The Origin of the Four Residues in the Seven Luminary and Four Residue Astrology

In ancient Chinese astronomy and divination, the 'Four Residues' were mostly related to the Sun and Moon on the celestial sphere, but after their introduction from India, their definitions gradually changed. Rahu (Rahu) and Ketu (Ketu) in the Four Residues are phonetic transliterations from Sanskrit. Legend has it that Rahu was originally a dragon leading various demons. He once united with the gods to fight against the demon Ananta. However, just as they were celebrating their victory, he took advantage of everyone's unpreparedness and secretly drank the nectar of immortality. Unexpectedly, his crime was seen by the Sun and the Moon, who reported it to the gods. Thus, the god Vishnu rushed over and fought hard to cut off Rahu's head. However, at this moment, the nectar had already taken effect in Rahu's body, allowing him to become eternal like a celestial body. Since then, Rahu's head and his body—Ketu—became the uncompromising enemies of the Sun and the Moon. Whenever the environment permitted, they attempted to swallow the Sun and the Moon, causing solar and lunar eclipses. Ketu's tail sometimes also appeared before the world in the form of a comet.

In China, the name Rahu is generally believed to have first appeared in the 6th year of Kaiyuan (718) in the translation of the 'Jiuzhi Li' (Nine Stems Calendar) by Qutan Xida, who was commissioned by the emperor. The so-called 'Nine Stems' refer to the 'Nine Luminaries', which indicate the Seven Luminary (Sun, Moon, and five planets) and the two hidden luminaries Rahu and Ketu. In the 8th year of Kaiyuan, Nangong Shuo memorialized that in his compiled 'Jiuyao Zhanshu' (Nine Luminaries Divination Book), it was necessary to measure and calibrate stellar phenomena, so he requested the creation of a celestial globe, which Emperor Xuanzong approved. Nangong Shuo was a famous astronomer of his time; later, he worked on the 'Yisi Yuanli' (Yisi Original Calendar) and served as the Deputy Director of the Directorate of Astronomy. From the title of the 'Jiuyao Zhanshu', it can be known that the ecliptic longitude of Rahu and Ketu and their astrological significance should have gradually been valued by contemporary Chinese astronomers. Another famous astronomer, the monk Yixing (683-727), is said to have extensively introduced Buddhist methods of divination using Rahu and Ketu to predict good and bad fortune. For example, in the 'Xuzangjing' (Continued Canon), which includes works attributed to Yixing such as 'Fantian Huoluo Jiuyao' (Brahma Fire Rahu Nine Luminaries), 'Beidou Qixing Huma Fa' (Big Dipper Seven Stars Homa Ritual), and 'Chishengguang Yao Fa' (Essential Ritual of Blazing Light), it is detailed how to draw shapes in secret places to offer worship to these two luminaries to avert disasters.

The 'Qiyao Rangzai Jue' (Seven Luminary Disaster-Averting Decisions), compiled by the Brahmin monk Jin Juzha from the Western Tianzhu Kingdom, was introduced and translated into China in the early 9th century. This book proposed quite specific definitions for the two luminaries Rahu and Ketu, stating:

Rahu, Eluoshi, also known as Huangfan, also known as the Head of the Eclipse God, also known as Fu, also known as the Head of the Sun. It often moves invisibly and is not seen. When it encounters the Sun or Moon, it causes an eclipse. If it encounters them during the new or full moon, there must be an eclipse; if it is opposite the Sun or Moon, there is also an eclipse... When it enters the person's natal palace, there will be disasters, or it may hide and block, causing the heaviest misfortune. It often moves retrograde in the sky, with no variation in speed. It moves one degree in nineteen days, one degree and six-tenths in a month, nineteen degrees and one-third in a year, once in a year and a half, and completes a cycle of the sky minus eleven degrees and two-thirds in eighteen years, finally completing a major cycle in ninety-three years and starting again.

From the description 'when it encounters the Sun or Moon, it causes an eclipse', we can infer that this book should have regarded Rahu as one of the two points where the ecliptic and the lunar orbit intersect.

Due to the bulging equatorial part of the Earth, it generates a perturbation on the Moon's orbit, leading to the rotation of the line connecting the nodes of the ecliptic and lunar orbit. The direction is opposite to the direction of the Moon's revolution around the Earth. This is what modern astronomy calls the Regression of the Nodes, and its period is the so-called Nutation Period, currently measured at approximately 18.61 years. This phenomenon of the westward regression of the ecliptic-lunar nodes was discovered as early as the late Eastern Han Dynasty by Liu Hong. Liu Hong believed that this point regressed at a uniform rate of 27683/1488 (= 18.60417) days per degree, so it took 18.60 years. The 'one year moves nineteen degrees and one-third' mentioned in the 'Qiyao Rangzai Jue' corresponds to a cycle of 18.62 years, which is quite close to the nutation period, further confirming that Rahu is indeed one of the ecliptic-lunar nodes. Furthermore, based on the inference of Rahu's ecliptic longitude recorded in the book after the first year of Yuanhe (806), it was defined at that time that Rahu's location was the ascending node among the two ecliptic-lunar intersections.

In the 'Jiuzhi Li', Rahu is also called 'Axiu'. This calendar takes the first day of the second month of the 2nd year of Xianqing (657) as the beginning of the calendar. It records that the period of Axiu's motion is 6794 days (approximately 18.60 years) and states that at the beginning of the calendar, Axiu was located at 'five phases, twenty-four degrees, and forty minutes'. Since one phase is 30 degrees, this is approximately 174.7 degrees of ecliptic longitude. After calculating the position of the ascending node of the lunar orbit using modern astronomical knowledge, it was found that its mean longitude at the beginning of the calendar should be 175.9 degrees. Obviously, this calendar also regarded Rahu as the ascending node of the lunar orbit. Moreover, the Indian system calendar used by the ancient people of Yunnan province in China also defined Rahu as the ascending node of the lunar orbit.

As for the nature and motion of Ketu, the 'Qiyao Rangzai Jue' states: Ketu, Eluoshi, also known as Leopard Tail, also known as the Tail of the Eclipse God, also known as Yueboli, also known as the Head of the Moon. It often moves invisibly and is not seen. When it enters the person's natal palace, there will be disasters, or it may hide and block, causing the heaviest misfortune. It often moves direct in the sky, with no variation in speed. It moves one degree in nine days, three degrees and four-tenths in a month, once in nine months, forty degrees and seven-tenths in a year, and completes a cycle of the sky minus six degrees and three-tenths in nine years, and completes seven cycles of the sky minus three degrees and four-tenths in sixty-two years.

From the phrase 'often moves direct in the sky', it is known that the Ketu defined at that time should be unrelated to the ecliptic-lunar nodes. And the motion of Ketu completing a cycle of the sky every approximately nine years is very similar to the apsidal motion of the Moon's perigee or apogee. After comparing the ecliptic longitude of Ketu recorded in the 'Qiyao Rangzai Jue' for the 62 years after the first year of Yuanhe, it was found that Ketu should refer to the apogee.

This theory that Rahu is the ascending node of the lunar orbit and Ketu is the Moon's apogee was also introduced to Japan by monks studying abroad. There are currently two early celestial charts in Japan that used the positions of the Nine Luminaries for astrological divination, dated to 1113 and 1268 AD respectively. The ecliptic longitudes of Rahu and Ketu were calculated in a manner similar to the 'Qiyao Rangzai Jue'.

In the late 8th century, the practitioner Cao Shiwei, who created the 'Futian Li' (Futian Calendar) for popular use, is said to have written a book called 'Luoji Er Yin Yao Licheng Li' (Established Calendar for the Two Hidden Luminaries Rahu and Ketu). The 'Song Shi - Yiwenzhi' (History of Song - Treatise on Literature) also lists related titles such as 'Futian Jiuxing Suanfa' (Futian Nine Stars Calculation Method) and 'Futian Jiuyao Tongyuan Licheng Fa' (Futian Nine Luminaries Tongyuan Established Method). During the reign of Emperor Shizhou of Later Zhou (954-959), Wang Pu even stated in his memorial presenting the 'Qintian Li' (Qintian Calendar) that the Nine Luminaries had long become the 'constant formula for annotating calendars'. From these evidences, it can be known that the two luminaries Rahu and Ketu should have been quite valued by official and民间 calendar experts in the late Tang and Five Dynasties periods.

As for the two luminaries Yuebei (Moon's Apogee) and Ziqi (Purple Qi) in the Four Residues, their introduction to the Central Plains was slightly later. According to Xing Yunlu of the Ming Dynasty, the Four Residues all originated from the 'Dulai Yusi Jing' (Dulai Yusi Sutra). In the early Zhenyuan period (785-804), the Indian calendar techniques for calculating the ecliptic longitudes of eleven stars (referring to the Four Residues and Seven Luminary, also known as the Eleven Luminaries) from this book were completely translated by the Brahmin scholar Li Bigan. However, the method for calculating Yuebei had been introduced earlier, over a hundred years ago, by Li Chunfeng (602-670). The 'Dulai Yusi Jing' mentioned here may have been lost; it should be an alternative name for the 'Duli Yusi Jing'. According to the records in the 'Xin Tangshu - Yiwenzhi' (New Book of Tang - Treatise on Literature), the latter consisted of two volumes, transmitted by the Duli (referring to Duli water in Kangju city in the Western Regions) scholar Li Mikan from Western Tianzhu during the Zhenyuan period, and translated into Chinese by a person named Qu Gong. This Li Mikan should be the same person as Li Bigan mentioned by Xing Yunlu.

Whether Xing Yunlu's statement about the introduction of Yuebei to the Central Plains has exact evidence is unknown. After checking books attributed to Li Chunfeng such as 'Yisi Zhan' (Yisi Divination), 'Qiankun Bianyi Lu' (Record of Changes in Heaven and Earth), and 'Yuli Tongjing Jing' (Yuli Tongjing Sutra), or the 'Linde Li' (Linde Calendar) compiled by Li Chunfeng, there is no mention of the Four Residues. Moreover, in other calendar, mathematical, or divination literature from the 6th and 7th centuries, there is also no mention of Yuebei and Ziqi. Therefore, a more conservative view is that the two luminaries Yuebei and Ziqi in the Four Residues might have been introduced to the Central Plains at the same time as Li Mikan.

The theory of the Four Residues may have initially circulated only among the people, used for divination of good and bad fortune. This is why titles like 'Wuxing Liuyao Mianbu Jue' (Five Stars Six Luminaries Face Reading Decisions) can be seen in the 'Song Shi - Yiwenzhi'. In the calendars used by the government, the technique of calculating the Four Residues did not officially appear until the Ming Dynasty's 'Datong Li' (Great Unity Calendar). It is said that in the 'Daming Li' (Great Brightness Calendar) revised by Jin Zhaowei, there was also a method for calculating the Four Residues, but it was lost when the calendar was created in the early Yuan Dynasty. However, the Jesuit missionary Luo Yagu (Jacques Rho; 1590-1638) at the end of the Ming Dynasty refuted this claim, stating:

The Jin and Yuan dynasties are not far apart. The early Yuan dynasty originally inherited the Jin calendar. How could it be lost so suddenly? Thus, Zhao Zhiwei was so sloppy and vulgar. His techniques and the 'Zhuan Shen Li' (Transforming Spirit Calendar) are vulgar and absurd, likely shunned by Yelü Chucai, Wang Xun, Guo Shoujing, and others. What is there to recount!

The Four Residues introduced from India began to change their definitions at the latest in the 11th century. For example, in Shen Kuo's (1031-1095) 'Mengxi Bitan' (Dream Pool Essays) of the Northern Song Dynasty, when discussing the causes of solar and lunar eclipses, he mentioned Rahu and Ketu:

The ecliptic and the lunar orbit are like two rings overlapping with a slight difference. Whenever the Sun and Moon meet at the same degree, the Sun is eclipsed; when they are opposite by one degree, the Moon is diminished. Although they are at the same degree, if the lunar orbit and the ecliptic are not close, they do not interfere with each other; if they are at the same degree and also close to the intersection of the ecliptic and lunar orbit, the Sun and Moon meet and eclipse each other. When they are exactly at the intersection, the eclipse is total; if not fully at the intersection, it depends on the degree of their crossing... The intersection regresses by more than one degree every month. In total, after 249 intersections, one period is completed. Thus, the Western Heaven method calculates Rahu and Ketu in retrograde motion, which is the current intersection. The beginning of the intersection is called 'Rahu', and the middle of the intersection is called 'Ketu'.

Here, Shen Kuo directly points to the two luminaries Rahu and Ketu as the two intersection points of the ecliptic and lunar orbits. As for the number '249' in the quoted text, Dr. Joseph Needham believed it should be a mistake for '349'. He interpreted 'in total, after 349 intersections, one period is completed' as referring to the length of the eclipse year being 349 days, but this explanation seems far-fetched, because the original text's 'in total, after 249 intersections, one period is completed' refers to the phenomenon of the 'intersection (ecliptic-lunar node)' regressing at a rate of more than one degree per month, and returning to its original position after passing through 249 nodal months, which has nothing to do with the eclipse year.

In the book 'Gexiang Xinshu' (New Book on Changing Phenomena) written by the Yuan Dynasty folk scholar Zhao Youqin, the Four Residues were explicitly defined, such as defining Rahu as the descending node (the point where the moon passes from north of the ecliptic plane to south of the ecliptic plane, anciently named Initial Intersection, Regular Intersection, Heavenly Head, or Yang Calendar Mouth), and Ketu as the ascending node (the point where the moon passes from south of the ecliptic plane to north of the ecliptic plane, anciently named Middle Intersection, Heavenly Tail, or Yin Calendar Mouth). The text states:

The Moon's path does not follow the ecliptic, nor does it follow the celestial equator; it enters and exits the inside and outside of the ecliptic. To the north is the Purple Forbidden Enclosure, where the Emperor's Seat resides, so north is 'inside' and south is 'outside'... The intersection of the lunar orbit and the ecliptic is at the beginning of the two intersections, strongly named 'Rahu'; at the middle of the intersection, strongly named 'Ketu'. From the initial intersection to the middle intersection, the Moon is outside the ecliptic, named 'Yang Calendar', which is the place facing away from Rahu towards Ketu; from the middle intersection to the initial intersection, the Moon is inside the ecliptic, named 'Yin Calendar', which is the place facing away from Ketu towards Rahu.

The same book also details the motion of Yuebei and Ziqi, stating:

Yuebei is sought from the Moon's fullness and shrinkage. One full cycle of fullness and shrinkage covers 27 days, 55 minutes, and 46 seconds. The Moon travels 368 degrees, 37 minutes, and 4.5 seconds. Yuebei travels 3 degrees, 11 minutes, and 40.5 seconds. Combining the degrees of the ecliptic cycle with the number of degrees Yuebei travels gives the position of the Moon. Approximately every 62 years, it completes seven cycles of the sky, coinciding with the place where the Moon is slowest. Ziqi originates from the leap method. Approximately every 28 years, it completes a cycle of the sky. The 'Shoushi Li' (Time-Sharing Calendar) takes 10 days, 87 minutes, 53 seconds, 84 hundredths, and 84 ten-thousandths as the leap of the year. Ziqi travels 13 degrees, 5 minutes, 4 seconds, 60 hundredths, and 80 ten-thousandths in one year. Comparing the two numbers, it is an addition of two. In 28 years with ten leaps, Ziqi travels through the twelve palaces, also an addition of two.

If expressed in modern astronomical terms, this book defines Yuebei as the apogee of the Moon's elliptical orbit around the Earth (the place where the Moon is slowest). Obviously, at this time, Yuebei in the Chinese Four Residues had replaced the originally introduced Ketu from India.

As for the single luminary Ziqi, calendar experts mostly believed it was related to leap residues. Because there are approximately ten leaps in 28 years, Ziqi was defined as traveling one degree every 28 days, i.e., completing a cycle of the sky every 28 years. Since Ziqi is the only one among the Four Residues without a specific corresponding astronomical meaning, later scholars proposed other forced explanations. For example, You Yi, a mathematician and astronomer in the early Qing Dynasty, although proficient in Western methods, could not completely break away from tradition and discard Ziqi. He stated: 'Ziqi originates from leap residues. It may be the place where the Earth and Jupiter meet, or the ancients used it to record the direct year. Thus, the leap residue of the month generates a cycle of Qi, all of which are accumulated calculations for eclipses.' Another mathematician and astronomer in the early Qing Dynasty, Wang Xichan (1628-1682), who attempted to integrate Chinese and Western knowledge, tried to give Ziqi a more astronomical meaning, stating:

The Four Residues all arise from the separation of the Sun and Moon, and there are no actual stellar phenomena to point to. Some say Qi is the Moon's radiance and Bei is a comet, which is absurd! In recent times, some argued that there is no Ziqi in the sky, analyzing it in detail with more than ten points. Although the argument was sound, Qi is lost due to leap residues, and it is not without cause. But with the passage of time and the accumulation of numbers, if one does not explore the root and source, it is not easy to modify. Moreover, combining Qi and the New Moon creates a leap, so having Qi implies having the New Moon. Practitioners discard the New Moon and keep Qi, losing the thread of logic, thus becoming tangled and unmanageable. I previously deduced the solar New Moon as the root for modifying Ziqi, but as a commoner without the authority to create, I dared not lightly publish it.

Wang Xichan also listed contents such as 'Siyu Licheng' (Established Calendar for Four Residues), 'Siyu Jiaogong' (Four Residues Entering Palaces), and 'Siyu Jiefa Licheng' (Quick Method Established Calendar for Four Residues) in his book 'Lifa Biao' (Calendar Table), using traditional uniform motion to replace small variations.

Although traditional Chinese views regarded the Four Residues as invisible stars, they believed that under special circumstances, they could also be measured like the Seven Luminary. For example, it was said that Rahu and Ketu, by blocking the light of the Sun and Moon to form eclipse phenomena, could be measured during eclipses. And when 'the King's virtue reaches Heaven', Ziqi might also be visible, which is the so-called Jingxing (Scenery Star) or Dexing (Virtue Star), shaped like a half-moon, usually appearing on the last or first day of the month, 'assisting the Moon's brightness'. Furthermore, when monsters appear, the ancients believed that Yuebei might also be visible. Its star belongs to comets, but its light radiates in all directions, different from ordinary comets whose light sweeps to one side. Its appearance signifies disaster, such as the records in the 'Spring and Autumn Annals' like 'Bei appears in the Great Chen' or 'Bei enters the Big Dipper'.

In his book 'Gujin Luli Kao' (Examination of Ancient and Modern Calendars and Astronomy), Xing Yunlu specifically discussed how to verify whether the cycle of the sky of Ziqi (or Yuebei) was correct based on the position and time of two appearances. The text states: Qi is born from ten leaps, Yue is born from the Moon's slowness. This was an ancient saying, but both are hidden luminaries. Bei stars occasionally appear in historical records, so one should take an ancient Bei, noting its constellation degree, day, and time of appearance, and compare it with a current Bei, noting its constellation degree, day, and time of appearance. Using the accumulated years, months, days, and time, and dividing by the cycle of Yuebei, one might obtain the result. As for Ziqi, those seen in ancient times are few. One must also wait for its two appearances before and after, and calculate it using the method for Yuebei, which will also work. However, since it is rarely seen, one can only establish a method to wait for its appearance and measure it.

But because Xing Yunlu considered the two luminaries Ziqi and Yuebei extremely rare, his theory remained at the theoretical level and was never put into empirical practice.