According to the ancient calendar, the sun shadow moves once a day, 360 degrees a year. How is this angle calculated?

What is a calendar? Simply put, it is a long-term punctuality system created by people for the needs of social production practice. Specifically, it is the arrangement of the year, month and day. The unit of measurement of time, like the units of measurement such as length and weight, is artificially specified. However, people's practice tells us that it will bring great convenience for people to measure time by using the changing law of natural phenomena closely related to production practice as the scale of natural measurement time. In this way, the tropical year, the new moon and the solar day, which reflect the seasonal variation, constitute three time measurement units with appropriate sizes. The calendar in which these three units of measurement are used together is called yin-yang calendar (such as lunar calendar); The solar calendar (such as the current Gregorian calendar) only considers the change of the tropic year; The lunar calendar is called the lunar calendar, and twelve lunar months are fixed as a year.

The origin of ancient calendars in China is very early. If there is primitive agriculture and animal husbandry, there should be primitive calendar. The early calendar now only leaves some legends, so it is difficult to study it in depth. The written calendar started from the weekend to the ancient calendar in the early Han Dynasty. After many calendar changes, it has been constantly improved and improved in the reform and struggle, reaching a fairly high scientific level and achieving success one after another. Most ancient calendars in China used the traditional lunar calendar, but it not only included the arrangement of years, months and days, but also included the calculation of the five-star positions of the sun, the moon and the moon, the forecast of solar eclipse, the arrangement of solar terms and so on. Calendar reform includes the proposal of new theories, the determination of accurate astronomical data, the improvement of calculation methods and so on. The achievements of ancient calendars in China occupy a very important position in the history of world astronomy. Here are just a few main aspects.

Study on the apparent motion of the sun

Because the earth's rotation axis is not perpendicular to its orbital plane, the angle between the ecliptic and the equator is about 23.5 degrees, which makes the solar horizon in the same area change regularly, forming a cold and warm climate in this area. Therefore, the study of the apparent motion of the sun is directly related to the formulation of the calendar. There are two methods to study the apparent motion of the sun: one is to measure the change of the length of the noon shadow to determine the seasonal change and determine the length of the tropical year (called "the year of the year" in ancient times). The instrument used in this observation is a standard watch. One is to determine the position of the sun in the stars, study the change of the sun's movement speed in a year, and determine the annual change value from winter solstice (called "precession"). The instruments used are angle measuring instruments such as armillary sphere. There are three main points here:

Determination of winter solstice time and calculation of tropical annual length

Before the Warring States Period, the winter solstice and the summer solstice were called "the southern solstice" and "the northern solstice", indicating that the winter solstice is the day when China and Japan are in the lowest position in the southern sky and the sun shadow is the longest, while the summer solstice is the day when the sun is in the highest position and the sun shadow is the shortest. Because the shadow of winter solstice is long and that of summer solstice is short, the measurement result of winter solstice is more accurate than that of summer solstice.

Only when the time of winter solstice is accurately measured can the season be accurately predicted; With a few accurate winter solstice moments, we can get a year's value. Therefore, it is an important task for ancient calendar workers in China to determine the exact time of winter solstice. The earliest observation records of the winter solstice in China were in the five years (655 BC) and twenty years (522 BC) of the Spring and Autumn Period.

Theoretically, the length of tropical year can be obtained by measuring two adjacent winter solstice moments. However, because the date of winter solstice determined by simply observing the change of sun shadow with soil gauge is not very accurate, there may be an error of one or two days; On the other hand, every time we arrive at the winter solstice, it's not just at noon. It is impossible to get the exact time of the winter solstice in 4 1 day by simply observing with the map. In order to make up for this shortcoming, the ancients tried to use the observation records of winter solstice separated by many years to reduce the error caused by observation error in calculating the length of tropical year.

At the end of the Spring and Autumn Period (5th century BC), China began to use the ancient calendar, and its actual year was 365.25, which was the most accurate value used in the world at that time. The Galileo calendar in Greece is equivalent to the quarter calendar in ancient China, but it is about 100 years later than in China. The ancient four-year calendar stipulated that there were seven leap months in 19 years, that is, there were exactly 235 moons in 19 tropic years, so one moon was equal to two 953085 days, which was quite accurate. The invention of the ancient calendar is a great contribution with world significance.

With the progress of society and the development of science, people put forward more and more accurate requirements for calendars. After the ancient calendar was used for a period of time, people found that the calendar gradually fell behind the actual astronomical phenomena. In order to avoid this phenomenon, we must use a new calendar to correct this mistake. Taichu calendar in the Western Han Dynasty and Four-year Calendar in the Eastern Han Dynasty corrected these mistakes by changing yuan. However, people gradually understand that continuous reform cannot fundamentally solve the problem. At the end of the Eastern Han Dynasty, Liu Hong realized that the error was caused by excessive calendar years, so he reduced it for the first time, thus improving the accuracy of the calendar.

In order to further improve the accuracy of the calendar, we must improve the measurement method of winter solstice time. After the Western Han Dynasty, people have become accustomed to using eight-foot meters to determine the date of winter solstice. However, the ideal results can not be obtained simply by using the eight-foot altimeter. So people have thought of many ways to improve it. During the Southern and Northern Dynasties, Zu Chongzhi (429-500) first improved the observation technology to improve the observation accuracy. Because the change of shadow length around the winter solstice is not obvious, it is difficult to accurately determine the time of winter solstice. More importantly, simple measurement can only get the date of winter solstice, but can't get when it is winter solstice in a day. Zu Chongzhi came up with a new method. He did not directly observe the length of the shadow on the day of the winter solstice, but observed the length of the shadow on the 234th day before and after the winter solstice, and then averaged them to find out the date and time of the winter solstice. Moreover, the farther away from the winter solstice, the faster the shadow changes, so this method improves the accuracy of determining the winter solstice time. The date of Zu Chongzhi's Da Li Ming is 365 years and 2428 years, which was very accurate at that time. Only after the Southern Song Dynasty can it reach or surpass his level. In the Song Dynasty, Houtianli adopted more observation points, which increased the interval between the two winter solstice moments, thus reducing the error of the tropical year and improving the accuracy.

Guo Shoujing in Yuan Dynasty was a scientist who made great contributions to mathematics, astronomy and instrument manufacturing. He not only inherited the previous method of observing the sun shadow, but also created an instrument called "mirror volt", which solved the problem of blurred boundary of the sun shadow. The principle of landscape symbol is to let the sunlight shining on the surface of the gauge first pass through a small hole, and then hit the surface of the gauge to form a light spot the size of a grain of rice, and its edge is very clear, so that the shadow length can be accurately measured. It stands to reason that lengthening the table height can make the change of the sun shadow more obvious, but after the table height is increased, the boundary blur is more serious, and the purpose of improving the accuracy cannot be achieved. By solving the problem of the clarity of the sun's shadow edge, the height of the watch can be increased to increase the accuracy of measuring the winter solstice time. The existing observation deck in Dengfeng, Henan Province is a model of a huge masonry structure built by Guo Shoujing. This watch is four feet high, which is five times the traditional height. In the Southern Song Dynasty, Yang Zhongfu carefully analyzed and studied the historical observation data, and used the exact chronology value of 365-2425 for the first time in his Unified Calendar, which was compiled in the fifth year of Qingyuan in Song Ningzong (A.D. 1 199). Guo Shoujing confirmed that 365-2425 is the most accurate value used in the history of China based on his own accurate determination of the winter solstice time for many times and the observation data of the six winter solstice times since Zu Chongzhi's Da Li Ming. The use of this value is the earliest in the history of the world. The famous European calendar "Gregorian calendar" also uses this value, but it is about 400 years later than the "unified calendar". At the end of the Ming Dynasty, Xing Yunlu increased the table height to 60 feet, and the measured age was 365.243.2 19 days, which was only 0.00027 days smaller than the value calculated by modern theory, and the accuracy exceeded the level of European astronomy at that time.

Determination of winter solstice and precession

The winter solstice refers to the position of the sun in the star during the winter solstice, which is expressed by right ascension and right latitude in modern times. In ancient China, it was expressed by the difference of right ascension (called the degree of entering the star) and the extreme degree of leaving the star at 28 nights.

The winter solstice measured by the ancient four-year calendar in the Warring States period began in Petunia. Beginning of spring, the Zhuan Xu calendar used by the Qin Dynasty, was in Wudu Camp. According to ancient calculations, the position of the sun's winter solstice is also in the morning glory. This should be the earliest measured data of winter solstice in China.

Because the position of the sun cannot be measured directly, it can only be inferred indirectly. In the early days, the method of determining the winter solstice was to use the given winter solstice at that time to get the midnight time through missing, and then get the position of the sun from the transit degree of the stars at that time. It is difficult to accurately determine the time of missing engraving by this method, so the measured value is also rough.

Before the Jin Dynasty, Chinese astronomers did not know precession. There is no difference between a day and a week. It is considered that from the solstice in winter to the solstice in winter, one year (year week) is the day when the sun moves among the stars (day week). Therefore, when the quarter calendar sets the length of the tropic year to 365 1/4 days, it also divides Sunday into 365 1/4 degrees. They believe that once the location of the winter solstice is determined, it will never change, so most of the information they use is that from the Warring States to the Western Han Dynasty, the winter solstice is at the beginning of the morning glory. Taichu calendar, written in the first year of Emperor Wu of the Western Han Dynasty (BC 104), holds that the sun makes stars on the winter solstice, but it has not changed. In the second year of He Sui in the Western Han Dynasty (7 BC), Liu Xin said that the position of the winter solstice was four degrees and five minutes before the ox, which was ambiguous. It was not until the second year of Zhang Yuanhe in the Eastern Han Dynasty (AD 85) that Jia Kui made it clear that the sun fought twenty-one and a quarter on the winter solstice. People in Han dynasty didn't know the precession phenomenon, but through actual observation, it has clearly reflected that the position of winter solstice is changing. At the end of the Northern and Southern Dynasties, Jiang Ji came up with an ingenious method, that is, measuring the position of the moon during the eclipse, thus calculating the position of the sun, and then calculating the position of the sun from winter to the sun. This method is quite accurate. He measured the position of the winter solstice at 17 degrees.

The earth is an ellipsoid. Because the axis of rotation is inclined to the ecliptic plane, the protruding part at the equator of the earth is attracted by the sun and the moon, which causes the axis of the earth to move slowly around the ecliptic for about 26,000 years, resulting in precession. This change is slow. According to the degrees used in ancient China, the annual westward shift of the winter solstice on the equator is about 77 years.

When Jin proclaimed himself emperor (around 330 A.D.), compared with the observation results of winter in history (about 284 A.D.-about 360 A.D.), it was clearly put forward for the first time that winter moves slowly, and it was recognized that the sun's movement on the celestial sphere for one week does not mean that winter is one year old, but should be "one day is one day, and one year is one year old". From the solstice of winter to the solstice of the next winter, the sun did not return to its original position in the stars. According to Yu's own analysis results, he thinks that the winter solstice in 50 years moved westward along the equator by 1 degree. Although his discovery was about 450 years later than that of the Iba Valley in ancient Greece, it was more accurate than that of the Iba Valley, which moved westward along the ecliptic every hundred years from the winter solstice (calculated according to the ancient system of China, it was equivalent to moving westward along the equator from the winter solstice 109 years or more). Shortly after Xi, precession was applied to the calculation of calendar.

He Chengtian also discussed the precession phenomenon. He thinks that the equatorial precession is once every one hundred years, but it is not used in his calendar. Zu Chongzhi was the first person to use it to improve the calendar. He measured that the winter solstice was 15 degrees, which was two degrees short of Jiang's 17 degrees in less than a hundred years, so it was one degree short of forty-five years and eleven months. Although the data obtained by Zu Chongzhi is not very accurate, it is a great innovation that he quoted precession in the calendar. Liu Zhuo of Sui Dynasty used the precession of the ecliptic in his calendar with a difference of 75 years, which was quite accurate at that time, but at that time, western countries still used the value of the Eba Valley. After the Tang and Song Dynasties, the calculation of equatorial time difference was more accurate. For example, Zhou Yan's calendar for tomorrow, Huang Juqing's calendar for observing the sky and Chen Deyi's calendar for Tong Yuan all adopted values with a difference of more than 457 years, reaching a high level.

Calm qi and stable qi.

The earth moves around the sun along an ellipse with a small eccentricity, and the actual distance traveled every day is unequal, but because of the small eccentricity, there is little difference. In ancient times, limited by observation instruments, I didn't know that the apparent motion of the sun was uneven before the Northern and Southern Dynasties, and thought that the sun walked at the same angle every day. At that time, I didn't know there was precession, so the sun walked once a week, 365 and a quarter days a year, and a week was 365 and a quarter degrees, and the sun walked once a day. Each solar term occupies the same number of days, which is 15.2 days. Later generations called this method of defining solar terms flat solar terms or constant solar terms.

Zhang Zi's credit armillary sphere in the Northern Qi Dynasty was measured on the island for many years, only to find that the apparent motion of the sun was uneven, and it was found that "the daily trip was later after the vernal equinox and faster after the autumnal equinox" (Sui Shu Tian Wenzhi). In fact, the winter solstice at that time was not far from the perigee of the ecliptic, and the perigee was only more than ten degrees before the winter solstice, which was roughly in line with the actual situation. With the passage of time, the winter solstice and perigee are getting closer and closer, and they coincide around the Southern Song Dynasty.

Zhang Zixin's discovery is of great significance to the improvement of the calendar, and will soon be applied to the calendar. Liu Zhuo and others began to suggest that solar terms should be determined by dividing the days of the week into 24 equal parts in the calendar, which would cost more than 15 degrees per breath. This method of dividing solar terms is called fixed breathing. Because the sun moves at different speeds on the ecliptic every day, it takes different days for the sun to move in one breath. Around the winter, the days passed quickly, only 14.7 18 days at a time; From summer to around, the days passed slowly, reaching 15.732 days in one breath. He separated the vernal equinox from the autumnal equinox, making the winter solstice more than 88 days and the summer solstice more than 93 days. However, the numerical value of the sun's speed given by him is inconsistent with the reality. Da Yan Li written by monks and their followers is more in line with the actual situation. They think that the day near the winter solstice is the fastest, so the time between the two gases is the shortest, while the day near the summer solstice is the slowest, so the time between the two gases is the longest. Da Yan Li pointed out the correct law of daily travel speed and corrected Liu Zhuo's misunderstanding. In fact, the fastest day of the solar calendar is between the heavy snow and the winter solstice, when the actual perigee is nine degrees before the winter solstice, so the data of the solar calendar is accurate. "Dayan Calendar" also calculates that from the winter solstice to the vernal equinox, there are six fixed air chambers * * * August and August, and the daily line is a quadrant; From the vernal equinox to the summer solstice, six fixed air periods * * * September 1 day and July 3rd are also a quadrant. The situation is the same before and after the autumnal equinox and before and after the vernal equinox.

Guo Shoujing's "Timing Calendar" defines the fastest one-day tour as the winter solstice. The perigee of the creation time of Shoushi Calendar is actually less than one degree from winter to the future, so the numerical value it uses is very accurate. According to the actual measurement, the Calendar of Time Service shows that from the winter solstice to the first three days of the vernal equinox (vernal equinox), the journey of a quadrant only needs 9 1 August; From three days before the vernal equinox to the summer solstice of 1 July 19931day, the sun is also a quadrant; It is the same before and after the autumnal equinox.

From this, we can see that there has been a more accurate method to calculate the position of the sun and the steady air since the Da Yan Li. However, due to the habit of using flat gas for a long time, flat gas is still used to arrange solar terms, and the concept of fixed gas is only used in calculating the daily travel degree and intersection time. It was not until the Qing Dynasty that the gas was changed to fixed gas.

The study of the movement of the moon.

In the ancient calendar of China, the moon's profit and loss was regarded as the unit to record the moon, and the observation and research on the moon's movement had been paid attention to for a long time. In ancient China, special attention was paid to the forecast of solar and lunar eclipses, so the study of lunar tours was more serious. In the ancient calendar at the end of the Spring and Autumn Period, the length of Wang Shuo's moon (called "Shuo Juan" in ancient times) has been mastered quite accurately, which is about 300 years less than the true value. In the calendar before the Sui Dynasty, the calendar days of each month were always calculated according to the length of the new moon. The first day of each month is called "the new moon", which means that the combination of the sun and the moon will take place on the first day of the first month. Because the length of Wang Shuo's moon is slightly longer than that of February 5, it is usually adjusted as follows: the big moon is on the 30 th, the small moon is on the 29 th, and the big and small moons appear alternately, with an interval of about 17 months.

Because the position of the sun on the celestial sphere is also moving, the new moon does not mean that the moon goes around the sky. China can distinguish these two concepts very early. "Astronomical Training of Huai Forty-seven Nanzi" records that there are daily trips, monthly trips 13 degrees 7/ 19 (1) and weekly trips of 27.32 19 days, and establishes the concept of "stellar moon" (the number of days required for the moon to move from a fixed position on the celestial sphere to return to its original position is called constancy.

The earth moves elliptically in its orbit, and the moon moves elliptically in its own orbit, so the speed of the moon changes periodically. The moon moves fastest when passing through perigee and slowest when passing through apogee. The number of days required for the moon to move from the fastest time to the fastest time is called "near moon". It is not equal to the length of the moon, which makes the time required for the moon to be full or short actually unequal. Therefore, the first month of the lunar calendar is only the average number of days required for a lunar phase change cycle. According to the length of the new moon, the time of the new moon is called Pingshuo.

The stone gods in the Warring States period may have known the change of the speed of the moon, but the record is very short. Liu Xiang of the Western Han Dynasty (77 BC-6 BC) recorded nine ways of traveling to the moon in Hong Fan's Biography of Five Elements. Jia Kui in the Eastern Han Dynasty also achieved the speed of traveling to the moon. He thinks that the speed of the moon is caused by the distance between the paths of the moon, and he knows that after one month, the perigee advances three times. Based on this calculation, it will not return to its original place until September 18, when the perigee is February 27, 550 1 year. Zhang Heng also advocates using nine methods. In ancient literature, nine pictures of the moon were also recorded, which shows that the theory of nine moons was very popular in Han Dynasty. Nine Techniques is China's early understanding of apogee change. According to the nine-skill calendar, there will be three big months and two small months in succession. Although the nine methods are rough, they are more accurate than the Pingshuo method, which does not consider the speed of the moon.

Liu Hong included the speed of the moon in the calendar for the first time. He assumes that the perigee of each perigee month advances by three degrees and four minutes (19 is divided into one degree), from which it can be concluded that the perigee month is 275336, which is not far from the current measured 275455. The degree of the moon in recent months is measured in the dry calendar, and the "profit and loss rate" table of the moon's daily speed exceeding or not reaching the average speed is given. The "profit and loss rates" add up one after another and are called "profit and loss products". To find the realization degree of the moon on a certain day, use the parallel value of the moon plus the surplus and contraction product from perigee to the previous day. The dry calendar uses an interpolation method to find the time when the sun and the moon meet. The problem of calculating the speed of the moon in the Dry Calendar is mainly to calculate the time and position of the solar eclipse, so not only the longitude when the sun shines, but also the time when the solar eclipse occurs can be calculated.

In order to study the need of copulation, the ancients also did a lot of research on the length of "intersecting moon". The number of days required for the moon to run once from the ascending (descending) intersection of yellow and white eclipses and then return to the ascending (descending) intersection is called the intersection month. Zu Chongzhi's "Da Li Ming" first deduced the value of the lunar intersection as 2721223, which is only one in 100,000 compared with the current measured value. In the future, almost all calendars will calculate the length of intersecting months with high accuracy.

Zhang Zixin's discovery that the sun moves fast and slow provided a good condition for Ding Shuo's further study. Starting from Liu Zhuoxuan and Zhang in Sui Dynasty, it is a great progress in the legal history of China to consider the disunity of the sun and the moon when calculating the date of the new moon in the calendar. When calculating the new moon, Liu Zhuo founded the quadratic difference interpolation formula with equal spacing, and introduced the advanced achievements of ancient mathematics in China into the calendar. I don't know the reference/thread-12192-1-1.html.