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Encyclopedia of India–China Cultural Contacts v1.c6.s1: A. Mathematics and Astronomy

Encyclopedia of India–China Cultural Contacts v1.c6.s1

Language
English

Contents

A. Mathematics and Astronomy

Ancient Indian Mathematics and Astronomy

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A kind of knowledge hierarchy which was applied to observing and describing astronomical phenomena, as well as to giving explanations and predictions via mathematical methods was developed in ancient India. According to the opinions of D. Pingree, Indian ancient astronomy can be divided into five periods orderly as Vedic Period (about 1000-400 BCE), Babylonian Period (about 400-200 BCE), Hellenistic Babylonian Period (about 200-400 CE), Greek Period (from 400-1600 CE) and Islamic Period (from 1600-1800 CE).

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The astronomical data of the Vedic Period is mainly preserved in Vedas such as Rig-Veda, thus constituting the most indigenous astronomy in India. During the Babylonian Period, a large

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Stone inscription in Gwalior, India from 9th century CE

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number of Mesopotamian astronomical contents were blended in Indian astronomy, including the method of describing day length changes, other parameters, mathematical models, time units, instruments etc derived from Babylon. Then, the astronomy of Babylonian Seleucid Period which was adapted by Greeks, was introduced into the Hellenistic Babylonian Period including the description of the planetary configuration, the geometric calculations etc concerned to eclipses, shadow length etc and meanwhile some contents were preserved in Pañcasiddhāntikā completed around 575 CE. Successively, a lot of astronomical talents came out in the Greek Period that the Indian astronomy lasted for more than one thousand years, many classic works were created and Pakṣa was also formed. Therein, Brāhmapakṣa was first established (400 CE) and lasted for the longest time (more than

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Ancient text books on Mathematics

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1,500 years) thus having a far-reaching influence and the rest are as follows: Āryapakṣa established in 500 CE, Ārdharātikapakṣa established in 500 CE, Saurapakṣa established in 800 CE and Gaṇeṣapakṣa established in 1500 CE. Along with the introduction of Islam into India, India experienced an Islamic Period between 1600 CE and 1800 CE, and after that the country came in direct contact with western modern astronomy.

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According to history, though foreign astronomy made inroads in India several times, Vedic astronomy as the indigeneous Indian astronomy was still preserved though various methods while blending with other external astronomy. Meanwhile, such Indian ancient astronomy blended with other external astronomy spread to China along with the eastern spread of Buddhism. Moreover, the detailed description of the spread of Indian ancient astronomy knowledge to China can be found in Chinese Buddhist Scriptures and various other historical records, and the contents thereof can be divided as follows: 1. Cosmology 2. Constellation system 3. Running of the sun and the moon, eclipses 4. Season and calendar 5. Running of planets and calendar thereof.

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The cosmology knowledge introduced along with Buddhism shall belong to early India mainly including the concept of Kalpa, cyclic destruction and creation of cosmos, size and measurement of cosmos etc. Buddhism world view believes that at the beginning of Kalpa, the cosmos with certain structure is constructed via wind while the whole world is destroyed completely at the end of Kalpa thus forming one cycle and then recreating the world. The Indian ancient cosmic structure is quantitatively described in Buddhist scriptures as follows: All things in the cosmos are located on a large wind wheel without edge, with the thickness thereof as 1,600,000 *yojanas*. A water wheel is set on the wind wheel, with the thickness as 1,120,000 *yojanas* and the surface of 320,000 *yojanas* is coagulated to form gold and land wheel. The

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New Delhi's Jantar Mantar observatory is located at Sansad Marg, New Delhi in India. It was built in 1724.

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Diagram of the four divisions of the twenty eight constellations, namely, Azure Dragon, White Tiger, Vermilion Bird, and Black Tortoise.

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diameter of the water wheel and the gold wheel are both 1,203,450 *yojanas* and the perimeters thereof are three times the diameters. Sumeru is located centrally on the gold wheel with the outflow height and the underwater penetration both as 84,000 *yojanas* and with seven gold hills orderly embracing externally; four great regions are located outside the seven gold hills while Cakkava is located outside the four great regions. The underwater penetration of the seven hills and Cakkava is the same with that of the Sumeru but the outflow height is orderly halved, and the widths of the hills are equal to the outflow height. Among the above nine hills, there are eight seas with the first seven seas as the inland seas, the width of the innermost sea is 80,000 *yojanas* and the perimeter thereof is 240,000 *yojanas*. The widths of the rest of the six seas are orderly halved and the eighth sea is open sea with the width as 321,000 *yojanas*. This Indian ancient cosmos model is similar with the Chinese ancient canopy heavens cosmos model but there is still no definite evidence to prove the correlation.

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We need to pay attention to the name translations, the Citrā issue, the constellation amount, the constellation width, the configuration etc of the Indian constellation system in Chinese Buddhist Scriptures. Before the Indian constellation system was introduced to China, a set of similar constellation system had already formed in China. The name translations of the Indian constellation in Chinese translation process have the following three conditions: 1. Most of the constellations are translated into Chinese existing constellations 2. Some of the constellations are translated into Chinese via transliteration according to Sanskrit pronunciations 3. Only one example which is translated into Chinese according to the literal meaning of the Sanskrit constellation name can be found in Chinese Buddhist Scriptures. The Citrā of Indian constellation system is different from that in ancient China and besides Spica, Kṛttikā and Asvini are also taken as Citrāin Chinese Buddhist scriptures. Obviously, Spica is taken as Citrā in order to remain the same with Chinese ancient constellation system. Moreover, Kṛttikā is taken as Citrā diagnostically in ancient India meaning the constellation where the spring equinox is located but taking Asvini as Citrā successively in Chinese Buddhist scriptures aims at reflecting the adjustment due to precession. There are two kinds of constellation amount, namely 28 and 27, in Chinese Buddhist scriptures. The 28 constellation system is consistent with ancient China on amount, however, the 27 constellation system excluding Abhijil is the Indian local constellation system which is the adjusted product aiming at matching with the signs of the zodiac. The most complete and clearest descriptions for the 27 constellation system in Chinese Buddhist scriptures first appeared in *Sutra on Taking Auspices by Celestial Constallations and Bodies* translated by Tang Bukong. However, the width of the constellations in Chinese Buddhist Scriptures division is mainly an uneven division and the unevenness is not ruleless as that in ancient China and can be roughly divided into wide, secondary wide and narrow. From *Sutra on Taking Auspices by Celestial Constellations and Bodies* and other Buddhist scriptures, we also can find a trend of unifying the widths of the constellations which shall be influenced by the introduction of signs of the zodiac into India. Additionally, in Buddhist Scriptures, the star numbers of the constellation systems are greatly different from that in ancient China and the star numbers of only five constellations are totally the same. Meanwhile, the names of the implements or the products used for describing the constellation configuration also have an Indian characteristic.

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Diagram on astronomy, Qing Xiao Yun Cong

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Indian Observatory at Jaipur, located in Rajasthan, India

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In regard to the running of the sun and the moon, several Chinese Buddhist Scriptures mention that the sun and the moon run around Sumeru while being supported and blown by the wind wheel and the running height is half height of Sumeru, namely 42,000 *yojanas*. The diameter of the sun is 51 *yojanas* while that of the moon is 50 *yojanas*. The four continents experience day and night alternative as well as seasonal variation along with the running of the sun around Sumeru. Volume V “Orbits of The Sun And The Moon 19th” in *Abhidharma Theory* mentions: the sun has 180 orbits while the moon has 15 orbits. The sun continuously transits from one sun orbit to another sun orbit every day in half-a-year and then returns to the innermost sun orbit from the outermost sun orbit orderly which can reasonably explain annual variation, such as Sunrise azimuth change, day and night length change etc. As for the orbits of the moon, one moon orbit is equivalent to 12 sun orbits, the sun and the moon repeatedly run between the most southern sun orbit and the most northern sun orbit between which the distance is 290 *yojanas*. Meanwhile, it also mentions that solar radiation shapes in a sphere with the diameter as 721,200 *yojanas*.

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According to Chinese Buddhist scriptures, eclipses were caused by Asurindaka Rahu with his hands covering the sun and meanwhile six eclipse periods were mentioned. From other document literatures except Chinese Buddhist scriptures, we can know that the Indian astronomy introduced in China featured the eclipse theory. Moreover, “Kasyapa Harvey Indian Method” which is an Indian method for calculating eclipses is attached to the article of Eclipse Theory in *Linde Calendar* of Volume 33 of *Old Book of Tang* (refer to article “Kasyapa Harvey” for details) and one paragraph of Indian eclipse method which is “Eclipse Cutting Method” spread by Indian Buddhist Kumāra is also attached to the article of Eclipse Theory in *Taien Calendar* of Volume 34 of *Old Book of Tang* (refer to article “*Kumāra*” for details). In addition, the Indian eclipse method translated by Gautama Siddha clearly introduces the Indian eclipse calculation method (refer to article ‘Jiuzhi Calendar’ for details).

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The contents related to the season and the calendar of ancient India are abundantly recorded in Chinese Buddhist scriptures mainly including season division, month name, concept of the white moon and black moon, arrangement of a solar month of 31 days and a solar month of 30 days, inserting of leap month, day and night length change, division and denomination of day and night etc. Four kinds of different definitions for the month are given in Volume 48 of *Great Treatise on the Perfection of Wisdom* namely “Solar Month”, “Karma Month”, “Lunation Month”, “Nakshatra Month”. Therein, “Solar Month” is one-twelfth the length of the tropical year, “Karma Month” is an ideal year (360 days as an ideal year), “Lunation Month” is namely the synodic month and “Nakshatra Month” is namely the sidereal month. And several sheets of the shadow length change during seasonal variation are also given in Buddhist Scriptures and *Time and Non-Time* has the most detailed data. Except Jiuzhi Calendar, Futian Calendar which originated from Indian astronomy and was popular among Chinese folks during the late Tang dynasty is a representative of the complete writings similar with Indian ancient siddhanta, but the main contents have been lost. In addition, Guangzhai Calendar written by Gautama Ambrosius is presumed to be a kind of astronomical table for calculating astronomical phenomena according to Indian astronomy.

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The following planet knowledge is mainly involved in Chinese Buddhist scriptures: planet name and order, planetary scale, travelling range, planetary calendar etc. Especially, Rahu and Ketu are also taken as planets. Sometimes Chinese ancient names are adopted as the names of the planets but sometimes the transliteration names are adopted. And the orders of the planets are sometimes determined by the order of Chinese ancient generation among five elements and sometimes determined by the brightness thereof, and meanwhile the orders “From Fast to Slow” are also given in *Sutra on Taking Auspices by Celestial Constellations and Bodies*. In addition, the travelling range of the planets and the calendar thereof are intensively described in *Expelling the Seven Planets’ Fated Calamities* (refer to *Expelling the Seven Planets’ Fated Calamities* for details).

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Since Indian astronomy was introduced in China in the early times and had a large time span (from 3^rd to 10^th centuries CE), thus the related materials are piecemeal preserved till now and it is also difficult to establish the correlation between the astronomical contents thereof and certain astronomical classics of Pakṣa in different Indian periods. However, we are still able to trace the impact of Indian astronomy on Chinese astronomy and such impact most seriously influenced Chinese astronomy in Tang dynasty. Li Qiao mentioned in *Shenlong Calendar Serial* that the Indian and Chinese astronomers were different from each other but they had the interlinked astronomy thoughts and fought for modifying the calendar together. The most classic works shall be *Taien calendar* as Japanese Buddhist Saicho who came to Tang for Buddha dharma described in *Dharma Blood Spectrum* that “Tang and Brahman Buddhists communicate and reach an agreement” thus completing the classic works in Chinese calendar history and becoming the model of later calendars.

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(*Chen Ming* & *Niu Weixing*)

Terms, Items

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Zero

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Zero is a numerical code denoting “nil”. Although the decimal system was adopted in ancient China quite early, there was still no special symbol to denote zero and in case of zero in number writing, the position of zero was replaced with a blank. When translating Jiuzhi Calendar in the sixth year of Emperor Kaiyuan (718 CE), Gautama Siddhartha introduced the Indian numerical symbol of zero to China at the beginning of the Jiuzhi Calendar in “Number Accounting Method”. He said that when counting up to 10 place one dot in the blank as a mark in order to make the blanks have marks and thus prevent mistakes. According to the opinion of Joseph Needham, the earliest dot or zero symbols, appeared in two inscriptions in Cambodia and Sumatra. This year was taken to be the starting point of Sakha calendar and the inscription belonged to Sakha 605th year. Therefore, if the Sakha Calendar started from 78 CE then the inscription belonged to 683 CE. According to another view, if the Sakha Calendar started from 128 CE then the inscription belonged to 733 CE, which is a little later than the compilation time of Jiuzhi Calendar. In nine chapters of Mathematical Book written by Qin Jiushao of Song Dynasty (1247 CE), zero was changed into a hollow circle which is the same as its modern form. At present, there is still no clear evidence to indicate that using the hollow circle as the zero symbol in the ancient Chinese mathematics books originated from India.

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(*Niu Wexing*)

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Portraits of Constellation Gods

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Painting of constellations was a kind of subject for painting that started in India and also an implement used for sacrifice via the introduction of Esoterism.

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As communications between India and China flourished since the Tang Dynasty, the paintings of constellations prevalent in India gradually spread to China. Zhang Sengyao was a painter in Liang dynasty whose three pieces of paintings were said to be the earliest record of constellation paintings ie *Painting of Nine Constellations*, *Painting of Saturn*, *Paintings of Five Planes and Twenty-eight Constellations*. If these records are true, they show that as early as the Liang Dynasty, China began to take in Indian star-god pictures. However, scholars mostly deem that there was some doubt that the star-god paintings emerged in Liang Dynasty and the real author of those paintings should be someone in the Tang Dynasty instead.

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The paintings of celestial gods were divided into two categories, one of which was regarded as such artistic works as *Paintings of Five Planes and Twenty-eight Constellations*. The other of which was used in sacrifice like the paintings of various star-gods and Buddha in Buddhist texts.

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Tang dynasty Liang Lin Zan’s ‘Diagram of the five stars and twenty Tibetan deities’. This picture is preserved at the art gallery in the city of Osaka, Japan.

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*Painting of Five Planes and Twenty-eight Constellations*, in Chinese art history, was passed as an exclusive painting subject, the content of which included pictures of five planes and those of 28 constellations. There were five types available in the world which were listed in order of authors’ times respectively – Tang, Song, Yuan and Ming, *Paintings of Five Planes and Twenty-eight Constellations* collected in Osaka Metropolitan Museum of Art, Painting Rolls by Song People collected in the Palace Museum, Constellation Paintings by Yuanzixia and Constellation paintings by Qiuying collected in America Metropolitan Museum of Art. The picture of Mars in *Painting of Five Planes and Twenty-Eight Constellations* was collected in Osaka Metropolitan Museum of Art.

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There was a large number of Buddhist texts which illustrated the paintings of star-gods. *Brahma Navagraha Fire*, *Seven Constellations Resisting Calamities*, *Picture of Fire* were all collected in Tripitake and they illustrate the paintings of nine constellation gods. However, Buddhist texts collecting paintings of 28 constellations and zodiacs were rare with an exception of *Picture of Fire* which collected a combination of paintings of various constellation gods.

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There were numerous paintings of star-gods available in Dunhuang accounts. Their popularity was due to their increasing inroads in folk customs and beliefs as well as prevalence of Esoterism. People believed that the unfavourable effects arising from vicious constellations could be warded off or weakened by means of enshrining pictures of constellation gods and holding proper rituals. Calamities could be diminished by prayers and blessings. The folk belief thus accelerated the popularity of constellation paintings.

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Picture of Fire, Collected in Toyota temple, in Tripitake, at the seventh roll, pages 693-704

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*Picture of Buddha* was a painting assembling Buddha and various constellation gods. It shared the effect of diminishing calamities and praying blessings. However, in this painting, Buddha dwelt at the centre and the other constellation gods gathered around.

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Painting of Buddha and Five Planes, collected in British Museum, NO. 1919,0101,0.31

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Constellation-god paintings that were made as art works and those out of Chinese Buddhist texts were obviously closely related but differed in properties. For example, a prominent change often is reflected in alteration of the constellation gods’ riding animals in painting of *Five Planes and Twenty-eight Constellations* to their crowns in Buddhist texts.

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Until today, constellation paintings that were introduced in China cannot match Indian pass-on paintings. The first reason was that star-god painting has been a long-evolving subject. These paintings, after being introduced into China, could be only regarded as the paintings of that certain stage, while those paintings in India had evolved into other forms. The second reason was that, after the introduction into China, they were transmitted secretly so not much was exposed to the world besides the limited number of paintings were incapable of representing the whole characteristics of India-originating paintings. The third reason was that the Buddhist texts and art works probably were endowed with some elements of Central Asia.

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(*Li Hui*)

Personalities

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Zhen Luan

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Zhen Luan was a Chinese mathematician from Wuji, a county in present day Hebei Province in the Northern Zhou of the Southern and Northern Dynasties. His dates of birth and death are unknown. Zhen Luan was a military officer and a governor in Hanzhong city before and after the middle of 6^th century CE. He was a Buddhist and good at arithmetic. He wrote three books on *Theory Against Taoism*, two books on *Five Classics of Arithmetics*, five books on *Five Cao Arithmetics*, two books on *Seven Luminaries of Calendar Art*, nine books on *Nine Chapters of Mathematics Art*, one book on the *Notes of Zhoubi Mathematics Art* etc. The name of the books that Zhen Luan noted are three books of Xia Houyang Mathematics Art and *Zhang Qiujian Mathematics Art*; three books of *Sun-Tzu Mathematics*; one book of the *Summary of Mathematics*; and one book of *Three Kinds of Mathematics*. He developed the Tianhe Calender which was implemented in 566 CE, the year of Tianhe in Northern Zhou and was used for the next 12 years.

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In the fifth year of Tianhe ie 570 CE, Zhen Luan was ordered by the Emperor Wu to learn Buddhism and Taoism in detail to confirm their length and determine their true and fictional portions. This is the primary content of his book, *Theory against Taoism*. Because Zhen Luan was a Buddhist, he was seriously against all kinds of points of Taoism and in 36 articles of the book, he referred to some of India's universe/cosmos theories, such as inexorable doom, universe of 1,000,000,000 universes and trayastrimsa.

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A picture of the book, 'Legacy of the Record of the technique of counting', an ancient block-printed edition of Mao Jin Ji of the Chongzhen reign period of the Ming Dynasty.

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The *Summary of Mathematics* was written by Xuyue in Han dynasty and is noted by Zhen Luan. Qian Baocong thought that the book was both written and noted by Zhen Luan and its authorship is inadvertently attributed to Xuyue but some other scholars believe that it was truly authored by Xuyue and noted by Zhen Luan. The writer narrates in first person that he met Liuhong when travelled to Mountain Tai and Liuhong told him that he had learned three kinds of scale system and 14 kinds of noting numbers from Tianmu whom he encountered in mountains.

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What Liuhong mentions is the three kinds of scale system. The bottom in 10 multiply 10, the middle in 10,000 multiply 10,000 and the top in number changed when there is no number. Ten multiply 10 is 100, 10 multiply 100 is 1000 and 10 multiply 1000 is 10,000 and so on. Ten thousand multiply 10,000 means 10,000 multiply 10,000 is a hundred million and then billion, a hundred billion etc. The number changes when there is no number means a hundred million, billion and a hundred billion. This so-called number changes is the same as the point 45 in *Buddhavatamsaka-Mahavaipulya-Sutra* translated by Siksananda in Tang dynasty; the article 25 of *The King of Hearts Bodhisattva Asks the Monk Named A Seng* in book 29 translated by Buddhabhadra in Eastern Jin Dynasty and the *Bodhisattva Arithmetics* in book 10 named *Never Thinking of Liberated World When All of the Elites Doing Good* translated by Tang monk Prajnaparamite Kawmira. If the *Summary of Mathematics* was truly written and noted by Zhen Luan, the way to getting involved in the arithmetics is easy for him.

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There is a sentence that reads, if you don't know how the *Ksana* rushes, how could you know peasant woman in ups and downs of time; if you don't know the small, how could you know Buddhist cosmology in the body of the *Summary of Mathematics*. In this sentence, the words *Ksana* and Buddhist cosmology are used in Buddhism, and *Ksana* hadn't been translated by its pronunciation into Buddhist *Sutra* in Xuyue's time so there is a strong evidence of it being treated as Zhen Luan's book and the fake one being written by Xuyue. Zhen Luan referred the time and length units in Lankavatara *Sutra*, four concepts of the world in *Buddhavatamsaka-Mahavaipulya-Sutra* and the small, middle and the big Buddhist cosmology when he noted the sentence.

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(*Niu Weixing*)

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Yixing

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Yixing was born in 683 CE and died in 727 CE. His common name was Zhangsui and the Buddhist name was Jingxian. He was also called Monk Yixing, Acharya Yixing, or Honored Yigong. He was a Chinese monk and astronomer in Tang dynasty. He and an Indian monk, Subhakarasimha, translated seven books of the *Great Buddhist Blessing Sutra* that is *Mahavairocana Tantra*. Besides this, Yixing wrote 20 books of the *Notes of the Great Buddhist Blessing* that is the *Notes on Mahavairocana Tantra*. Ordered by Xuanzong Emperor, he developed the Dayan calendar. Also, for calculation of auspicious and inauspicious time, four books on Buddhist tantrism are classified under Yixing's name which include one book each on *Timing Ritual*, *Another Line Method of Seven Telecom Stars*, *Methods of Protecting the Big Dipper* and *ROM: Obsidian Brahma Fire*.

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Yixing was born in a very reputed family. His great grandfather, Zhang Gongjin, had assisted Emperor Taizong Li Shimin and was involved in the Xuanwu Gate Incident. He was one of the 24 *Meritorious Statesmen of Lingyan Pavilion* but as time went by the fortune of his family declined. After his parents' death, Yixing became a monk and went to Song mountain to observe Shramana's silence. In order to avoid being called by the government, he hid in Mt Dangyang in Jingzhou, and travelled to the south. In fifth year of Kaiyuan, 717, Yixing was called into the capital by Xuanzong Emperor and translated the Indian Tantrik Classics together

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*Yixing*

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with Subhakarasimha and Vajrabodhi. In ninth year of Kaiyuan, 721, he made new calendar under the order of Xuanzong Emperor. In the 12^th^ year of Emperor Kaiyuan (724), Huangdaoyouyi was made and declared a national astronomical measurement standard. In the 15^th^ year of Kaiyuan (727), when the draft of the new calendar was made, Yixing had died. Zhangshuo and the Calendar Officer Chen Xuanjing made up the draft into 52 books and presented them to the government. In the 17ᵗʰ year of Kaiyuan, 729, the new calendar, the Dayan Calendar, came into use.

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Yixing had made a lot of innovations in the Dayan Calendar. These reflect the influence of the foreign knowledge on astronomy. Firstly, Yixing made a correct description for the uneven movement of the sun for the first time in *Few Examples of the Sun*

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[Figure: *Astronomical diagram/figure of the* *Song Dynasty on a stone tablet*]

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*Full and Empty in Dayan Calendar* and the shape of Sun in the Dayan Calendar with the maximum differential centre value calculated as 143’.29. Some scholar point that this result is almost the same as the differential center value 143’.38. Probably the latter was influenced in some ways by the Indian astronomy as mediation. Secondly, Yixing made a Tangent Table in Dayan Calendar which was treated as the first one in the history. Some scholars thought it was transformed from the Tangent Table in Jiuzhi Calendar by Yixing but some others insist that it was his own invention in the form of the Chinese aboriginal astronomy and calendar. Thirdly, while introducing the uneven movement of the *Five-Star Anniversary*, Yixing referred to the Indian calendar. He thought that every star has its specific love and hate relations when it meets good stars, it moves faster; when it meets bad ones, it moves slower. Fourthly, the *Calculating Eclipse Arithmetic* mentioned in *Few Examples of Sun Eclipse of Dayan Calendar* is corresponding with the method of calculation in Jiuzhi Calendar. Fifthly, the method of calculation of vanishing and offsetting in Dayan Calendar is obviously originated from India. For example, book five in the *Abhidharma Theory* has expressed the similar idea. Sixthly, some results in Dayan Calendar were based on the adoption of the division of 360 degree for the whole day. The *Moon* *Moving Table* is calculated starting from the farthest point of earth.

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If the influence of the India astronomy is only vaguely observed in Yixing, it cannot be denied that, the way that the idea of India Calendar or the Western Calendar he directly referred from *Mahavairocana* was used to define the terms of astronomy and calendar showed that Yixing had grasped the India calendar sufficiently. In other way, the name of the Dayan Calendar came from the way of great changing in the Confucianism classic, *Book of Change*. Yixing had read Taoist classics intensively when he was young, so we can see that Yixing had gotten the classics of Confucianism, Buddhism and Taoism. Also we can see the content of astrology of Taoism in his *ROM. Obsidian Brahma Fire*.

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We have no historical records of the way Yixing received the knowledge of the Indian astronomical calendar. But as both Chinese and Indian scholars had personal exchanges at that time, there must be books on astronomy coming into Tang dynasty. For example, the book 198 of *Tang Annals* recorded, Kawmira had sent ambassadors to Tang in seventh year of Kaiyuan (719), and paid in tribute an astronomy book, some secret prescriptions and foreign medicine. Further, the book 971 in *Ce Fu Yuan Gui* read that in June of the seventh year of Kaiyuan, Tuhuoluo Country had paid tribute to astronomers. After setting things right in the state, Xuanzong Emperor paid more attention to the cultural construction and amendment of the calendar was one of the great measures undertaken then.

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In sixth year of Kaiyuan, 718, Xuanzong Emperor ordered Zhaitan Xida to translate Jiuzhi Calendar. Iin seventh year of Kaiyuan, some books on astronomy came from Jibin country and astronomers from Tuhuoluo country joined them. All of these could have prepared the groundwork for the amendment of the calendar. Xuanzong Emperor thought highly of Yixing, and gave him full authority to pursue the work of amendment. Therefore, Yixing, who was proficient on ancient India language, could check up the foreign astronomy books kept in the library of royal household and conveniently consult the astronomers.

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(*Niu Wexing*)

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Dharmacandra

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Dharmacandra (652-743 CE) was an eminent monk from eastern India who came to China and disseminated Dharma in the Tang Dynasty. His abbreviated name is Damozhan which was called Fayue in Tang Dynasty and his personal deeds were recorded in the 14ᵗʰ volume of *New Directory for Buddhism in Zhenyuan Period* written by Yuanzhao. He once took a trip to central India and was deemed as a citizen of Magadha. Dharmacandra made great contributions to the cultural exchange between India and China by the transmission of ancient books and records and through translation of Buddhist scriptures and texts on Arthashastra, medical science, astrology, pictures of esoteric Buddhism and occult techniques.

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Dharmacandra has a good knowledge of Tripitaka and medical science. He used the materials according to specific conditions and travelled far and wide to disseminate dharma. In Qiuci (Kuqa in Sinkiang now), he met his apprentice, Satyacandra

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*Sanskrit text of 'Favourable Opinions' (Li lun)*

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(Zhenyue in Tang Dynasty; the surname is Bunaxian which is also called Liyan) and then devoted themselves to translation together. In the 18ᵗʰ year of Kaiyuan period (730), they were recommended to enter court by Lu Xiulin, the military governor in Anxi. In the 20ᵗʰ year of Kaiyuan period (732), they arrived in Chang'an and entered the court. There they presented the court with occult techniques, medical prescriptions, Buddhist books, medical herbs and sutras etc. In the years that followed, the translated versions of *Medical Prescriptions* and *Chinese Herbal Medicine* were published. They not only offered material objects of medical herbs but also helped translate books on medical prescriptions and Chinese herbal medicine. These medicines and books which were derived from India or the Western Regions, were a great event for exchange of Chinese and Western medicine. During the years when they stayed in Chang'an, they also translated Buddhist scriptures, such as *Prajna Paramita Heart Sutra for Ordinary Buddhist Monks*. In the 29ᵗʰ year of Kaiyuan period (741), they left Chang'an and returned to India, by way of the Shaqi, Western Liang. Unfortunately, he caught an illness halfway and died in Cakra Temple of Khotan in the second year of Tianbao Period (743). The 971ˢᵗ volume, *Wai Chen Bu – Tribute Four*, *Ce Fu Yuan Gui* recorded that Dharmacandra, the eminent monk proficient in Tripitaka, had come to offer medicine, new conjuration, various kinds of scriptures in Sanskrit, *Theory for Governing the Country*, *Astrology Records*, and *Various Prescriptions in Sanskrit*. Dharmacandra had paid tribute to the court many times. His *Theory for Governing the Country* referred to *Artha÷àstra* written by Kauñilya, a minister of Chandragupta of the Maurya Dynasty in India. The *Astrology Records* is, without any doubt, a work for astrology, while “*Various Prescriptions in Sanskrit*”, just as its name implies included books on medical prescriptions in Sanskrit. The *Incantation – Preface of Thousand-eyes and Thousand-arms Avalokiteshvara Dharani*, translated by Zhitong of Tang Dynasty recorded that Damozhantuo, a monk in Brahman of Udyana country (present-day Swat Valley, Pakistan), “translate sutras according to the orders” and offered Bodhisattva and incantation as a tribute. Damozhantuo mentioned there was actually Dharmacandra.

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(Chen Ming)

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Kasyapa

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Kasyapa Xiaowei was an astronomer who worked in the astronomy institution in the Tang Dynasty. He belonged to the family of *Kasyapa*, one of the three great Indian astronomical families that came to China in the Middle Ages. There is not much description about Kasyapa Xiaowei's deeds. After the Linde Calendar in *Old book of Tang History* narrated the method for calculating the eclipse, there were 400 words involved in the Indian method that he preached. This mainly explained that the Indian Eclipse Calculation Method calculated eclipse points according to the speed of sun and moon. This is a different method from what is recorded in Linde Calendar that calculates eclipse points using a different formula created according to shuowang (lunar) great eclipse points. Additionally, *Old book of Tang* also narrated presages about lunar eclipse, such as “the moon is wet as sweat”, it's different from Chinese astronomical narrations and compares with Indian Sanskrit astronomical documents. The *Indian Eclipse Method* recorded here is close to Vasistha Paulisa's system for calculating lunar eclipse. There is no lunar eclipse from 13 degree to 15 degree but the lunar eclipse takes place within 13 degree (12 degree 15 cents in Chinese). Vasistha Paulisa's calculation method was recorded in verse 1 through 10, chapter VI in *Five Great Siddhantas* (*Pañca Siddhanta*) by Varaha Mihira.

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There were another two Kasyapas who worked for Tang who could be identified. One is Kasyapa Ji who was an astronomical officer in Jingyuan in the Period of Zhenguan (627-649 CE). Another one is Kasyapas Zhong (or Zhi Zhong) who was a historiographer for Right General of Royal Guards and he was rewarded for dedicating an extol song to Queen Wei in the second year of *Period of Jinglong* (708 CE).

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Gautama Siddhartha

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Gautama Siddhartha was a popular Indian astronomer who came to China and joined the Tang state as the Imperial astronomer in the early period of 8^th century CE. Dates of his birth and death are unknown. As an imperial astronomer (the highest position in imperial astronomical organisation), he compiled Jiuzhi Calendar and *Kai Yuan Zhan Jing*, comprising in total of 120 volumes.

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From the epitaph of Gautama tomb explored by Xi’an Historical Relic Administrative Office in May 1977, we know about Gautama Siddhartha’s progenitor and later generations with names in the following order Gautama Yi, Gautama Luo, Gautama Siddhartha, Gautama Zhuan (the fourth son of Gautama Siddhartha). Gautama Zhuan had six sons, with the first names in order of seniority being as Sheng, Bian, Yu, Huang, Yan and Mao. When referring to Gautama’s progenitor, it said on the epitaph that he moved from central Indian region to China and then settled in Chang’an (today’s Xi’an, Shaan’xi). The epitaph also mentions that Gautama Zhuan’s great-grandfather, Gautama Yi, was an eminent monk but entered the imperial court. But from Gautama Siddhartha’s father, namely Gautama Luo on, all the four successive generations held posts of imperial astronomer, astronomical supervisor, judge etc in astronomical organisation of Tang totally for more than 100 years. At that time, people called Gautama Siddhartha as “Gautama Supervisor” and called the Indian calendar of Gautama’s family as “Gautama Calendar”. When noting *Sutra on Taking*

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Kaiyuan Zhanjing

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*Auspices by Celestial Constellations and Bodies*, Yang Jingfeng mentioned that currently there were three Indian calendars respectively – from Mahakasyapa family, Gautama family and Kumara – and stored in Court Historian Library but the calendar from Gautama family was frequently used and considered most appropriate.

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In the Gautama family, Gautama Siddhartha had made outstanding contributions to astronomy. According to the records in Volume 1 of *Kai Yuan Zhan Jing*, in Jing Yun second year (711), Emperor Ruizong of Tang, Gautama Siddhartha was ordered to hold the repair work for the armillary sphere which was built by Emperor Ruizong of Northern Wei Dynasty, which he completed in Xiantian second year (713 CE) of Emperor Xuanzong. *Old Book of Tang-Astronomical Log* recorded that Gautama Siddhartha was ordered to translate the Indian calendar Jiuzhi Calendar in Kaiyuan second year (718) of Emperor Xuanzong (refer to Article *Kai Yuan Zhan Jing*). According to textual research, Gautama Siddhartha was ordered to compile *Kai Yuan Zhan Jing* after February, Kaiyuan second year (714), and he completed the compilation before Kaiyuan 14^th Year (726). *Kai Yuan Zhan Jing* compiled by Gautama Siddhartha collected many astronomical data during ancient and mediaeval times in China. Additionally, the translation and Jiuzhi Calendar collected and recorded in Kai Yuan Zhan Jing are the Chinese Indian Calendar preserved till now.

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(Niu Weixing)

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Kumara

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Jumoluo (Kumāra) was an astronomer who worked in an astronomy institution in the Tang Dynasty. His Sanskrit name was Kumāra which means boy or prince and was one of the commonest names in India. Both the books of Tang history mentioned Jumoluo once as an appendix for recording the eclipse calculation method of Dayan calendar. The main content was to introduce the eclipse calculation method by judging the sun preached by him, which indicated that he was good at eclipse calculation method and his method could be used with Dashu. The constellation’s name *Yuche* in *Indian Astronomy Method* was preached by him and it corresponded to *Number Jianglou* in Chinese and Constellation *Yuche* was Aries. *Yuche* sounded similar to “iku” in Akkadian, and E. KUE in Sumerian of Aries. It probably provided proof that *Chinese Twelve Numbers* was the evolution of Babylon’s *Twelve Constellation*. Further, the *Jumoluo Occult Sciences at West Gate* was mentioned in the contents of *Indian Astronomy* in *General History* was probably written by Jumuoluo.

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(Zhou Liqun)

Works

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Kai Yuan Zhan Jing

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Kai Yuan Zhan Jing is a Chinese ancient astronomy and astrology work. With the full name as *Da Tang* *Kai Yuan Zhan Jing* was compiled by Gautama Siddhartha between 718-726 CE. *Kai Yuan Zhan Jing* was once lost after the Tang Dynasty but was rediscovered at the end of the Ming Dynasty and then handed down up to now.

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*A picture of the book Kaiyuan Zhanjing*

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*Kai Yuan Zhan Jing* totally includes 120 volumes wherein the cosmic theories of various scholars are collected and recorded in volumes 1 and 2. Various phenomena related to the sky and earth and the corresponding divinations were collected and recorded in volumes 3 & 4. Phenomena related to the sun and divinations thereof are collected and recorded in volumes 5-10. The astronomical phenomena related to the moon and the divinations thereof were collected and recorded in volumes 11-17. The treatises pertaining to the astronomical phenomena, such as condition of the five planets, apparent motion, planet-occulting/ approaching-fixed star, mutual occulting/ approaching of planets and divinations thereof were collected and recorded in volumes 18-59. The astronomical phenomena pertaining to the fixed stars including 28 constellations and three asterisms and the divinations thereof were collected and recorded in volumes 60-70. The descriptions pertaining to the occasional astronomical phenomena such as shooting star, guest star, comet etc and the corresponding divinations were collected and recorded in volumes 71-90. The descriptions pertaining to various climatic phenomena and the corresponding divinations were collected and recorded in volumes 91-102. *Lin De Li Jing* written by Li Chunfeng of Tang Dynasty has been copied in volume 103. Jiuzhi Calendar was recorded in volume 104. Shang Yuan Ji Nian and other basic constants of 29 calendars from 6^th Pre-Qin period to Shenlong Calender of Tang Dynasty were collected and recorded in volume 105. The ancient fixed star map provided with an introduction for the position difference between currently measured positions and the old star map was collected and recorded in volumes 106-110. The descriptions pertaining to various vegetation, birds, beasts, humans, ghosts, implements, etc and the corresponding divinations were collected and recorded in volumes 111-120.

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On the whole, *Kai Yuan Zhan Jing* is a classic Chinese ancient astrological pandect. Such Chinese traditional astrological pandect was organised for compilation by an Indian astronomer who came and took a post in Tang. So this work fully reflected the depth and width of Chinese and Indian astronomy exchange – undoubtedly Gautama Siddhartha well mastered these Chinese traditional astronomical and astrological contents. Jiuzhi Calendar collected and recorded in volume 104 was an only exception for the Chinese traditional astronomical astrology. Jiuzhi Calendar is an Indian calendar which was translated by Gautama Siddhartha in Kaiyuan sixth year (718 CE) according to the imperial order and it has also not been recorded in other Chinese ancient historical materials. So *Kai Yuan Zhan Jing* is the only resource for it. Jiuzhi Calendar is the first to introduce the writing symbol for zero (refer to the entry on “zero”) and the arithmetic for sine function and sine function table in trigonometry. It also distinguished perigee and winter solstice as well as apogee and summer solstice via ecliptic system of coordinates and geometrical method and introduced the method for calculating the visible diameter change of the moon (refer to the entry on “Jiuzhi Calendar”) etc. All the above-mentioned were the newly introduced materials for Chinese mathematics and astronomy in those times. Moreover, Jiuzhi Calendar is an important exhibit historically for the scientific communication between India and China and has been well preserved till now by virtue of *Kai Yuan Zhan Jing*.

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(Niu Weixing)

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Jiuzhi Calendar

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Jiuzhi Calendar is the only Chinese translation for Indian calendar preserved till now which was translated by Indian astronomer Gautama Siddhartha in Emperor Xuanzong the sixth Year of Emperor Kaiyuan (718 CE) in the Tang Dynasty according to the imperial order (refer to entry on “Gautama Siddhartha”) and was collected and recorded in Volume 104 of *Kai Yuan Zhan Jing* compiled by himself (refer to entry on *Kai Yuan Zhan Jing*).

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Currently, there is no conclusive proof on the existence of a Sanskrit original manuscript for Jiuzhi Calendar. However, according to the researches of scholars, the contents thereof are closely related to the Indian astronomical work *Pancasidhantika* (about 550 CE which to a large extent relates to *Khandakhadyka*, 665 CE).

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The Jiuzhi Calendar was the carrier for the spread of many mathematical astronomical contents into China. For example, the Indian mathematical concept for zero (refer to the entry on “zero”) entered China produces some unfocussed impacts on date calendar and divination. A saying of “Seven Luminaries Are Directly Used for Prediction, this Divination Is Special” is included therein. Additionally, “Seven Luminaries Are Directly Used for Prediction” herein refers to a method for selecting auspicious days and fortune telling on the basis of seven luminaries – namely “Week” as stated by later generations (refer to entry on *Sutra on Taking Auspices by Celestial Constellations and Bodies*). Meanwhile, this record in *Jiuzhi Calendar* indicates to the first reference of the appearance of week calendar mode known till now.

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Jiu yao in Guangsheng xiasi frescoes/mural paintings (Yuan Dynasty). It is now preserved in Horatio Nelson's museum.

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(*Li Hui*)

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Sardulakarnavadana

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Modengjia Scripture (Śārdūlakarṇāvadāna) is a Chinese translation of Buddhist metaphor scripture involving knowledge of Indian astronomy. It was translated by Zhiqiangong and Zhuluyan in the period of Three Kingdoms. There are two versions of this translation in volumes 2 and 3. It narrates a story that a Chandala girl picked up Ananda but she was finally converted by Buddha. The story includes rich content on constellation and divination. Besides Chinese, there are Tibetan, Sanskrit and more than 10 different versions.

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The name of the original Sanskrit version is *Śārdūlakarṇāvadāna* while for Chinese, it is *Modenjia*. Tibetan version’s is called *sTag rNa’i rTogs pa brJod pa* and its free translation is *Tiger Ear Metaphor Scripture* abbreviated as *Tiger Ear Scripture*. *Modengjia* in *Modengjia Scripture* is the transliteration of *Mātaṅga* in Sanskrit. It is the name of Chandala girl’s mother in the story. The part of pre-life story in the metaphor scripture’s Sanskrit version narrates that Brahman the Great Puspakārin questioned the King of Chandala Triśaṅku in the ancient time. The latter gained appreciation of Brahman the Great by his erudition, reinterpretation of caste, retrospect of Vaishya’s root and listing of horoscope prediction. The conversation between Brahman the Great and King of Chandala involved all kinds of discussion on horoscope and early Indian calendar etc including names and features of constellations; degrees of day and night, seasons of long and short, degrees of instant; units of length, units of gold’s weight and units of grain’s bulk. They also discussed fate of those who are born under each of the constellations; prediction about cities constructed under constellations; prediction about rain in the last month in summer under constellations; prediction about lunar eclipse under constellations; suitable and unsuitable things under constellations; degrees of syzygy and day and night, length of shadow, changing of seasons; prediction about earthquakes’ result; prediction about releasing of

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for the first time via this Indian calendar. In terms of astronomical calculation, Jiuzhi Calendar, at least, introduces six western astronomical elements to China: 1. 360°circumference division 2. Counting method for 60 binary 3. Ecliptic coordinates system 4. Definition of distinctions between perigee and winter as well as between apogee and summer solstice 5. Method for calculating the visible diameter change of the Moon 6. Sine function calculation method and sine function table.

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The translated contents of Jiuzhi Calendar include the motions of the Sun and the Moon and the eclipse portion but are not related to planetary motion, so it is difficult to speculate whether the Sanskrit original manuscript includes planetary theory or not. However, as far as the eclipse part is concerned as the Chinese ancient astronomy does not include any geometrical method, it always lacked in terms of eclipse calculation. The appearance of Jiuzhi Calendar accordingly provides a totally new technical reference for Chinese ancient calendar compilation.

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As far as the impact of Jiuzhi Calendar on Chinese calendar is concerned, there is a “Public Record” in Chinese astronomy history in *New Book of Tang* that informs that “Taien Calendar is an incomplete copy of Jiuzhi Calendar”. Taien Calendar is one of the important calendars in the history of China, created by Buddhist astronomer Yi Xing (refer to the entry on “Yi Xing”). In consideration of the Buddhist identification, generally, the most subsequent Chinese ancient astronomers and modern researchers for astronomical history believe that Taien Calendar refers to or adapts some contents of Jiuzhi Calendar.

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Jiuzhi Calendar not only greatly influences Chinese mathematical astronomy but also probably prisoners and features of good and evil occurred under constellations, among others.

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Illustration of 'Leng Yan Jing Shujie Meng Chao'

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Among so many Chinese versions of *Tiger Ear Metaphor Scripture*, the most complete and most closest to the original version is *Modengjia Scripture*. *Modengjia Scripture* consists of seven collections: *Women Collection of Conversion*, *Karma Collection of Going*, *Actual Collection of Truth Showing*, *Question Collection of All Human*, *Picture Collection of Constellation*, *Auspicious Collection of Calamity Perceiving* and *Discern Collection of Lightening*. The content of the last three collections are about horoscope and take up 50 per cent of the whole scripture. They depict all kinds of predictions that occurr in light of the moon's position in different constellations by themes. When *Modengjia Scripture* depicts horoscope, its depiction method is that of gathering a certain number of prediction themes into one collection with the translator adding the name of the collection. But the Sanskrit version of *Tiger Ear Metaphor Scripture*'s depiction method is according to chapters, its division method is same as the Indian classical divination book, *Guangji* (Bṛhatsaṃhitā) which is divided into chapters (Adhyāya). In *Book of Sui History and Confucian Classics*, there is the earliest recording about the names of seven Indian astronomy books that came to China and *Modengjia Scripture*, *Picture Collection of Constellation* were recorded as a separate document of *Picture Collection of Constellation* from *Modengjia Scripture*.

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Historically, materials on horoscopy is included in Chinese translations of Buddhist scriptures. These are referred in *Constellation Divination Method of Esoteric Buddhism*, which primarily have been discussed in *Constellation Scripture of Auspicious and Inauspicioul, Day and Time, Kind and Wick Talked by Manjusri Bodhisattva and other Immortals*, *Modengjia Scripture*, and *Twenty-Eight Constellations Scripture of Prince Shetoujian*. *Twenty-Eight Constellations Scripture of Prince Shetoujian* is another translated version of *Modengjia Scripture*. This includes rich material on horoscopy and the translation date of *Twenty-eight Constellations Scripture of Prince Shetoujian* can be identified to the Western Jin Dynasty by Zhufahu. Therefore, the Chinese versions series of *Tiger Ear Scripture* can be counted as early Chinese translation copies of Buddhist horoscopy material. In the area of Dunhuang and Turpan, fragmentary transcript of *Modengjia Scripture* (S.3374 etc), fragmentary transcripts of Sanskrit version (Or.15010/6 and Or.15010/20 etc) and also the relevant *Twenty-Eight (seven) Constellation Horoscopy Book* were unearthed. The documents can prove that there are basis for mixed explanation of divinations of lunar eclipse, interfluves and earthquake in *Tiger Ear Scripture*'s Sanskrit version. Meanwhile, it also illustrates that *Tiger Ear Scripture* influenced people's lives in the Western Regions.

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