Śulbasūtras

Posted: August 22, 2016 in inspiring, Mathematics, philosophy, sanskrit
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Śulbasūtras are collection of sanskrit works from the vedic period which supplements kalpa as appendices. kalpa is one of the six veda-angas which deals with procedures to perform vedic rituals. Śulbasūtras provide as the source of ancient Indian mathematics in the area of geometry developed during the vedic period.

The mathematics in the vedic period should not be confused with the 20th century work titled “Vedic Mathematics” by former Shankaracharya of Puri, the late Jagadguru Swami Shri Bharati Krishna Tirthaji Maharaj. Tirthaji’s “Vedic Mathematics” is neither vedic nor mathematics of any significant importance except for some alternate methods in arithmetics and elementary algebra. The title of the work is actually a misleading one, and people without understanding the facts propagate it as something from ancient Indian epistemology. The claim that the sanskrit aphorisms mentioned in his text were from the appendix (parishishta) of Atharva-Veda is controversial, and so far, no versions of Atharva-Veda contained such aphorisms. Since the book was published posthumously, we are not sure whether the author or the editor is to be blamed for such a misleading book title. For a more detailed discussion about this topic, please refer to the article titled “Myths and reality : On ‘Vedic mathematics’ ” by S.G. Dani, a renowned mathematician at Tata Institute of Fundamental Research, India.

The importance of mathematics were indeed well cherished in ancient Indian mathematical works, and the Jyothisha-Vedanga (attributed with Rig-Veda) glorifies mathematics as follows:

yatha shikha mayurānam nāgānām maṇayo yatḥa |
taḍvad vedāṅga shāstrāṅām gañitham mūrdhin stḥitḥam ||

“Like the crest of the peacock, like the gem in the hood of the king cobra, so is mathematics the top-head of all branches of science/knowledge”.

The geometry in Śulbasūtras particularly laid out details for the design and construction of fire altars for vedic rituals. The vast corpus of works developed in Śulbasūtras are mainly attributed to Baudhāyana, Mānava, Āpastamba and Kātyāyana. The oldest being developed by Baudhāyana during 800 BCE, and the youngest by Kātyāyana during 200 BCE.

One of the most significant work which gained popularity among contemporary mathematician is the statement about hypotenuse theorem (which is currently called as Pythagoras Theorem) contained in Baudhāyana Śulbasūtras which belongs to Taittiriya branch of the Krishna Yajur-Veda.  Though, Baudhāyana did not wrote proof to his theorem, he laid out the sūtra as follows:

dīrgha chaturasrasya akṣaṇayā rajjuḥ pārśvamānī tiryagmānī cha
yat pṛthagbhūte kurutah tat ubhayāṅ karoti. (Chapter 1, sutra 12)

A rope stretched along the diagonal of a rectangle makes a squared length which is made by the squared lengths of the horizontal and vertical sides of the rectangle together.

Other important concepts contained in Śulbasūtras are as follows:
1) Pythagorean triples.
2) Formula to find square roots.
3) Finding a circle whose area is same as a square.
4) Diagonals of rectangle bisecting each other.
5) Diagonals of rhombus bisecting at right angles.
6) Areas associated with squares, rectangles and rhombus.
7) Methodology to handle fractions.

Further reading:
http://www.math.cornell.edu/~dwh/papers/sulba/sulba.html
http://www-history.mcs.st-and.ac.uk/Projects/Pearce/Chapters/Ch4_2.html

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