Shulba Sutras

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The Shulba Sutras (Sanskrit śulba: "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to altar construction, including the problem of squaring the circle.

They are are part of the larger corpus of Shrauta Sutras, considered to be 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 mathematically the most significant, are those composed by Baudhayana, Manava, Apastamba and Katyayana. The texts have been dated from around 800 BCE to 200 BCE^[1], and thus are approximately coeval to Pythagoras (c. 572 - 497 BCE).

These Sutras include what may be the first 'use' of irrational numbers.^{[citation needed]} Other equations from this early period of Indian mathematics include examples of quadratic equations of the form $a x 2 = c$ and $a x 2 + b x = c$ .^{[citation needed]}

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^[2] and Baudhayana.^[3] The Satapatha Brahmana and the Taittiriya Samhita were probably also aware of the Pythagoras theorem.^[4] 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".^[5]. Seidenberg suggested that this source might be Sumerian and may predate 1700 BC.

Pythagorean triples are found in Apastamba's rules for altar construction. The complete list is: $(3,4,5)$ , $(5,12,13)$ , $(8,15,17)$ , $(7,24,25)$ , and $(12,35,37).$ ^[6]

One of the Sulba Sutras later estimates the value of pi as 3.16049.^{[citation needed]} 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:

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

The true value is $1.414213...$ Although this formula arose as a result of geometric measurements involved in altar construction, the result can be seen as a first order Taylor approximation:^[7]

$\sqrt{a^2+r} = \sqrt{\left(a+\frac{r}{2a}\right)^2-\left(\frac{r}{2a}\right)^2}$ $\approx a + \frac{r}{2a} - \frac{(r/2a)^2}{2(a+\frac{r}{2a})},$ with

a = 4 / 3

and

r = 2 / 9

^[8]

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.^{[citation needed]}

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. This result easily follows from the Taylor approximation in the above paragraph, or, alternately, can be derived from the Binomial theorem for fractional indices.

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.

The following Shulba Sutras exist in print or manuscript

1. Apastamba 2. Baudhayana 3. Manava 4. Katyayana 5. Maitrayaniya (somewhat similar to Manava text) 6. Varaha (in manuscript) 7. Vadhula (in manuscript) 8. Hiranyakeshin (similar to Apastamba Shulba Sutras)

[edit] 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.

[edit] Notes

^ [1]
^ The rule in the Apastamba cannot be derived from Old Babylon (Cf. Bryant 2001:263)
^ Cf. Seidenberg 1983, 98.
^ Seidenberg 1983. Bryant 2001:262
^ Seidenberg 1983, 121
^ Joseph, G. G. 2000. The Crest of the Peacock: The Non-European Roots of Mathematics. Princeton University Press. 416 pages. ISBN 0691006598. page 229.
^ For $f(x) = \sqrt{x}$ , let $y = \left(a+\frac{r}{2a}\right)^2$ and $h = -\left(\frac{r}{2a}\right)^2$ . Now expand $f(y+h) = f(y) + h\cdot f'(y) + O(h^2)$
^ Cooke, R. 2005. The History of Mathematics: A Brief Course. Wiley-Interscience. 632 pages. ISBN 0471444596. page 200.