Sulba Sutras
The Sulba Sutras or Sulva Sutras are texts of the Hindu canon dealing with the geometry of altar construction. They are parts of larger works called Dharma Sutras which are appendices to the Vedas elaborating sacrificial rituals, the conduct of marriage, the sacred law and such. The Sulbasutras are our only source of knowledge of Indian mathematics of the Vedic period. It is not clear if mathematics was pursued for its own sake or was purely a by-product of religious pursuits. It is also not known how much of the Sulbasutras constitute original work by their authors and how much of it is merely a transcription of existing mathematical knowledge. Seidenberg has calculated the date of the Sulba Sutras to be approx. 2200 BCE or earlier than 1700 BCE.Bryant 2001:262-3The name "Sulba Sutra" means rule of chords, which is another name for geometry. Of the Sulvas so far 'uncovered', the four major and most mathematically significant are those composed by Baudhayana, Manava, Apastamba and Katyayana. These Sulba Sutras have been dated from around 800-140 BC [1]. These Sutras include what may be the first 'use' of irrational numbers. Other equations from the pre-Hellenistic period of Indian mathematics include examples of quadratic equations of the form and . The sutras also contain discussion and non-axiomatic demonstrations of cases of the Pythagorean theorem and Pythagorean triples that predate or roughly co-eval with Pythagoras (c 572 - 497 BC).
Pythagoras's theorem is first found in it full generality with non-axiomatic demonstration in the Katyayana sutra. It is also implied and cases presented in the earlier work of ApastambaThe rule in the Apastamba cannot be derived from Old Babylon (Cf. Bryant 2001:263) and Baudhayana.Cf. Seidenberg 1983, 98. The Satapatha Brahmana and the Taittiriya Samhita were probably also aware of the Pythagoras theorem.Seidenberg 1983. Bryant 2001:262 Scholars like Seidenberg have 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 1983, 121 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:
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 repeated application of the formula: , r being small.
Before the period of the Sulbasutras was at an end, the Brahmi numerals had definitely begun to appear (c. 300BC) 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 BC) 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.