Astronomy and Mathematics at Vedic era in Ancient India

'Vedic practices provided the inspiration for advances in astronomy and mathematics'
(Excerpted from an article by B.V.Subbarayyappa in the book India 1000 to 2000, Editor : T.J.S.George, published in December 1999 by Express Publications (Madurai) Ltd, Express Estates, Anna Salai, Chennai - 600 002. The excerpt was also published in The New Indian Express on Sunday in the FYI column on April 8, 2001.)
Jyothisha (astronomy) was one of the six auxiliaries of the Vedas and the earliest Indian astronomical text goes by the name of Vedanga Jyotisha. Year-long sacrifices commenced from the day following the winter solstice and Vedic knowledge of both winter and summer solstices was fairly accurate. The Vedanga Jyotisha had developed a concept of a cycle of 5 years (one Yuga) for luni-solar and other time adjustments with intercalation at regular intervals.
Indian mathematics too owes its primary inspiration to Vedic practices. The Shulba sutras, part of another Vedic auxiliary called the Kalpa sutras, deal with the construction of several types of brick altars and in that context with certain geometrical problems including the Pythagorean theorem, squaring a circle, irrational numbers and the like. Yet another Vedic auxiliary, Metrics (chandah), postulated a triangular array for determining the type of combinations of 'n' syllables of long and short sounds for metrical chanting. This was mathematically developed by Halayudha who lived in Karnataka (10th Century) into a pyramidal expansion of numbers. Such an exercise appeared six centuries later in Europe, known as Pascal's triangle. Vedic mathematics and astronomy were pragmatic and integrated with Vedic religio-philosophical life.
But such an approach was not to last long. During the three centuries before and after the Christian era, there were new impulses. Astronomy became mathematics-based. In the succeeding centuries, while astronomy assimilated Hellenic ideas to some extent mathematics was really innovative. Indian astronomers were able mathematicians too. The doyen among them, Aryabhatta I (b.476 A.D.) gave the value of pi (3.1416 approx., a value used even today) worked out trigonometrical tables, areas of triangles and other plane figures, arithmetical progression, summation of series, indeterminate equations of the first order and the like. He expounded that the earth rotates about its own axis and the period of one sidereal rotation given by him is equivalent to 23h 56m 4s.1, while the modern value is 23h 56m 4s.091. He discarded the mythical Rahu-Ketu postulate concerning eclipses in favour of a scientific explanation.
Aryabhatta's junior contemporary Varahamihira, was well known for his compendium, the Panchasiddhantika, a compilation of the then extant five astronomical works called the Siddhantha- Surya, Paulisha, Romaka, Vasishta, and Paitamaha. Of them, the Suryasiddhanta, which he regarded as the most accurate, underwent revisions from time to time and continues to be an important text for computing pancangas.
Brahmagupta was a noted astronomer mathematician of the 7th Century. His remarkable contribution was his equation for solving indeterminate equations of the second order - an equation that appeared in Europe a thousand years later known as Pell's equation. His lemmas in this connection were rediscovered by Euler (1764) and Lagrange (1768). Brahmagupta was also the first to enunciate a formula for the area of a rational cyclic quadrilateral. In the latter half of the first millenium A.D. there were other noted astronomers and mathematicians like Bhaskara I, Lalla, Pruthudakasvamin, Vateshvara, Munjala, Mahavira (Jaina mathematician), Shripati, Shridhara, Aryabhatta II , and Vijayanandin. The tradition of astronomy and mathematics continued unabated - determination of procession of equinoxes, parallax, mean and true motions of planet, permutations and combinations, solving quadratic equations, square root of a negative number and the like.
Using nine digits and zero, the decimal place value system had established itself by about the 4th century A.D. Says historian of science, George Sarton, "Our numbers and the use of zero were invented by the Hindus and transmitted by Arabs, hence the name Arabic numerals which we often give them.' Brahmagupta's Brahmasphuta Siddhanta and Khandakhadyaka were also rendered into Arabic in the 9th-10th century. The Brahmi numerical forms with some modifications along with the decimal place-value system developed in India have since become universal.
The beginning of the second millenium A.D. witnessed the emergence of the notable astronomer-mathematician, Bhaskaracharya II (b.1114). His cyclic (cakravala) method for solving indeterminate equations of the second order has been hailed by the German mathematician 'Henkel', as the finest thing achieved in the theory of numbers before Lagrange. Bhaskaracharya II had also developed basic Calculus. Between the 14th and 18th Centuries, there were schools of astronomers-mathematicians in Kerala and Maharashtra, Ganesha Daivajna simplified methods of computation for almanac makers. The Kerala school was well known for its keen observations of eclipses over 55 years. Parameshwara (1360-1455) was the first in the history of mathematics to have given the exact formula for the circumradius of a cyclic quadrilateral; this was rediscovered in Europe by L'Huiler nearly 300 years later.
Adhering to the Aryabhatta tradition, other Kerala savants like Govinda Bhatta, Damodara, Nilakantha Somayaji, Jyesthadeva, Acyuta Pisharati and Putumana Somayaji added to both astronomy and mathematics. The leader of this intellectual lineage was Madhava (14th century), who formulated the approximations for pi, trigonometrical sine, cosine, arc tan power series (now known as Gregory series) that were rediscovered in Europe three centuries later. Nilakantha Somayaji provided a convergent infinite geometric progression. In north India, Narayana Pandita worked out a rule for finding out factors of divisors of a number, much before such an attempt was made in Europe. He was well known for his analysis of sets of numbers, magic squares and the like. As for astronomy, the astrolabe began to be used during the Muslim rule. Maharaja Sawai Jai Singh erected five observatories in Benares, Mathura, Ujjain, Delhi and Jaipur (early 18th Century). Though Jai Singh knew the use of telescope and European heliocentric astronomy, he clung to the traditional geo-centric calculations of Indian astronomy, but raised the level of observational mathematics.
- India 1000 to 2000