Baudhāyana, (fl. c. 800 BCE)[1] was an Indian mathematician, who was most likely also a priest. He is noted as the author of the earliest Sulba Sūtra—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 the other famous mathematician Āpastambha. He belongs to the Yajurveda school.
He is accredited with calculating the value of pi to some degree of precision, and with discovering what is now known as the Pythagorean theorem.
[ The sūtras of Baudhāyana
[ The Shrautasūtra
His shrauta sūtras related to performing Vedic sacrifices has followers in some Smārta brāhmaṇas (Iyers) and some Iyengars of Tamil Nadu, Kongu of Tamil nadu, Yajurvedis or Namboothiris of Kerala, Gurukkal brahmins, among others. The followers of this sūtra follow a different method and do 24 Tila-tarpaṇa, as Lord Kṛiṣhṇa had done tarpaṇa on the day before Amāvasyā; they call themselves Baudhāyana Amāvasya.
[ The Dharmasūtra
The Vivaraṇa of Govindasvami is an important commentary on the Dharmasūtra.
[ The mathematics in Sulbasūtra
[ Pythagorean theorem
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:
dīrghasyākṣaṇayā rajjuḥ pārśvamānī, tiryaḍam mānī,
cha yatpṛthagbhūte kurutastadubhayāṅ karoti.
- A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together.
Bodhayana also states that if a and b be the two sides and c be the hypotenuse, such that 'a' is divisible by 4( as in all pythogorean triplets one of the two shorter sides ateast is divisible by 4).Now, c = (a - a/8) + b/2 This method makes us solve without using squares and square roots.
This appears to be referring to a rectangle or a square(in some cases as interpreted by some people), 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.
Baudhāyana 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 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
.
- Then draw a circle with radius
, or 
, which equals 
.
- Now
, so the area 
.
[ 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 thirty-fourth. That is its diagonal approximately.
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 yajña bhūmikās—i.e. the altar on which a rituals were conducted, including fire offerings (yajña).
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] of the Pythagorean theorem.
[ Notes
- ^ O'Connor, "Baudhayana".
[ 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
- O'Connor, John J.; Robertson, Edmund F., "Baudhayana", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Baudhayana.html . St Andrews University, 2000.
- O'Connor, John J.; Robertson, Edmund F., "The Indian Sulbasutras", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_sulbasutras.html . St Andrews University, 2000.
- Ian G. Pearce. Sulba Sutras at the MacTutor archive. St Andrews University, 2002.
[ See also
