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{{Hinduism}}
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The '''{{IAST|Baudhāyana sūtras}}''' 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.<ref name="Plofker 27">{{cite book | first = Kim | last = Plofker | year = 2007 | title-link=Mathematics in India| title=Mathematics in India | page=[https://books.google.com/books?id=DHvThPNp9yMC&q=Is+also+rather+vague&pg=PA17 17]| isbn = 978-0691120676 }}. In relative chronology, they predate [[Apastamba|Āpastamba]], 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</ref>

The Baudhayana sūtras consist of six texts:
# the [[Baudhayana Shrauta Sutra|{{IAST|Śrautasûtra}}]], probably in 19 {{IAST|Praśnas}} (questions),
# the {{IAST|Karmāntasûtra}} in 20 {{IAST|Adhyāyas}} (chapters),
# the {{IAST|Dvaidhasûtra}} in 4 {{IAST|Praśnas}},
# the [[Grhyasutra|Grihyasutra]] in 4 {{IAST|Praśnas}},
# the [[Dharmasutra|{{IAST|Dharmasûtra}}]] in 4 {{IAST|Praśnas}} and
# the [[Sulba Sutras|{{IAST|Śulbasûtra}}]] in 3 {{IAST|Adhyāyas}}.<ref>[http://www.sacred-texts.com/hin/sbe14/sbe1403.htm ''Sacred Books of the East'', vol.14 – Introduction to Baudhayana]</ref>

The '''''{{IAST |Baudhāyana Śulbasûtra}}''''' is noted for containing several early mathematical results, including an approximation of the [[square root of 2]] and the statement of the [[Pythagorean theorem]].<ref>{{citation |first=Meera |last=Nanda |title=Hindutva's science envy |url=http://www.frontline.in/science-and-technology/hindutvas-science-envy/article9049883.ece |newspaper=Frontline |date=16 September 2016 |access-date=14 October 2016}}</ref>

==Baudhāyana Shrautasūtra==
{{main|Baudhayana Shrauta Sutra}}
His [[Śrauta|shrauta]] sūtras related to performing [[Vedas|Vedic]] [[sacrifice]]s has followers in some [[Smartism|Smārta]] [[brāhmaṇa]]s ([[Iyers]]) and some [[Iyengar]]s of [[Tamil Nadu]], [[Yajurveda|Yajurvedis]] or [[Namboothiri]]s of [[Kerala]], Gurukkal Brahmins (Aadi Saivas), among others. The followers of this sūtra follow a different method and do 24 Tila-tarpaṇa, as Lord [[Krishna]] had done tarpaṇa on the day before [[Amavasya|amāvāsyā]]; they call themselves Baudhāyana Amavasya.

==Baudhāyana Dharmasūtra==
The Dharmasūtra of Baudhāyana like that of [[Apastamba]] also forms a part of the larger [[Kalpa (Vedanga)|Kalpasutra]]. Likewise, it is composed of ''[[praśna]]s'' which literally means 'questions' or books. The structure of this Dharmasūtra 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 ''praśnas'' consist of the [[Srautasutra]] and other ritual treatises, the Sulvasutra which deals with vedic geometry, and the [[Grhyasutra]] which deals with domestic rituals.<ref name="Patrick Olivelle 1999 p.127">Patrick Olivelle, Dharmasūtras: The Law Codes of Ancient India, (Oxford World Classics, 1999), p. 127</ref>

There are no commentaries on this Dharmasūtra with the exception of [[Govindasvāmi]]n's ''Vivaraṇa''. 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 Āpastamba and Gautama.<ref name="Patrick Olivelle 1999">Patrick Olivelle, Dharmasūtras: The Law Codes of Ancient India, (Oxford World Classics, 1999), p. xxxi</ref>

This Dharmasūtra is divided into four books. Olivelle states that Book One and the first sixteen chapters of Book Two are the 'Proto-Baudhayana'<ref name="Patrick Olivelle 1999 p.127" /> even though this section has undergone alteration. Scholars like Bühler and Kane agree that the last two books of the Dharmasūtra are later additions. Chapter 17 and 18 in Book Two lays emphasis on various types of ascetics and acetic practices.<ref name="Patrick Olivelle 1999 p.127" />

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.<ref>Patrick Olivelle, Dharmasūtras: The Law Codes of Ancient India, (Oxford World Classics, 1999), pp. 128–131</ref>

==Baudhāyana Sulbasūtra==

===Pythagorean theorem===
The ''Baudhāyana Sulba Sūtra'' 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.<ref>*{{cite conference
| last=Høyrup
| first=Jens
| contribution=Pythagorean ‘Rule’ and ‘Theorem’ – Mirror of the Relation Between Babylonian and Greek Mathematics
| pages=393–407
| year=1998
| editor-last=Renger
| editor-first=Johannes
| title=Babylon: Focus mesopotamischer Geschichte, Wiege früher Gelehrsamkeit, Mythos in der Moderne. 2. Internationales Colloquium der Deutschen Orient-Gesellschaft 24.–26. März 1998 in Berlin
| publisher=Berlin: Deutsche Orient-Gesellschaft / Saarbrücken: SDV Saarbrücker Druckerei und Verlag
| url=http://akira.ruc.dk/~jensh/Publications/Pythrule.pdf
}}</ref> For the most part, the Sulbasūtra-s do not contain proofs of the rules which they describe. The rule stated in the ''Baudhāyana Sulba Sūtra'' is:

<blockquote>
दीर्घचतुरश्रस्याक्ष्णया रज्जु: पार्श्र्वमानी तिर्यग् मानी च यत् पृथग् भूते कुरूतस्तदुभयं करोति ॥ 
''dīrghachatursrasyākṣaṇayā rajjuḥ pārśvamānī, tiryagmānī,'' 
''cha yatpṛthagbhūte kurutastadubhayāṅ karoti.'' 
:A rope stretched along the length of the [[diagonal]] produces an [[area]] which the vertical and horizontal sides make together.<ref>[[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 </ref></blockquote>

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]].

Baudhāyana 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 Baudhāyana is that of finding a circle whose area is the same as that of a square (the reverse of [[squaring the circle]]). His sūtra 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 <math>x = {a \over 2}\sqrt{2}- {a \over 2}</math>.
*Then draw a circle with radius <math>{a \over 2} + {x \over 3}</math>, or <math>{a \over 2} + {a \over 6}(\sqrt{2}-1)</math>, which equals <math>{a \over 6}(2 + \sqrt{2})</math>.
* Now <math>(2+\sqrt{2})^2 \approx 11.66 \approx {36.6\over \pi}</math>, so the area <math>{\pi}r^2 \approx \pi \times {a^2 \over 6^2} \times {36.6\over \pi} \approx a^2</math>.

===Square root of 2===
Baudhāyana i.61-2 (elaborated in Āpastamba Sulbasūtra 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 dvikaraṇī. pramāṇaṃ tṛtīyena vardhayet tac caturthenātmacatustriṃśonena saviśeṣaḥ''

: 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.{{citation needed|date=December 2012}}

That is,

:<math>\sqrt{2} \approx 1 + \frac{1}{3} + \frac{1}{3 \cdot 4} - \frac{1}{3 \cdot4 \cdot 34} = \frac{577}{408} \approx 1.414216,</math>

which is correct to five decimals.<ref name="Baudhayana">O'Connor, "Baudhayana".</ref>

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 bhūmikā''s—i.e. the altar on which a rituals were conducted, including fire offerings (yajña). This is an aspect of [[Vastu Shastra|Vaastu Shastras]] and [[Shilpa Shastras]]. These theorems are derived from those texts.{{citation needed|reason=Most discussions of mathematical works in Sanskrit start with the Baudhāyana Sulbasūtra, and I have never heard that it is known to have been derived from any earlier work. Is there a reliable source that says this?|date=February 2020}}

==See also==
*[[Indian mathematics]]
*[[List of Indian mathematicians]]

==Notes==
{{reflist}}

==References==
* George Gheverghese Joseph. ''The Crest of the Peacock: Non-European Roots of Mathematics'', 2nd Edition. [[Penguin Books]], 2000. {{ISBN|0-14-027778-1}}.
* Vincent J. Katz. ''A History of Mathematics: An Introduction'', 2nd Edition. [[Addison-Wesley]], 1998. {{ISBN|0-321-01618-1}}
* [[S. Balachandra Rao]], ''Indian Mathematics and Astronomy: Some Landmarks''. Jnana Deep Publications, Bangalore, 1998. {{ISBN|81-900962-0-6}}
* {{MacTutor Biography|id=Baudhayana}} [[St Andrews University]], 2000.
* {{MacTutor Biography|id=Indian_sulbasutras|title=The Indian Sulbasutras|class=HistTopics}} St Andrews University, 2000.
* Ian G. Pearce. [https://web.archive.org/web/20131005003318/http://turnbull.mcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch4_2.html ''Sulba Sutras''] at the [[MacTutor archive]]. St Andrews University, 2002.
* B.B. Dutta."The Science of the Shulba".

== External links ==
* "The Śulvasútra of Baudháyana, with the commentary by Dvárakánáthayajvan", translated by [[George Thibaut]], was published in a series of issues of ''The Pandit. A Monthly Journal, of the Benares College, devoted to Sanskrit Literature'':
** (1875) '''9''' [https://books.google.com/books?id=KMAIAAAAQAAJ&pg=PA292#v=onepage&q&f=false (108): 292–298]
** (1875–1876) '''10''' [https://books.google.com/books?id=ICkJAAAAQAAJ&pg=PA17#v=onepage&q&f=false (109): 17–22], [https://books.google.com/books?id=ICkJAAAAQAAJ&pg=PA44#v=onepage&q&f=false (110): 44–50], [https://books.google.com/books?id=ICkJAAAAQAAJ&pg=PA72#v=onepage&q&f=false (111): 72–74], [https://books.google.com/books?id=ICkJAAAAQAAJ&pg=PA139#v=onepage&q&f=false (114): 139–146], [https://books.google.com/books?id=ICkJAAAAQAAJ&pg=PA166#v=onepage&q&f=false (115): 166–170], [https://books.google.com/books?id=ICkJAAAAQAAJ&pg=PA186#v=onepage&q&f=false (116): 186–194], [https://books.google.com/books?id=ICkJAAAAQAAJ&pg=PA209#v=onepage&q&f=false (117): 209–218]
** (new series) (1876–1877) '''1''' [https://books.google.com/books?id=jHxFAQAAIAAJ&pg=PA316#v=onepage&q&f=false (5): 316–322], [https://books.google.com/books?id=jHxFAQAAIAAJ&pg=PA556#v=onepage&q&f=false (9): 556–578], [https://books.google.com/books?id=jHxFAQAAIAAJ&pg=PA626#v=onepage&q&f=false (10): 626–642], [https://books.google.com/books?id=jHxFAQAAIAAJ&pg=PA692#v=onepage&q&f=false (11): 692–706], [https://books.google.com/books?id=jHxFAQAAIAAJ&pg=PA761#v=onepage&q&f=false (12): 761–770]

{{Indian mathematics}}

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[[Category:Ancient Indian mathematicians]]
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[[Category:Ancient Indian mathematical works]]

Baudhayana sutras - Revision history

>RG067 at 11:48, 9 October 2021