Shulba Sutras

The Shul ba Slut ras (Sanskrit śulba meaning "string, cord, rope") are sutra texts belonging to Śrauta ritual and containing geometrical calculations and attempts at squaring the circle related to altar construction.

They are parts of the larger corpus of Dharma Sutras which are considered appendices to the Vedas. The Shulba Sutras are our only source of knowledge of Indian mathematics of the Vedic period. The four major Shulba Sutras, and the most mathematically significant, are those composed by Baudhayana, Manava, Apastamba and Katyayana. These Sulba Sutras have been dated from around 800 BCE to 200 BCE, and they thus may predate or rank roughly co-eval to Pythagoras (c. 572 - 497 BCE).

These Sutras include what may be the first 'use' of irrational numbers. Other equations from this early period of Indian mathematics include examples of quadratic equations of the form $ax^{2}=c$ and $ax^{2}+bx=c$.

The sutras also contain discussion and non-axiomatic demonstrations of cases of the Pythagorean theorem and Pythagorean triples. The Pythagorean theorem is first found in its full generality with non-axiomatic demonstration in the Katyayana sutra. It is also implied and cases presented in the earlier work of Apastamba and Baudhayana. The Satapatha Brahmana and the Taittiriya Samhita were probably also aware of the Pythagoras theorem. Seidenberg (1983) argued that either "Old Babylonia got the theorem of Pythagoras from India or that Old Babylonia and India got it from a third source".. Seidenberg suggested that this source might be Sumerian and may predate 1700 BC.

Pythagorean triples are found in Apastamba's rules for altar construction. One of the Sulba Sutras later estimates the value of pi as 3.16049. Altar construction also led to the discovery of irrational numbers—a remarkable estimation of the square root of 2 is found in three of the sutras. The method for approximating the value of this number gives the following result:

${\sqrt {2}}\approx 1+{\frac {1}{3}}+{\frac {1}{3\cdot 4}}-{\frac {1}{3\cdot 4\cdot 34}}={\frac {577}{408}}=1.414215686...$

The result is correct to 5 decimal places. Elsewhere in Indian works however it is stated that various square root values cannot be exactly determined, which strongly suggests an initial knowledge of irrationality.

Indeed an early method for calculating square roots can be found in some Sutras, the method involves the recursive formula: ${\sqrt {x}}\approx {\sqrt {x-1}}+{\frac {1}{2\cdot {\sqrt {x-1}}}}$ for large values of x, which bases itself on the non-recursive identity ${\sqrt {a^{2}+r}}\approx a+{\frac {r}{2\cdot a}}$ for values of r extremely small relative to a.

Before the period of the Sulbasutras was at an end, the Brahmi numerals had definitely begun to appear (c. 300BCE) and the similarity with modern day numerals is clear to see. More importantly even still was the development of the concept of decimal place value. Certain rules given by the famous Indian grammarian Panini (c. 500 BCE) add a zero suffix (a suffix with no phonemes in it) to a base to form words, and this can be said somehow to imply the concept of the mathematical zero.

Further reading

• Seidenberg, A. 1983. "The Geometry of the Vedic Rituals." In The Vedic Ritual of the Fire Altar. Ed. Frits Staal. Berkeley: Asian Humanities Press.
• Sen, S.N., and A.K. Bag. 1983. The Sulbasutras. New Delhi: Indian National Science Academy.

External links

• Sulba Sutras
• The Indian Sulbasutras