Ancient Indian astronomy and mathematics

Rajesh KochharHonorary professor, Mathematics Department, Panjab

University, Chandigarh 160014Indian Institute of Science Education and Research, Mohali,

Punjabrkochhar2000@yahoo.com

Lecture delivered at SJB Institute of Technology, Bangalore, 5 May 2015

• Modern astronomy today is at the cutting edge of scientific enquiry. Although we are by definition part of the universe, we look at it as if from the outside. In earlier times, however, cosmic environment was seen as inseparable from the terrestrial environment and human affairs. The world was anthropo-centric. Earth stood still and the whole world revolved around it. As masters of the Earth, human beings considered themselves to be special in the eyes of the Almighty and worthy of his attention.

• The beginnings of astronomy 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, it had to be elaborate and well-timed, so that a careful distinction could be made between auspicious and inauspicious times.

• (Note that mathematical problems such as obtaining the square root of two and approximate value of pi ( circumference of a circle divided by its diameter) were taken up in the context of preparation of fire altars and are discussed in the Shrautasutras.)

• Since planetary motions provided a natural means of time keeping and were seen as couriers of divine signals. Skies were therefore regularly monitored. This was the beginning of astronomy as an intellectual discipline.

• The above considerations are general; they would apply to any geographical area or cultural group.

• We now come to ancient India. Before proceeding further, let us examine the nature and limitations of the source material available.

Source material: nature and limitations

• Scripts (Kharoshthi, Brahmi) were introduced into India about 3rd century BCE or somewhat earlier for writing Prakrit languages derived from Sanskrit. Script for Sanskrit itself, the language of Hindu scriptures, was adopted much later, the first use being for stone inscriptions. Writing material came from plants or trees and had a short life. Paper was not introduced into India till about 8th century CE. Paper in any case was not used for Brahminicaltexts.

• Texts were in the custody of specialist caste groups who memorized them and transmitted them to the next generation by word of mouth. The extant texts would have been supplemented with explanatory “notes” to serve an immediate purpose. What was not considered worth preserving at any point in time was lost for ever.

• Astronomical results were seen as a revelation rather than deduction. They were therefore recorded in the Rigvedic format, that is, in metrical poetry. An astronomer had to be a poet first and then an astronomer. Because of constraints of metre, synonyms or half-words had to be used or allusions made. This introduced vagueness and imprecision.

• These texts are not library texts. They were meant for a select group which knew the context. Their interpretation, out of context, is a difficult task.

• These texts are not complete. Knowledge which was considered to be the requisite background would not be mentioned. Absence of mention therefore does not constitute proof of absence. More specifically, we do not know how observational parameters were obtained ( borrowing? actual observations?)

• Although decimal system was invented in India, in astronomical texts, numbers are expressed in terms of real or artificial words or word parts, opening the door for deliberate or inadvertent mis-representation.

• Because of the sanctity of Vedic texts, elaborate schemes were devised to preserve them as a gramophone record. No such mechanism was employed for astronomy, with the result that astronomical ( and Ayurvedic) texts are like an audio cassette.

• The Vedic corpus once created was preserved as such. In the case of the Epics and Puranasadditions were made but no subtractions. But in the case of scientific texts, both deletions and additions were made. Once a new edition came, parts of old edition would be deleted for ever.

• Also, it is not possible to assign firm dates to any early event or development. It is therefore not possible to construct a connected account of any aspect of early India.

Vedic astronomy (1400-BCE-5th century CE)

• There are stray astronomical references in the Rigveda. For our purposes, Yajurveda is more useful because it was a manual for ritual . It thus lists 27-28 nakshatras ( bright star or star groups) which are seen near the Moon every night. Later, when an accurate luni-solar calendar was introduced, Vikrami month names were so chosen as to contain an important piece of information. For example, the month Vaisakha is so called because in this month, the Moon is full near the nakshatra, Vishakha.

• The oldest Indian text exclusively devoted to astronomy is the Vedangajyotisha (VJ). Because of the sanctity attached to the Vedic corpus, it has been faithfully preserved down the ages. But as a scientific text it was overtaken by new developments and became obsolete.No wonder then, VJ is the least understood of all Vedic texts. Remarkably, it remained in vogue for a very long period, till the 6th century CE.

• The oldest portions of VJ could be as old as 1400 BCE.

• Week days and zodiacal signs which are now integral part of Indian astronomy/ astrology were not known in the Vedic astronomy period.

• 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, all adhere to the Vedic astronomy.

• The Mahabharata text is a vast collection, of about 100,000 shlokas, composed over an extended period of time. And yet, the Mahabharata does not know of week days or zodiacal signs.

Valuable inputs were received, in stages, from the Greco-Babylonian world to the northwest of India.

It is surmised that the (old) Shaka [Saka] calendar was established by the Shakas in 123 CE to commemorate their victory over the Parthians in Bactria [Balkh in Cental Asia]. It was used by the Shaka emperors and Satraps in their Indian territories. 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 BaudhaGaya, dated c. 100 BCE.

• Weekdays were slow in making an entry. It has been suggested that they appeared in 4th or 5th century CE.

Aryabhata

• The modernization of Indian astronomy, by incorporating these and other inputs, took place at the hands of Aryabhatawhose influential work, Aryabhatiyam, appeared in 499CE. This was the beginning of the Siddhantic astronomy, so-called because the primary astro-mathematical texts were called Siddhanta[ proven in the end].

What was the situation before Aryabhata?

• 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; Romakaand Paulisha, which are obviously of foreign origin, slightly less so. The two older ones, Vasishtha Siddhanta and the PaitamahaSiddhanta, were the least accurate, the latter more so than the former. PaitamahaSiddhanta 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. Note that we learn about pre-Aryabhata works not from their own time but from later redactions.

• 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. 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, (arranged in order of increasing orbital period, they are the Moon, Mercury, Venus, Sun, Mars, Jupiter , and Saturn. ; 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.

• Illustrious 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); 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).

• 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 them from the reports of others such as Nilkantha who lived 100 years later. Madhava’spupil 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.

Development of mathematics

• 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.

• Western scholarship especially during the colonial period tended to deny antiquity or originality to ancient India. As a backlash, many researchers have tended to unduly stretch the chronology backwards. Also, receipt of inputs from others is an important ingredient of intellectual development. There should not be any self-consciousness while discussing this.

•Thank you