National Workshop on Ancient Indian Scientific HeritageKurukshetra University, 11 November 2014

Ancient Indian astronomical tradition: Characteristics and accomplishments

Rajesh KochharPresident IAU Commission 41: History of Astronomy

Hon. Prof., Panjab University, Mathematics Department, ChandigarhIndian Institute of Science Education and Research, Mohali

rkochhar2000@yahoo.com

Ever since human beings learnt to walk upright, they have looked at the sky and wondered. The sky has remained the same, but its meaning and significance have been changing.

Human beings have tried to comprehend their cosmic environment in their own cultural framework, and derive material benefit from knowledge so gained. At the same time this knowledge has been used to construct a theological and philosophical worldview.

We are of course part of the Universe. But today, we tend to look at it as if from the outside. In ancient past, cosmic environment was seen as an integral part of human life and affairs.

The beginnings of astronomy (and mathematics) are related to the requirements of the ritual in early cultures. Ritual was a means of securing divine approval and support for terrestrial actions. To be effective, the ritual had to be elaborate and well-timed. In India sacrificial altars were specified to be built of proper size and shape. Out of this requirement, came the development of geometry as well as arithmetic and algebra.

In addition, the ceremonies had to be performed at the proper time. Since planetary motions provided a natural means of time keeping, their refined study became important. Early astronomical knowledge went into the making of sacred literature, mythology and cosmogonical models.

Even when astronomy developed as a scientific discipline in its own right in India, it very consciously sought to retain links with the sacred texts.

The rather static Vedic astronomy, whose routes go the Rigveda itself, prevailed in India for more than a millennium. Its termination can be assigned a precise date, namely, 499 CE. This is the year when the terse Aryabhatiya, authored by Aryabhata (b. 476 CE), appeared on the scene. The work has remained influential ever since.

Since the basic astro-mathematical texts were called Siddhantas (proven in the end), this phase can be called Siddhantic.

The transition from Vedic astronomy to Siddhantic astronomy is rather poorly understood. Before discussing these phases and the intervening transitional period in some detail, it will be useful to review the nature and limitations of the available source material .

India had taken to the composition and preservation of texts long before writing began in India. Scripts like Kharoshthi and Brahmi were introduced into India in about third century BCE for writing Prakrit languages rather than Sanskrit. The oldest documents in Sanskrit are inscriptions from about first century BCE. Writing of sacred texts began later; the writing material came from plants or trees and had a short life.

Ancient texts that can serve as source of information for us are of five types: (i) Vedic corpus, (ii) Puranas and the epics of Ramayana and Mahabharata, (iii) Buddhist and Jain texts, (iv) astronomical texts, and (v) texts from other countries.

Each text in the vast Vedic corpus, from the archaic Rigveda to the relatively late Dharmashastras, has been preserved in its original form without any addition, deletion or alteration being made. This feature makes them an extremely valuable source of information.

The early Vedic texts constitute Hinduism’s heritage. Hinduism in action is represented by the Puranas and the Epics which were often recast to meet the contemporaneous requirements of their custodians and their audience. There are 18 major Puranas and various recensions of the Ramayana and historically the more important Mahabharata. At the level of individual texts and recensions there are additions as well as deletions.

But, if we take this corpus as a whole, we notice that whatever was composed at any time has survived in one text or the other so that over all there have been additions but no deletions.

In contrast to the sacred literature, astronomical texts underwent deletion as well as addition. New or revised texts appeared and many old ones either totally vanished or survived partially. In the following, I discuss some of the salient points of ancient astronomical literature.

In imitation of the Rigveda, astronomical texts were composed in metrical verse so that an astronomer had to be a poet also. Poetry is not the ideal vehicle for dissemination of scientific knowledge. Requirements of meter compelled the poet-astronomer to use synonyms or half-words and resort to allusions. This introduced vagueness and imprecision.

These texts were not composed for the purpose we are using them now. They were not designed as library books in the sense of self-contained self-study material. They required familiarity with the context and personal intervention of a teacher. In their time, there must have been background knowledge to go with these texts which was not preserved and is now lost. All ancient texts are valuable for what they expressly contain. Absence of mention does not necessarily mean that the thing did not exist.

Although decimal system was invented in India, astronomical texts express numbers in terms of real or artificial words or word parts, opening the door for deliberate or inadvertent mis-representation. While elaborate schemes were devised to prevent corruption of Vedic texts, no such mechanism was available in the case of scientific works. Since texts were written on plant material which had a very short life, old manuscripts had to be regularly copied. During the process, inadvertent errors could be introduced. At times, a word was deliberately changed here and there to suppress or modify the original meaning. Also, new material was added to old texts without recording that this was being done. To add to the confusion, entirely different texts have identical names.

Sacred literature can be considered timeless, but science advances, rendering older texts outdated. Because of the oral tradition, once a text fell into disuse because of arrival of a better or newer text, the old one was forgotten except for the excerpts that may have been incorporated in other texts. From the above, it is clear that it is not possible to construct a connected account of history of ancient astronomy in India. Chronology to an extent remains a problem and there are significant gaps in our understanding that cannot be filled.

The Vedic period

There are stray astronomical references in the Rigveda (Rv) including to a solar eclipse. Rv (5.40.5-9) describes how an asura, Svarbhanu by name, pierced the Sun ‘through and through with darkness’. The Sun appealed to rishi Atri who through his prayers ‘caused Svarbhanu’s magic arts to vanish’ and thus ‘found the Sun again’. This passage occurs in the Rigveda’s fifth mandala whose authorship is credited to the Atris. This ‘episode’ is mentioned and embellished at a number of places in the Vedic literature :

Kaushitaki (or Sankhayana Brahmana) of the Rigveda (24.3); Panchavimsha (or Tandya) Brahmana of the Samaveda (4.5.2; 4.6.13; 6.6.8; 14.11. 14-15; 23.16.2); Shatapatha Brahmana of the Shukla Yajurveda (5.3.2.2); and Gopatha Brahmana of the Atharvaveda (8.19) ( Dikshit 1896, Vol. 1, p. 58; Kane 1975, pp. 241-242). What Atri probably did was to chant mantras while the eclipse lasted. The Rigvedic description is significant. An eclipse was seen as the demon’s work in disrupting the cosmic order. Propitiation was needed to restore that order.

Later Vedic texts rename the eclipse-causing demon as Rahu. Note that the word Rahu does not occur in Rv. As we shall see, to maintain semblance of continuity with sacred texts, post-Aryabhata astronomy/astrology texts came to employ Rahu as a scientific term. There is thus a dichotomy in the use of the term Rahu before and after 499 CE.

The Yajurveda proper as well as its associated literature is a valuable source for a study of early history of mathematics and astronomy. As already noted, these intellectual disciplines arose from the postulated requirements of ritual for which the Yajurveda is the manual.

It introduces the concept of nakshatra, 27 (earlier 28) bright stars or star groups in the sky which are used as markers for the Moon and the Sun’s orbits.

Yajurveda also refers to the four colures, the two equinoxes and two solstices. There is an important phenomenon associated with the colures, known as precession of the equinoxes (ayanamsha). The position of the equinox (or solstice) is not fixed in sky but move in a retrograde manner completing the cycle in 26000 years.

The precession of equinox serves as a clock, with equinox or solstice as the hand of the clock and the background nakshatras as the digits on the dial. The phenomenon is thus a useful, though approximate, chronological tool.

The oldest exclusively astronomical text is the very concise Vedanga Jyotisha, which comes in two overlapping recensions, one attached to the Rigveda (attributed to one Lagadha) , the other to the Yajurveda.

Taken together, the two versions contain some 49 independent verses, some of which have not yet been interpreted. Of all the Vedic texts, it is the most obscure. This is not surprising. As a scientific text, it was made redundant, but being a Vedic text it was memorized and passed on from generation to generation as a relic. It contains an interesting observation which can be approximately dated. Both the versions say that the winter solstice took place at the nakshatra Shravishtha ( later called Dhanishtha ) . Making some reasonable assumptions, one can assign the date c. 1400 BCE to this observation.

How old the general contents of the two recension are is difficult to say. Vedanga Jyotisha describes a rather inexact calendar and does not mention zodiacal signs and weekdays which make their appearance in India two millennia later. Vedanga Jyotisha concepts remained in vogue for a very long time.

Shulvasutras, attached to the Yajurveda, which address the question of making of sacrificial altars, are the world’s oldest texts on geometry. Among other things, they make extensive use of what later came to be known as Pythagoras theorem.

One of the problems taken up in the Shulvasutraa is the construction of a circle or a square of twice the size of a given one. This led the authors to calculate a fairly correct value of . Similarly, attempts to construct a circle of the same area as a square or vice versa resulted in evaluating the ratio of a circle’s circumference to its diameter, Mahabharata simply sets pi =3. However by the time we come to Aryabhata, we get a value accurate to four decimal places.

Transitional period

The political vacuum in India caused by the collapse of the post-Ashokan Maurayan empire was filled by rulers who were based in Afghanistan and Central Asia. From our point of view, particularly important are the Greco-Bactrians and the Shakas (Sakas, or Indo-Scythians). Mahabharata (Vanaparva 188:34-36) and other texts call them mlechchhas and dub them as sinful and untruthful.

Unrighteous or not, they brought with them elements of Greco-Babylonian astronomy, which slowly got incorporated into the mainstream and brought about modernization of Indian astronomy. Most notably, India obtained an accurate luni-solar calendar. It is surmised that the old Shaka calendar was established by the Shakas in 123 CE to commemorate their victory over the Parthians in Bactria. It was used by the Shaka emperors and Satraps in their Indian territories.

It is surmised that in 78 CE, in Ujjain, the accumulated 200 years were dropped and the suitably Indianized new Shaka era was ushered in. There is direct archaeological evidence of the depiction of zodiacal signs at Baudha Gaya, dated c. 100 BCE. Weekdays were slow in making an entry. It has been suggested that they appeared in 4th or 5th century CE. 

A number of old texts adhere to the Vedic astronomy. Kautilya’s Arthashastra; the Ashokan edicts (3rd cent. BCE); the Buddhist Sanskrit text, Shardulakaranavadana ( 4th cent. CE); and the Jain works, Surya Pannati and Chand Pannati.

It is remarkable that the zodiac and the weekdays do not figure in the Mahabharata text. It is well known that additions were made to Mahabharata over an extended period of time, till it came to its present size of a hundred thousand shlokas.

It is reasonable to suppose that if zodiacal signs and weekdays had been in general vogue when the Mahabharata text was still open, they would have found a way in. Experts believe that the Mahabharata took its present form in about 400 CE. One can therefore say that the zodiac and weekdays, which later became an integral part of Siddhantic astronomy, were introduced into Indian mainstream in the fifth century CE.

In a significant scholarly exercise, Varahamihira (d. 587 CE), a junior contemporary of Aryabhata, made a comparative study of the five extant Siddhantas. The compendium, which came to be known as Panchasiddhantika, is actually a Karanagrantha; it omits all theory and provides concise rules for quick calculations.

Varahamihira grades the texts according to their accuracy. Surya Siddhanta is the most accurate; Romaka and Paulisa, which are obviously of foreign origin, slightly less so. The two older ones, Vasishtha Siddhanta and the Paitamaha Siddhanta, were the least accurate, the latter more so than the former. Paitamaha Siddhanta is based on Vedanga Jyotisha, and like it deals only with the Sun and Moon. While in the other cases, the epoch is 505 CE, in this case it is 80 CE. It was obviously included for its archival value.

It is not surprising that of the five, Surya Siddhanta was the most accurate; it was an old text only in name; it was recast in the light of Aryabhata’s work, not the Aryabhatiya, but another one since lost. Around 1000 CE, Surya Siddhanta was again recast; it is this version which is still in use for making panchangas, or traditional almanacs which depend on it except for timings of eclipses which they take from modern sources.

We know of three Surya Siddhantas: Pre-Varahamihira ( known only by name), Surya Siddhanta as redacted by Varahamihira after Aryabhata; the present Surya Siddhanta.

Interestingly, astronomical works as text books were known by their author. But when their elements were incorporated into astrology-oriented texts, they were given divine names to enhance their market value.

Siddhantic astronomy Siddhantic astronomy focused on the calculation of mean and true position of the (geo-centric) planets; time of rise and setting of planets; conjunction of planets; conjunction of a planet and a star; heliacal rising and setting of stars; instrumentation; etc. A notable achievement of it was the calculation of lunar and solar eclipses.

Siddhantic astronomy broke new ground. And yet, it tried to maintain continuity with sacred literature by borrowing terminology and concepts from the Vedic corpus. Aryabhata himself made astronomical use of the Vedic Yuga scheme, while the Vedic terms Rahu and Ketu were incorporated, presumably by Varahamihira, into astronomical/astrological literature pertaining to eclipses.

Yuga scheme

Manusmrti describes a Yuga scheme which postulates a universe without beginning or end that continually undergoes spells of creation and destruction. The scheme is further elaborated on in the Puranas. Complete description becomes available from Surya Siddhanta. The main points of the scheme are summarized below.

In the Vedic times, a year comprised 12 months and 360 days. A human year was said to be a day of the gods so that a divine year (Dyr) would consist of 360 human years (yr).

Four Yugas, Kali, Dvapara, Treta, Krta (or Satya), were defined with their duration in the ratio 1:2:3:4. Kaliyuga was the current one and the shortest. Numerically, it was set equal to 1200 Dyr. The four added together constitute a Chaturyuga [four-age] or a Mahayuga [great-age]. A Mahayuga thus consists of 12000 Dyr.

A still bigger unit called Brahma’s day or Kalpa was defined as equal to 1000 Mahayugas.

To combine the celestial with the terrestrial, a mythical ruler, Manu, was postu;lated who presides of a Manvantara ( Manu’s interval) comprising 71 Mahayuga. Since 1000 is not divisible by 71, there is no simple way by which Manvantara and Kalpa can be reconciled .The equation is set up as follows.

It will be convenient to use mathematical notation to properly understand the structure within a Brahma’s day. Let us denote the duration of a Kaliyuga by the symbol k; Dvapara, Treta and Krta are then 2k, 3k and 4k respectively.

Let a Mahayuga be denoted by m, so that 

m = k + 2k + 3k +4k = 10k.

Let us denote a Krtayuga (=4k) by s. Then 1 Brahma’s day=1000m= 994m + 6m=14 x 71m +15s=14 x 71m + 14 s+ s=s + 14(71m + s).

Recall that 71m is a Manvantara. We can now describe a Brahma’s day in words. A Brahma’s day begins with a dawn equal to a Krtayuga. This dawn is followed in succession by 14 Manvantaras, at the end of each of which there occurs a deluge (pralaya) lasting a Krtayuga. This complex scheme has perplexed many modern-day commentators.

Thus, Rev. Ebenezer Burgess in his famous 1860 annotated translation of the Surya Siddhanta wondered: ‘Why the factors fourteen and seventy - one were thus used in making up the Aeon [Kalpa] is not obvious’ (Burgess 1860:11). I think this scheme was constructed working backwards from the neat round figure of 1000.

To sum up so far, the three basic building blocks, expressed in human years, are as follows.

1 Kaliyuga=1200 Divine years=432,000 years.

1 Mahayuga=10 Kaliyuga=4.32 million years.

1 Brahma’s day or Kalpa = 1000 Mahayuga = 4.32 billion years .

For the sake of continuity with the scriptures, the Yuga scheme along with the nomenclature was borrowed by the astronomers. Instead of simply expressing revolutions in a million or a billion years, an astronomer would say that there were 146,568 revolutions of Saturn in a Mahayuga, implying an orbital period of 29.4743 years.

Interestingly Aryabhata boldly modifies the Vedic Yuga scheme to suit his purpose. He makes the four components of a Yuga equal in length. He next defines his Manvantara to comprise 72 Mahayugas and sets a Kalpa equal to 14 Manvantaras, so that his Kalpa consists of 1008 Mahayugas, rather than 1000.

Rahu and Ketu

In the Indian context, Aryabhata was the first person to enunciate the mathematical theory of eclipses. According to this theory, solar and lunar eclipses occur when the moon is at either of its orbital nodes. These theoretical points move in a direction opposite to that of the planets and complete an orbit in the rather short period of 18.6 years.

This development was immediately taken note of in astrological literature, which classified the two nodes as planets, implying that they were now amenable to mathematics. Since they were hypothetical they were dubbed shadow planets. The 6th century CE text Brihajjataka (2.2-3) by Varahamihira includes Rahu and Ketu in the list of planets, and even gives their synonyms The two nodes are 180 degrees apart so that specifying one fixes the other. It would thus have sufficed to include just one of them. Both were listed no doubt to bring the planetary number up to nine which was considered sacred.

For naming these nodes, Varahamihira turned to Vedic literature. The eclipse-causing Vedic demon Rahu now became the ascending node. The term Ketu was merely a common noun employed variously to describe comets, meteors, etc. It was now made into a proper noun to denote the descending node. The Rahu-Ketu theory travelled to China in course of time, where it was integrated into the mainstream.

Siddhantic astronomersIllustrious names in Indian astronomy following Aryabhata include Latadeva (505 CE) who was Aryabhata’s direct pupil; Varahamihira (already mentioned ) a compiler and integrator rather than an original scholar, and an expert on omens; Bhaskara I (c. 574); Aryabhata’s bête noire Brahmagupta (b. 598) whose works were very influential and were later translated into Arabic; Lalla (c. 638 or c. 768); Manjula or Munjala (932); Shripati (1039); and the celebrated Bhaskara II (b. 1114).

It has often been stated Siddhantic astronomy, on the basis of old scholarship that Indian mathematics went into decline after Bhaskara II. This is not true.

Indian astronomy and mathematics received a new lease of life with Madhava (c. 1340-1425), who founded what has come to be known as the Kerala School of Astronomy. His own mathematical works have been lost.

We know of Madhava’s work from the reports of others such as Nilakantha who lived 100 years later. Madhava’s pupil Parameshvara (1360-1455), in a career spanning more than half a century, timed many eclipses and planetary conjunctions. He then set out to devise mathematical means to bring calculated times closer to observations. His singular contribution is the construction of Drgganita ( Drk system of computations).

The unbroken tradition of eclipse calculation was alive till as recently as early 19th century. A Tamil astronomer computed for John Warren , a French astronomer in the service of British East India Company, the lunar eclipse of 1825 May 31-June 1 with an error of +4 minutes for the beginning,-23 minutes for the middle, and -52 minutes for the end ( Neugebauer 1983:435).

Critique

The most remarkable feature of ancient Indian astronomical tradition from Aryabhata to the Kerala school has been the development of mathematical tools for astronomical calculations.

The 19th and early 20th century Western historiography viewed mathematics as a triumph of pure thought and accepted ancient Greek as standard for judging the rest of the world. In such a framework, Indian contribution came to be belittled. There is now greater appreciation of cultural plurality and the realization that historical developments should be examined in their own framework. The earliest known systematic treatment of linear Diophantine equations in two variables was given by Aryabhata who proposed a continued-fraction like solution of ax+by=c. Subsequently, Brahmagupta ,

Bhaskara I, Bhaskara II and Parameshvara also considered special types of system of two linear Diophantine equations.  Brahmagupta found integer solution of many Pell equations x2-Ny2=1, but was not able to apply it uniformly to all values of N. The general solution was obtained by Bhaskara II.  Madhava discovered infinite series for sine, cosine and arctangent functions and for as early as 14th century. The European names associated with these ‘discoveries’, made more than 200 years later, are Colin Maclaurin, Isaac Newton, James Gregory and Gottfried Wilhelm Leibniz.

Mathematics was developed as a tool for planetary calculations. There was very little work on mathematics for its own sake. A notable full-time mathematician is Mahavira (9th century CE). He for example worked out how a number can be cubed using an arithmetical progression.  

As I pointed out earlier, Western appreciation of Indian mathematical achievements is a recent phenomenon. This calls for rewriting of the world history of mathematics. How Indian achievements influenced developments in Europe in their time can be seen from the etymology of terms. The numbers 0 to 9 came to be known as Arabic numerals, because Europe learnt them from the Arabs. In Arabic they are called Hind-se, from India.

Three terms in English, (trigonometrical) sine, algebra and algorithm come from the 12th century Latin translation of works of a noted 8th - 9th century Baghdad mathematician known by his short name al-Khwarizmi who became the conduit for transfer of Indian mathematical knowledge to Europe. He came from a small historical place called Khiva to the south of Aral Sea, which is now part of Uzbekistan and whose ancient name is Khwarizm. In the translation of his book on arithmetic, his name was Latinized to Algoritmi which in turn gave rise to algorithm.

The Latin/English term sine comes from his algebra. Indian astronomy introduced the term jya , which literally meant a bowstring and was given the technical meaning of half-chord. It was also called jiva, was rendered in Arabic as jaib. Now, jaib was an existing word in Arabic meaning fold of a dress; this was literally translated as sinus in Latin.

We have seen that requirements of ritual, astronomy and astrology gave a great fillip to the development of approximate methods in mathematics as a versatile tool for solving practical problems. Now that the historians are sensitive to the fact that different cultures had different characteristics and the developments in a particular cultural setting must be examined in context, there is greater appreciation the world over for Indian astronomical-mathematical tradition.

Thank you