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| {{Hinduism}}
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| '''Baudhāyana''', (fl. c. 800 BCE)<ref name="Baudhayana">O'Connor, "Baudhayana".</ref> was the author of the Baudhayana sūtras, which cover dharma, daily ritual, Vedic sacrifices, etc. He belongs to the [[Yajurveda]] school, and is older than the other sūtra author [[Apastamba|Āpastambha]]. | |
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| He was the author of the earliest ''[[Sulba Sutras|Sulba Sūtra]]''—appendices to the [[Vedas]] giving rules for the construction of [[altars]]—called the ''{{IAST |Baudhāyana Śulbasûtra}}''. These are notable from the point of view of mathematics, for containing several important mathematical results, including giving a value of [[pi]] to some degree of precision, and stating a version of what is now known as the [[Pythagorean theorem]].
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| ==The sūtras of Baudhāyana==
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| The {{IAST|Sûtras}} of {{IAST|Baudhāyana}} are associated with the ''Taittiriya'' {{IAST|Śākhā}} (branch) of Krishna (black) ''[[Yajurveda]]''. The sutras of {{IAST|Baudhāyana}} have six sections,
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| # the [[Srautasutra|{{IAST|Śrautasûtra}}]], probably in 19 {{IAST|Praśnas}} (questions),
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| # the {{IAST|Karmāntasûtra}} in 20 {{IAST|Adhyāyas}} (chapters),
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| # the {{IAST|Dvaidhasûtra}} in 4 {{IAST|Praśnas}},
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| # the [[Grhyasutra|Grihyasutra]] in 4 {{IAST|Praśnas}},
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| # the [[Dharmasutra|{{IAST|Dharmasûtra}}]] in 4 {{IAST|Praśnas}} and
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| # 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>
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| ===The Shrautasūtra=== | | '''MathML''' |
| {{main|Baudhayana Shrauta Sutra}}
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| His [[Śrauta|shrauta]] [[sūtra]]s related to performing [[Vedas|Vedic]] [[sacrifice]]s has followers in some [[Smartism|Smārta]] [[brāhmaṇa]]s ([[Iyers]]) and some [[Iyengar]]s and [[Gounder|Kongu]] of [[Tamil Nadu]], [[Yajurveda|Yajurvedis]] or [[Namboothiri]]s of [[Kerala]], Gurukkal Brahmins, 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.
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| ===The Dharmasūtra===
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| The Dharmasūtra of Baudhāyana like that of [[Apastamba]] also forms a part of the larger [[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>
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| ====Authorship and Dates====
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| Āpastamba and Baudhāyana come from the [[Taittiriya]] branch vedic school dedicated to the study of the [[Black Yajurveda]]. [[Robert Lingat]] states that Baudhāyana was the first to compose the Kalpasūtra collection of the Taittiriya school followed by Āpastamba.<ref>Robert Lingat, The Classical Law of India, (Munshiram Manoharlal Publishers Pvt Ltd, 1993), p.20</ref> Kane assigns this Dharmasūtra an approximate date between 500 to 200 BC.<ref name="Patrick Olivelle 1999">Patrick Olivelle, Dharmasūtras: The Law Codes of Ancient India, (Oxford World Classics, 1999), p.xxxi</ref>
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| ====Commentaries==== | | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
| There are no commentaries on this Dharmasūtra with the exception of [[Govindasvāmin]]'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"/>
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| ====Organization and Contents==== | | ==Demos== |
| 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" />
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| 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), p.128-131</ref>
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| ==The mathematics in Sulbasūtra==
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| ===Pythagorean theorem===
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| The most notable of the rules (the Sulbasūtra-s do not contain any proofs of the rules which they describe, since they are sūtra-s, formulae, concise) in the ''Baudhāyana Sulba Sūtra'' says:
| | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| <blockquote>
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| ''dīrghasyākṣaṇayā rajjuḥ pārśvamānī, tiryaḍam mānī,''<br>
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| ''cha yatpṛthagbhūte kurutastadubhayāṅ karoti.''<br>
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| :A rope stretched along the length of the [[diagonal]] produces an [[area]] which the vertical and horizontal sides make together.''{{Citation needed|date=November 2013}} | | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| </blockquote>
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| This appears to be referring to a rectangle, although some interpretations consider this to refer to a [[Square (geometry)|square]]. In either case, it states that the square of the [[hypotenuse]] equals the sum of the squares of the sides. If restricted to right-angled isosceles triangles, however, it would constitute a less general claim, but the text seems to be quite open to unequal sides.
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| If this refers to a rectangle, it is the earliest recorded statement of the [[Pythagorean theorem]].
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| Baudhāyana also provides a non-axiomatic demonstration using a rope measure of the reduced form of the Pythagorean theorem for an isosceles [[right triangle]]:
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| :''The cord which is stretched across a square produces an area double the size of the original square.''
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| ===Circling the square===
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| 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:
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| :''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. ''
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| Explanation:
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| *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>.
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| * 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>.
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| ===Square root of 2===
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| Baudhāyana i.61-2 (elaborated in Āpastamba Sulbasūtra i.6)
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| 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:
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| :''samasya dvikaraṇī. pramāṇaṃ tṛtīyena vardhayet <br> tac caturthenātmacatustriṃśonena saviśeṣaḥ''
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| : 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}}
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| That is,
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| :<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>
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| which is correct to five decimals.<ref name="Baudhayana"/>
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| Other theorems include: diagonals of rectangle bisect each other,
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| diagonals of rhombus bisect at right angles, area of a square formed
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| by joining the middle points of a square is half of original, the
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| midpoints of a rectangle joined forms a rhombus whose area is half the
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| rectangle, etc.
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| Note the emphasis on rectangles and squares; this arises from the need
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| to specify ''yajña bhūmikā''s—i.e. the altar on which a rituals were
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| conducted, including fire offerings (yajña).
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| [[Āpastamba]] (c. 600 BC) and [[Kātyāyana]] (c. 200 BC), authors of other sulba sūtras, extend some of Baudhāyana's ideas. Āpastamba provides a more general proof{{Citation needed|date=February 2007}} of the Pythagorean theorem.
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| ==Notes==
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| <div class="references-small">
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| <references/>
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| ==See also==
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| *[[Indian mathematics]]
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| *[[Indian mathematicians]]
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| *[[Sulba Sutras]]
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| ==References== | |
| * George Gheverghese Joseph. ''The Crest of the Peacock: Non-European Roots of Mathematics'', 2nd Edition. [[Penguin Books]], 2000. ISBN 0-14-027778-1.
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| * Vincent J. Katz. ''A History of Mathematics: An Introduction'', 2nd Edition. [[Addison-Wesley]], 1998. ISBN 0-321-01618-1
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| * [[S. Balachandra Rao]], ''Indian Mathematics and Astronomy: Some Landmarks''. Jnana Deep Publications, Bangalore, 1998. ISBN 81-900962-0-6
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| * {{MacTutor Biography|id=Baudhayana}} [[St Andrews University]], 2000.
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| * {{MacTutor Biography|id=Indian_sulbasutras|title=The Indian Sulbasutras|class=HistTopics}} St Andrews University, 2000.
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| * Ian G. Pearce. [http://turnbull.mcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch4_2.html ''Sulba Sutras''] at the [[MacTutor archive]]. St Andrews University, 2002.
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| </div>
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| *B.B.DUTTA."THE SCIENCE OF THE SULBA".
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| {{Indian mathematics}}
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| {{DEFAULTSORT:Baudhayana}}
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| [[Category:Ancient Indian mathematicians]]
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| [[Category:Pi]]
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| [[Category:Indian_mathematics]]
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