Baudhayana

Baudhāyana, (fl. ca. 800 BCE)cite web
last = O'Connor
first = J J
authorlink =
coauthors =E F Robertson
title =Baudhayana
work =
publisher = School of Mathematics and Statistics, University of St Andrews, Scotland
month = November | year = 2000
url =http://www-history.mcs.st-and.ac.uk/~history/Biographies/Baudhayana.html
format =
doi =
accessdate =2007-06-09 ] was an Indian mathematician, whowas 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 "IAST |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 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, 1. the IAST|Śrautasûtra, probably in 19 IAST|Praśnas (chapters), 2. the IAST|Karmāntasûtra in 20 IAST|Adhyāyas (chapters), 3. the IAST|Dvaidhasûtra in 4 IAST|Praśnas, 4. the Grihyasutra in 4 IAST|Praśnas, 5. the IAST|Dharmasûtra in 4 IAST|Praśnas and 6. the IAST|Śulbasûtra in 3 IAST|Adhyāyas [ [http://www.sacred-texts.com/hin/sbe14/sbe1403.htm "Sacred Books of the East", vol.14 – Introduction to Baudhayana] ] .

The Shrautasutra

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 IAST |"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 $x\; =\; \{a\; over\; 2\}sqrt\{2\}-\; \{a\; over\; 2\}$.
*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 $a^2\; imes\; \{pi\; over\; 4\}\; imes\; \{11.66\; over\; 9\}$ which is about $a^2$.

quare 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"

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

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 formedby joining the middle points of a square is half of original, themidpoints of a rectangle joined forms a rhombus whose area is half therectangle, etc.

Note the emphasis on rectangles and squares; this arises from the needto specify "yajNa bhUmikA"s -- i.e. the altar on which a rituals wereconducted, 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 proofFact|date=February 2007 of the Pythagorean theorem.

Notes

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 8190096206
* St Andrews University, 2000.
* J. J. O'Connor and E. F. Robertson. [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html "The Indian Sulbasutras"] at the MacTutor archive. St Andrews University, 2000.
* 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.

ee also

*Indian mathematics
*Indian mathematicians
*Sulba Sutras