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.

The 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 800-500 BC and include first 'use' of irrational numbers, quadratic equations of the form $ax^2\; =\; c$ and $ax^2\; +\; bx\; =\; c$, unarguable evidence of the use of the Pythagorean theorem and Pythagorean triples, predating, or roughly co-eval with, Pythagoras (c 572 - 497 BC), and evidence of a number of geometrical proofs.

Pythagoras's theorem is first found in the Baudhayana sutra. It is also implied in the later work of Apastamba, and Pythagorean triples are found in his rules for altar construction. One of the Sulba Sutras 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:

$\backslash sqrt\{2\}\; =\; 1\; +\; \backslash frac\{1\}\{3\}\; +\; \backslash frac\{1\}\{3\; \backslash cdot\; 4\}\; -\; \backslash frac\{1\}\{3\; \backslash cdot4\; \backslash cdot\; 34\}\; =\; \backslash frac\{577\}\{408\}\; \backslash approx\; 1.414215686$

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 repeated application of the formula: $\backslash sqrt\{a\}\; \backslash approx\; \backslash sqrt\{a\; +\; r\}\; \backslash approx\; a\; +\; \backslash frac\{r\}\{2\; \backslash cdot\; a\}$, 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.

External links

• Sulba Sutras
• The Indian Sulbasutras

Mathematical literature | Indian literature | Hindu texts