days from the winter solstice to the new moon (the first day of the lunar calendar), subtract 1, then add the previously calculated accumulated degrees (the number obtained in the previous paragraph). Then, add the winter solstice remainder to the accumulated degree remainder; if it exceeds the degrees of one day, subtract the degrees of one day and add 1. At this point, 'accumulated degrees' is redefined as 'accumulated daily degrees,' and the 'accumulated degree remainder' becomes 'daily remainder.' Starting from the eleventh month of the lunar calendar, subtract according to the number of days in each month; if it does not complete a full month, it can be disregarded, and what remains is the time when the stars, moon, and sun align. If there is a leap month, then calculate based on the number of days in the leap month.
Paragraph three: This paragraph explains how to calculate the time of the next alignment of the stars, moon, and sun. Combine the days and remainder from the last alignment with the months and remainder of the upcoming alignment. If the total exceeds one day, subtract one day and then add 1. Then subtract according to the number of days in each month, starting from the last alignment time to calculate the time of the next alignment. The calculation method for Venus and Mercury requires adding their daily degrees and remainder, adding to the evening if they meet in the morning, and adding to the morning if they meet in the evening.
Paragraph four: This paragraph explains how to calculate the degrees of the next celestial body's movement. Add the remainder of the celestial body to the degrees and remainder from the last movement. If the total exceeds one day, subtract one day. Calculate from the last movement position, subtract based on the degrees of the constellation, and if it’s less than a full constellation degree, ignore it, and the remaining value will be the next movement’s remainder. Then divide this remainder by a number (4117) to get a fraction.
Paragraph five and six: These two paragraphs provide detailed data about Jupiter, including the total days, remainder, speed, direction of daily movement, and the time and position of its alignment with the sun in different scenarios. This data is highly specific and is used for practical calculations. While these numbers may seem complex, they are the results of long-term observations and summaries by ancient astronomers, reflecting their understanding of the laws of celestial motion.
The situation of Mars' movement is as follows: It aligns perfectly with the sun a total of 779 times; the remaining days after overlapping with the sun total 15143 days; it takes 1717 days to complete one week of movement; it travels 49 degrees, leaving a remainder of 699 degrees.
Next is another orbital period of Mars: Mars conjuncts with the sun in the morning, then hides behind the sun, lasting for 71 days, with 16,001 days remaining, moving 55 degrees, with 13,943 degrees remaining. Then, in the morning, Mars can be seen in the east, moving forward at a fast speed, moving 13/14 of a degree each day, traversing 112 degrees over 184 days. The speed then slows down, moving 1/12 of a degree each day, covering 48 degrees in 92 days before stopping. After stopping for 11 days, it starts moving retrograde, moving 1/17 of a degree each day, moving 6 degrees in 102 days. It stops again for 11 days, then starts moving forward again, at a slower speed, moving 1/12 of a degree each day, covering 48 degrees in 92 days. The speed increases again, moving 13/14 of a degree each day, traversing 112 degrees over 184 days. At this point, it is in front of the sun, hiding in the west in the evening, moving forward, lasting for 71 days, with 16,002 days remaining, moving 55 degrees, with 13,943 degrees remaining, and finally conjuncts with the sun again.
The orbital situation of Venus is as follows: It conjuncts with the sun a total of 378 times; the total remaining days after each conjunction with the sun is 981 days; the time it takes to complete one orbit is 15,879 days; it moves 12 degrees; and the remaining degrees are 13,724 degrees.
Another orbital period of Venus: Venus conjuncts with the sun in the morning, then hides behind the sun, lasting for 18 days, with 490 days remaining, moving 2 degrees, with 6,862 degrees remaining. Then, in the morning, Venus can be seen in the east, moving forward at 1/12 of a degree each day, covering 7 degrees in 84 days before stopping. After stopping for 36 days, it starts moving retrograde, moving 1/17 of a degree each day, moving 6 degrees in 102 days. It stops again for 36 days, then starts moving forward again, moving 1/12 of a degree each day, covering 7 degrees in 84 days. At this point, it is in front of the sun, hiding in the west in the evening, moving forward, lasting for 18 days, with 491 days remaining, moving 2 degrees, with 6,862 degrees remaining, and finally conjuncts with the sun again.
The orbital situation of Mercury is as follows: It conjuncts with the sun a total of 583 times; the total remaining days after each conjunction with the sun is 14,502 days; the time it takes to complete one orbit is 2,358 days; it moves 291 degrees; and the remaining degrees are 15,681 degrees. (This also represents the number of days for one conjunction with the sun.)
Lao Bai (Tai Bai Xing) said: When it appears together with the sun at night, it is in front of the sun, remaining hidden for 41 days, approximately 15,681 time segments, moving 51 degrees, also approximately 15,681 time segments. When seen in the west at night, it moves ahead rapidly, traveling one degree and thirteen minutes daily, covering 112 degrees in 91 days. Then its speed decreases to one degree and thirteen minutes per day, covering 115 degrees in 91 days. Then the speed increases again, traveling one degree and fifteen minutes per day, covering 33 degrees in 45 days and then stopping. After an eight-day pause, it resumes movement. This time it reverses direction, moving three-quarters of a degree per day, retreating six degrees in nine days. In front of the sun, it remains concealed in the west at night, hiding for six days, retreating four degrees, and then appearing together with the sun the next morning.
Next, Lao Bai said: When it appears together with the sun in the morning, it is behind the sun, remaining hidden for six days, retreating four degrees. When seen in the east in the morning, it reverses, moving three-quarters of a degree per day, retreating six degrees in nine days and then stopping. After stopping for eight days, it starts moving forward, traveling one degree and fifteen minutes per day, covering 33 degrees in 45 days. Its speed increases, traveling one degree and thirteen minutes per day, covering 115 degrees in 91 days. Its speed increases again, traveling one degree and thirteen minutes per day, covering 112 degrees in 91 days. Behind the sun, it hides in the east in the morning, moving forward, remaining hidden for 41 days, approximately 15,681 time segments, moving 51 degrees, also approximately 15,681 time segments, and then appearing together with the sun at night.
Next is the Chen star: It appears alongside the sun for a total of 115 days, approximately 14,818 time segments, completing one revolution in 2,044 days and moving 57 degrees, also totaling 115 days. Approximately 15,838 time segments.
Chénxīng said: When it appears with the sun at night, it is in front of the sun, hiding for seventeen days, approximately fifteen thousand eight hundred and forty-eight ke. At night, it can be seen in the west, moving quickly, traveling one degree and one-third in a day, covering twenty-four degrees in eighteen days. Then the speed slows down, moving five-sevenths of a degree in a day, covering five degrees in seven days before remaining stationary for four days. In front of the sun, it hides in the west at night and moves backward, retreating six degrees in eleven days, then appears again with the sun the next morning.
Finally, Chénxīng said: When it appears with the sun in the morning, it is behind the sun, hiding for eleven days, retreating six degrees. It can be seen in the east in the morning, then remains stationary for four days. After that, it begins to move forward at a slow pace, traveling five-sevenths of a degree in a day, covering five degrees in seven days. Its speed picks up, moving one degree and one-third in a day, covering twenty-four degrees in eighteen days. Behind the sun, it hides in the east in the morning, moves forward, hiding for seventeen days, approximately fifteen thousand eight hundred and thirty-eight ke, traveling thirty-four degrees, approximately fifteen thousand eight hundred and forty-eight ke, then appears again with the sun at night.
Let’s first talk about the steps of this Five-Star Calendar, which states: Using a specific method to calculate both the degrees traveled and the remaining degrees of the planet's daily movement, if the remaining degrees reach a full day (sixteen thousand eight hundred and sixty), it is equivalent to completing a circle, and then starting the calculation from the beginning. This way, one can determine the visible and remaining degrees of the planet on any given day. Then, multiply the denominator of the planet's movement by the visible degrees and the fractional part, treating the full day as 1, to obtain the fractional part. If the fractional part is less than 0.5, it is discarded; if it is greater than or equal to 0.5, it is rounded up by adding 1 and then added to the already calculated movement fraction. If the fractional part exceeds the denominator, it carries over and counts as an additional degree. The denominators for direct and retrograde motion are different and need to be calculated based on the current denominator of the planet's movement to determine the previous fraction, treating the current denominator as 1 to calculate the current movement fraction. If the planet is in a stationary (stopping) state, it inherits the previous degrees; if it is in retrograde, it must be subtracted. If the calculated degrees are less than a full degree, it is divided by the unit of measure, using the denominator of the planet's movement as the ratio, where the fractional part can increase or decrease, influencing each other in the process.
Next, let's discuss how to calculate the specific position of the five stars: multiply the orbital factors of the planet by the number of days it has been moving, then divide by the appropriate denominator, and the result is the degree of the planet's position on that day.
