NationMaster - Encyclopedia: Baudhayana

Baudhāyana, (fl. ca. 800 BC)^[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 Sulba Sutra, which contained several important mathematical results. He is older than other famous mathematician Apastambha. He belongs to Yajurveda school. The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... Veda redirects here. ... Picture of an altar from the Meyer Encyclopaedia An altar, (Hebrew mizbeah, from a word meaning to slay) is any structure on which sacrifices known as the korbanot or incense offerings are offered for religious purposes. ...

1 Shrauta (Dharma) sutras
2 References
3 See also

Shrauta (Dharma) sutras

He is the author of shrauta sutras related to performing to Vedic sacrifices which 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 Åšrauta refers to the tradition of the Åšruti in Vaidika Dharma. ... SÅ«tra (sex) (Sanskrit) or Sutta (PÄli) 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 a manual. ... Veda redirects here. ... Marcus Aurelius and members of the Imperial family offer sacrifice in gratitude for success against Germanic tribes: contemporary bas-relief, Capitoline Museum, Rome 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... Smartism[1], (or Smarta Sampradaya, Smarta Tradition, as termed in Sanskrit), is a denomination of the Hindu religion. ... A Brahmin (anglicised from the Sanskrit word IAST ; Devanagari ), also known as Vipra, Dvija, Dvijottama (best of the Dvijas), (god on Earth) is the highest caste in Indian caste system within Hindu society. ... Iyer is the name given to a community of Brahmins (members of the priestly class / caste) of India whose members profess the advaita philosophy propounded by sri Shankaracharya and whose ancestors have had strong ties with the Tamil region,for many centuries. ... Tamil Nadu (தமிழ் நாடு, Land of the Tamils) is a state at the southern tip of India. ... The Yajurveda (Sanskrit , a tatpurusha compound of sacrifice + veda knowledge) is one of the four Hindu Vedas. ... The Namboothiris (Malayalam :à´¨à´®àµà´ªàµ‚à´¤à´¿à´°à´¿) are the Brahmins of Kerala, thought to be the most orthodox brahmins in India. ... (IPA: ; , Written as àµ‡à´•à´°à´³à´‚ in the native language Malayalam) is a state on the Western Coast of south-western India. ... // Gurukkal or Sivacharya or Adisaiva Brahmins Orgins A sub-sect of Vadamas not recognized as one amongst them and whose duties are to worship at temples. ... Krishna with Radharani, 18th C Rajasthani painting Krishna (à¤•à¥ƒà¤·à¥à¤£ in Devanagari, in IAST ), according to various Hindu traditions, is the eighth avatar of Vishnu. ...

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. In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and matrix terminology. ... Area is a physical quantity expressing the size of a part of a surface. ... An object is in a vertical position when it is aligned in an up-down direction, perpendicular to the horizon. ... Horizontal is an orientation relating to, or in parallel with the horizon, and thus perpendicular to the vertical. ... In plane (Euclidean) geometry, a square is a polygon with four equal sides, four right angles, and parallel opposite sides. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

If this refers to a rectangle, it is the earliest recorded statement of the Pythagorean theorem. In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry among the three sides of a right triangle. ...

Baudhayana also provides a non-axiomatic demonstration using a rope measure of the reduced form of the Pythagorean theorem for an isosceles right triangle: For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

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

In plane (Euclidean) geometry, a square is a polygon with four equal sides, four right angles, and parallel opposite sides. ...

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: This square and circle have the same area. ...

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 $x = {a over 2}sqrt{2}- {a over 2}$ .
Then draw a circle with radius ${a over 2} + {x over 3}$ , or ${a over 2} + {a over 6}(sqrt{2}-1)$ , which equals ${a over 6}(2 + sqrt{2})$ .
Now $(2+sqrt{2})^2 = 11.66 approx {36over pi}$ , so this turns out to be $a^2 times {pi over 4} times {11.66 over 9}$ which is about $a 2$ .

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.

$sqrt{2} = 1 + frac{1}{3} + frac{1}{3 cdot 4} - frac{1}{3 cdot4 cdot 34} = frac{577}{408} approx 1.414216$

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^{[citation needed]} of the Pythagorean theorem. Apastamba (c. ... Centuries: 8th century BC - 7th century BC - 6th century BC Decades: 650s BC 640s BC 630s BC 620s BC 610s BC - 600s BC - 590s BC 580s BC 570s BC 560s BC 550s BC Events and Trends Fall of the Assyrian Empire and Rise of Babylon 609 BC _ King Josiah... Katyayana was probably a priest who lived in India around 200 BC. Like Baudhayana, he composed Shulba Sutra, or sacred mathematical texts. ... Centuries: 3rd century BC - 2nd century BC - 1st century BC Decades: 250s BC 240s BC 230s BC 220s BC 210s BC - 200s BC - 190s BC 180s BC 170s BC 160s BC 150s BC Years: 205 BC 204 BC 203 BC 202 BC 201 BC - 200 BC - 199 BC 198 BC...

References

^ 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.

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 8190096206
O'Connor, John J., and Edmund F. Robertson. "Baudhayana". MacTutor History of Mathematics archive. St Andrews University, 2000.
J. J. O'Connor and E. F. Robertson. The Indian Sulbasutras at the MacTutor archive. St Andrews University, 2000.
Ian G. Pearce. Sulba Sutras at the MacTutor archive. St Andrews University, 2002.

2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the Anno Domini (common) era. ... June 9 is the 160th day of the year in the Gregorian calendar (161st in leap years), with 205 days remaining. ... Penguin Books is a British publisher founded in 1935 by Allen Lane. ... Pearson can mean Pearson PLC the media conglomerate. ... University of St Andrews The University of St Andrews was founded between 1410-1413 and is the oldest university in Scotland and the third oldest in the United Kingdom. ... BITCH!111 ... BITCH!111 ...

Contents

Shrauta (Dharma) sutras GA_googleFillSlot("encyclopedia_square");

Pythagorean theorem

Circling the Square

Square root of 2

References

See also

Shrauta (Dharma) sutras