Indian Mathematics in Vedic Period ( 1500 BCE -400 BCE)

Ancient Indians were involved in different forms of Mathematical study.The effort on understanding the deeper patterns of quantity,structure,space and change, in the Mathematical framework, reached its extremity in the Vedic period. Though there are other great Mathematicians who came after wards and followed the tradition.

The religious texts of the Vedic Period provide evidence for the use of large numbers . By the time of the last Veda, the Yajurvedasamhita (1200-900 BCE), numbers as high as 10¹² were being included in the texts. For example, the Mantra (sacrificial formula) at the end of the annahoma ("food-oblation rite") performed during the asvamedha ("horse sacrifice"), and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to a trillion:

"Hail to śata ("hundred," 10²), hail to sahasra ("thousand," 10³), hail to ayuta ("ten thousand," 10⁴), hail to niyuta ("hundred thousand," 10⁵), hail to prayuta ("million," 10⁶), hail to arbuda ("ten million," 10⁷), hail to nyarbuda ("hundred million," 10⁸), hail to samudra ("billion," 10⁹, literally "ocean"), hail to madhya ("ten billion," 10¹⁰, literally "middle"), hail to anta ("hundred billion," 10¹¹, lit., "end"), hail to parārdha ("one trillion," 10¹² lit., "beyond parts"), hail to the dawn (uśas), hail to the twilight (vyuṣṭi), hail to the one which is going to rise (udeṣyat), hail to the one which is rising (udyat), hail to the one which has just risen (udita), hail to the heaven (svarga), hail to the world (loka), hail to all."

This shows the rigor on the collective study on Mathematics in the Vedic Age, which is prominently known for its spiritual study.Further, Satapatha Brahmana (9th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.

The Sulba Sutras (literally, "Aphorisms of the Chords" in Vedic Sanskrit ) (c. 700-400 BCE) list rules for the construction of sacrificial fire altars. Most mathematical problems considered in the Śulba Sūtras spring from "a single theological requirement," that of constructing fire altars which have different shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks.

According to ( Hayashi 2005 p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world."

"The diagonal rope (akṣṇayā-rajju) of an oblong (rectangle) produces both which the flank (pārśvamāni) and the horizontal (tiryaṇmānī) produce separately."

Since the statement is a sūtra, it is necessarily compressed and what the ropes produce is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student.They contain lists of Pythagorean triples , which are particular cases of Diophantine equations . They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square."

Baudhyana (c. 8th century BCE) composed the Baudhayana Sulba Sutra, the best-known Sulba Sutra, which contains examples of simple Pythagorean triples, such as: (3,4,5), (5,12,13), (8,15,17), (7,24,25), and (12,35,37) as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square." It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together."

The main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the Sulvasutras. The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time.

In all three Sulba Sutras were composed. The remaining two, the Manava Sulba Sutra composed by Manava (fl. 750-650 BCE) and the Apastamba Sulba Sutra, composed by Apastamba (c. 600 BCE), contained results similar to the Baudhayana Sulba Sutra.

The tradition Sulba Sutras also proved the fact that the religious ritual carried out in the Vedic times had wide usage mathmetical principals for perfecting the material requirement of the rituals.