Baudhayana

Baudhāyana, (fl. ca. 800 BCE)[1] was an Indian mathematician, who was most likely also a priest. He is noted as the author of the earliest Sulba Sutra — appendices to the Vedas giving rules for the construction of altars — called the Baudhāyana Śulbasûtra, which contained several important mathematical results. He is older than other famous mathematician Apastambha. He belongs to Yajurveda school.

The sutras of Baudhayana

The Sûtras of Baudhāyana are associated with the Taittiriya Śākhā (branch) of Krishna (black) Yajurveda. The sutras of Baudhāyana have six sections, 1. the Śrautasûtra, probably in 19 Praśnas (chapters), 2. the Karmāntasûtra in 20 Adhyāyas (chapters), 3. the Dvaidhasûtra in 4 Praśnas, 4. the Grihyasutra in 4 Praśnas, 5. the Dharmasûtra in 4 Praśnas and 6. the Śulbasûtra in 3 Adhyāyas[2].

The Shrautasutra

Main article: Baudhayana Shrauta Sutra

His shrauta sutras related to performing to Vedic sacrifices has followers in some Smartha brahmins (Iyers)And some iyengars of Tamil Nadu, Yajurvedis or Namboothiris of Kerala, Gurukkal brahmins, among others. The followers of this sutra follow different method and do 24 thilatharpanam which his because of lord krishna who had done tharpanam on the day before amavasaya and they call themself as baudhayana amavasaya

The Dharmasutra

The Vivarana of Govindasvami is an important commentary on the Dharmasûtra.

The mathematics in Shulbasutra

Pythagorean theorem

The most notable of the rules (the Sulbasutras do not contain any proofs of the rules which they describe) in the Baudhāyana Sulba Sutra says:
dīrghasyākṣaṇayā rajjuH pārśvamānī, tiryaDaM mānī,
cha yatpṛthagbhUte kurutastadubhayāṅ karoti.
A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together.''


This appears to be referring to a rectangle, although some interpretations consider this to refer to a 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.

If this refers to a rectangle, it is the earliest recorded statement of the Pythagorean theorem.

Baudhayana also provides a non-axiomatic demonstration 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 Baudhayana is that of finding a circle whose area is the same as that of a square (the reverse of squaring the circle). His sutra 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 .
  • Then draw a circle with radius , or , which equals .
  • Now , so this turns out to be which is about .

Square root of 2

Baudhayana i.61-2 (elaborated in Apastamba Sulbasutra i.6) gives this formula for square root of two:

''samasya dvikaraNI. pramANaM tritIyena vardhayet
tachchaturthAnAtma chatusastriMshenena savisheShaH.


Translation Requested



which is correct to five decimals.

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 yajNa bhUmikAs -- i.e. the altar on which a rituals were conducted, including fire offerings (yajNa).

Apastamba (c. 600 BC) and Katyayana (c. 200 BC), authors of other sulba sutras, extend some of Baudhayana's ideas. Apastamba provides a more general proof of the Pythagorean theorem.

Notes

1. ^ O'Connor, J J ; E F Robertson (November 2000 ). Baudhayana. School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved on 2007-06-09.
2. ^ Sacred Books of the East, vol.14 – Introduction to Baudhayana

References

See also

The Shulba Sutras (Sanskrit śulba
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Vedas (Sanskrit véda वेद
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worldwide view of the subject.
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An altar is any structure upon which sacrifices or other offerings are offered for religious purposes, or some other sacred place where ceremonies take place.
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Apastamba (fl. ca. 5th century BCE), was an Indian priest/scholar associated with Krishna (Black) Yajur Veda (the others being Baudhayana, Vaikhanasa, Satyasadha, Bharadhvaja and Agnivesa).
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The Yajurveda (Sanskrit यजुर्वेदः
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The Baudhayana Shrauta Sutra (BSS) is a Hindu text. It was probably written in eastern Uttar Pradesh. It belongs to the Black Yajurveda. It was first published in English in 1904-23 by the Asiatic Society of Bengal.
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Sūtra (Sanskrit, Devanagari सूत्र) or Sutta (Pāli), literally means a rope or thread that holds things together, and more metaphorically refers to an aphorism (or line, rule, formula), or a collection of such aphorisms in the form of
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Vedas (Sanskrit véda वेद
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Sacrifice (from a Middle English verb meaning "to make sacred", from Old French, from Latin sacrificium: sacer, sacred; sacred + facere, to make) is commonly known as the practice of offering food, or the lives of animals or people to the gods, as an act of propitiation or worship.
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Smartism (or Smarta Sampradaya, Smarta Tradition, as termed in Sanskrit), a newly-coined term derived from the word smarta by the shaivite guru Sivaya Subramuniyaswami[1], refers to a denomination of the Hindu religion.
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brahmin moths, see Brahmaeidae.


The term Brahmin denotes both a member of the priestly class in the Hindu varna system, and a member of the highest caste in the caste system of Hindu society.
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Iyengar (also Ayyangar, Aiyangar, Ayengar, Iengar) is the name of a community of Tamil Brahmins of South India whose members subscribe to the Visishtadvaita philosophy codified by Ramanuja.
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Tamil Nadu (Tamil: தமிழ்நாடு pronunciation  
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The Yajurveda (Sanskrit यजुर्वेदः
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The Namboothiris (Malayalam :നമ്പൂതിരി) are the upper class Brahmins of Kerala, who consider themselves the most orthodox Brahmins in India. They perform Pooja in temples of Kerala based on Tantra Vidhi.
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Kerala (/span>]] ?· i ; Malayalam: ; Kēraḷaṁ) is a state on the Malabar Coast of southwestern India.
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Gurukkal or Sivacharya or Adisaiva Brahmins are a sub-sect of Vadamas not recognized as one amongst them and whose duties are to worship at temples. Gurukkal is a corrupt form of Gurukul is a sect of Brahmins who worship in the temples of Lord Shiva.
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Krishna (कृष्ण in Devanagari, kṛṣṇa
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diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. The word "diagonal" was originally from the Greek διαγωνιος (diagonios
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Area is a physical quantity expressing the size of a part of a surface. The term Surface area is the summation of the areas of the exposed sides of an object.

Units

Units for measuring surface area include:
square metre = SI derived unit

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Horizontal may refer to:
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In plane (Euclidean) geometry, a square is circle with four sides.

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A square is a regular quadrilateral. Likewise it is also a special case of a rhombus, kite, parallelogram, and trapezoid.
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hypotenuse of a right triangle is the triangle's longest side; the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares
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In mathematics, the Pythagorean theorem (AmE) or Pythagoras' theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and
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A triangle is one of the basic shapes of geometry: a polygon with three corners or and three sides or edges which are straight line segments.

In Euclidean geometry any three non-collinear points determine a triangle and a unique plane, i.e.
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In plane (Euclidean) geometry, a square is circle with four sides.

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A square is a regular quadrilateral. Likewise it is also a special case of a rhombus, kite, parallelogram, and trapezoid.
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Squaring the circle is a problem proposed by ancient geometers. It is the challenge to construct a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.
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Apastamba (fl. ca. 5th century BCE), was an Indian priest/scholar associated with Krishna (Black) Yajur Veda (the others being Baudhayana, Vaikhanasa, Satyasadha, Bharadhvaja and Agnivesa).
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