Baudhayana
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Baudhayana

The Baudh?yana s?tras are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics, etc. They belong to the Taittiriya branch of the Krishna Yajurveda school and are among the earliest texts of the genre, perhaps compiled in the 8th to 6th centuries BCE.[1]

The Baudhayana s?tras consist of six texts:

1. the ?rautasûtra, probably in 19 Pra?nas (questions),
2. the Karm?ntasûtra in 20 Adhy?yas (chapters),
3. the Dvaidhasûtra in 4 Pra?nas,
4. the Grihyasutra in 4 Pra?nas,
5. the Dharmasûtra in 4 Pra?nas and
6. the ?ulbasûtra in 3 Adhy?yas.[2]

The Baudh?yana ?ulbasûtra is noted for containing several early mathematical results, including an approximation of the square root of 2 and the statement of the Pythagorean theorem.[3]

## Baudh?yana Shrautas?tra

His shrauta s?tras related to performing Vedic sacrifices has followers in some Sm?rta br?hma?as (Iyers) and some Iyengars of Tamil Nadu, Yajurvedis or Namboothiris of Kerala, Gurukkal Brahmins (Aadi Saivas), among others. The followers of this s?tra follow a different method and do 24 Tila-tarpa?a, as Lord Krishna had done tarpa?a on the day before am?v?sy?; they call themselves Baudh?yana Amavasya.

## Baudh?yana Dharmas?tra

The Dharmas?tra of Baudh?yana like that of Apastamba also forms a part of the larger Kalpasutra. Likewise, it is composed of pra?nas which literally means 'questions' or books. The structure of this Dharmas?tra is not very clear because it came down in an incomplete manner. Moreover, the text has undergone alterations in the form of additions and explanations over a period of time. The pra?nas consist of the Srautasutra and other ritual treatises, the Sulvasutra which deals with vedic geometry, and the Grhyasutra which deals with domestic rituals.[4]

There are no commentaries on this Dharmas?tra with the exception of Govindasv?min's Vivara?a. The date of the commentary is uncertain but according to Olivelle it is not very ancient. Also the commentary is inferior in comparison to that of Haradatta on ?pastamba and Gautama.[5]

This Dharmas?tra is divided into four books. Olivelle states that Book One and the first sixteen chapters of Book Two are the 'Proto-Baudhayana'[4] even though this section has undergone alteration. Scholars like Bühler and Kane agree that the last two books of the Dharmas?tra are later additions. Chapter 17 and 18 in Book Two lays emphasis on various types of ascetics and acetic practices.[4]

The first book is primarily devoted to the student and deals in topics related to studentship. It also refers to social classes, the role of the king, marriage, and suspension of Vedic recitation. Book two refers to penances, inheritance, women, householder, orders of life, ancestral offerings. Book three refers to holy householders, forest hermit and penances. Book four primarily refers to the yogic practices and penances along with offenses regarding marriage.[6]

## Baudh?yana Sulbas?tra

### Pythagorean theorem

The Baudh?yana Sulba S?tra states the rule referred to today in most of the world as the Pythagorean Theorem. The rule was known to a number of ancient civilizations, including also the Greek and the Chinese, and was recorded in Mesopotamia as far back as 1800 BCE.[7] For the most part, the Sulbas?tra-s do not contain proofs of the rules which they describe. The rule stated in the Baudh?yana Sulba S?tra is:

? ? ? ? ? ?

d?rghachatursrasy?k?a?ay? rajju? p?r?vam?n?, tiryagm?n?,

A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together.[8]

The diagonal and sides referred to are those of a rectangle, and the areas are those of the squares having these line segments as their sides. Since the diagonal of a rectangle is the hypotenuse of the right triangle formed by two adjacent sides, the statement is seen to be equivalent to the Pythagorean theorem.

Baudh?yana also provides a statement using a rope measure of the reduced form of the Pythagorean theorem for an isosceles right triangle:

The cord which is stretched across a square produces an area double the size of the original square.

### Circling the square

Another problem tackled by Baudh?yana is that of finding a circle whose area is the same as that of a square (the reverse of squaring the circle). His s?tra i.58 gives this construction:

Draw half its diagonal about the centre towards the East-West line; then describe a circle together with a third part of that which lies outside the square.

Explanation:

• Draw the half-diagonal of the square, which is larger than the half-side by ${\displaystyle x={a \over 2}{\sqrt {2}}-{a \over 2}}$.
• Then draw a circle with radius ${\displaystyle {a \over 2}+{x \over 3}}$, or ${\displaystyle {a \over 2}+{a \over 6}({\sqrt {2}}-1)}$, which equals ${\displaystyle {a \over 6}(2+{\sqrt {2}})}$.
• Now ${\displaystyle (2+{\sqrt {2}})^{2}\approx 11.66\approx {36.6 \over \pi }}$, so the area ${\displaystyle {\pi }r^{2}\approx \pi \times {a^{2} \over 6^{2}}\times {36.6 \over \pi }\approx a^{2}}$.

### Square root of 2

Baudh?yana i.61-2 (elaborated in ?pastamba Sulbas?tra i.6) gives the length of the diagonal of a square in terms of its sides, which is equivalent to a formula for the square root of 2:

samasya dvikara. prama? t?t?yena vardhayet
tac caturthen?tmacatustrionena savi?e?a?
The diagonal [lit. "doubler"] of a square. The measure is to be increased by a third and by a fourth decreased by the 34th. That is its diagonal approximately.[]

That is,

${\displaystyle {\sqrt {2}}\approx 1+{\frac {1}{3}}+{\frac {1}{3\cdot 4}}-{\frac {1}{3\cdot 4\cdot 34}}={\frac {577}{408}}\approx 1.414216,}$

which is correct to five decimals.[9]

Other theorems include: diagonals of rectangle bisect each other, diagonals of rhombus bisect at right angles, area of a square formed by joining the middle points of a square is half of original, the midpoints of a rectangle joined forms a rhombus whose area is half the rectangle, etc.

Note the emphasis on rectangles and squares; this arises from the need to specify yajña bh?mik?s--i.e. the altar on which a rituals were conducted, including fire offerings (yajña). This is an aspect of Vaastu Shastras and Shilpa Shastras. These theorems are derived from those texts.[]

## Notes

1. ^ Plofker, Kim (2007). Mathematics in India. p. 17. ISBN 978-0691120676.. In relative chronology, they predate ?pastamba, which is dated by Robert Lingat to the sutra period proper, between c. 500 to 200 BCE. Robert Lingat, The Classical Law of India, (Munshiram Manoharlal Publishers Pvt Ltd, 1993), p. 20
2. ^ Sacred Books of the East, vol.14 - Introduction to Baudhayana
3. ^ Nanda, Meera (16 September 2016), "Hindutva's science envy", Frontline, retrieved 2016
4. ^ a b c Patrick Olivelle, Dharmas?tras: The Law Codes of Ancient India, (Oxford World Classics, 1999), p. 127
5. ^ Patrick Olivelle, Dharmas?tras: The Law Codes of Ancient India, (Oxford World Classics, 1999), p. xxxi
6. ^ Patrick Olivelle, Dharmas?tras: The Law Codes of Ancient India, (Oxford World Classics, 1999), pp. 128-131
7. ^ *Høyrup, Jens (1998). "Pythagorean 'Rule' and 'Theorem' - Mirror of the Relation Between Babylonian and Greek Mathematics". In Renger, Johannes (ed.). Babylon: Focus mesopotamischer Geschichte, Wiege früher Gelehrsamkeit, Mythos in der Moderne. 2. Internationales Colloquium der Deutschen Orient-Gesellschaft 24.-26. März 1998 in Berlin (PDF). Berlin: Deutsche Orient-Gesellschaft / Saarbrücken: SDV Saarbrücker Druckerei und Verlag. pp. 393-407.
8. ^ Subhash Kak, Pythagorean Triples and Cryptographic Coding, https://arxiv.org/find/all/1/all:+kak/0/1/0/all/0/1?skip=25&query_id=a7b95a2782affe4b
9. ^ O'Connor, "Baudhayana".