India’s Contribution to GeometryShulba Sutras (Geometrical constructions)

ByDr. Bina Das

L.A.D. College for Women, Nagpur

GEOMETRICAL CONSTRUCTIONS

• The Shulba Sutras are part of the larger corpus of texts called the Shrauta Sutras considered to be appendices to the Vedas.

• They are the only sources of knowledge of Indian Mathematics from the Vedic period.

• The four major Shulba Sutras, which are mathematically the most significant, are those composed by Baudhayana, Manava, Apastamba and Katyayana, about whom very little is known.

• The text have been dated from around 800 BCE to 200 CE, with the oldest being the sutra that was attributed to Baudhayana around 800 BCE to 600 BCE.

• According to the theory of the ritual origins of geometry, different shapes symbolized different religious ideas.

• The Baudhayana Shulba sutra gives the construction of geometric shapes such as squares and rectangles. It also gives, sometimes approximate, geometric area-preserving transformations from one geometric shape to another.

Baudhāyana, (800 BC), a great hindu mathematician and author of Baudhāyana Śulbasûtra that gave life to geometry

Baudhayana• Baudhayana(800 BCE) was an Indian Mathematician, who

was most likely also a priest. He is noted as the author of the earliest Shulba Sutra—appendices to the Vedas giving rules for the construction of altars—called the Baudhayana Sulbasutra, which contained several important mathematical results. He is older than the other famous mathematician Apastamba. He belongs to the Yajurveda school.

• He is accredited with calculating the value of ‘pi' before Pythagoras, and with discovering what is known as the Pythagorean theorem.

In Baudhayana the rules are given as follows: (i) The diagonal of a square produces double the area (of the square). (ii) The areas (of the squares) produced separately by the lengths of the breadth of a rectangle together equal the area (of the square) produced by the diagonal. (iii) This is observed in rectangles having sides 3 and 4, 12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 36.

Squaring the circle

Squaring the circle: the areas of this square and this circle are equal.

In 1882, it was proven that this figure cannot be constructed in a finite

number of steps with an idealized compass and straightedge.

Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.

In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi (π) is a transcendental, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients.

So the area

Now

Half-diagonal of the square is larger than the half-side by

radius = or , which equals

O

Area of the circle = 1.019 a2

= a2 + (.019) a2

Thus the area of circle is slightly larger than that of square by a2(.019). The

reason of this inequality is that the value of π is irrational (and not exact).

Baudhayana gives a better result. He says: if sides of square is ‘a’ and

diameter of approximate circle is ‘d’ then

How he has obtained this result is a surprise?

A K L B

D M E C

H F G

Conversion of a rectangle into a square.

ABCD is a rectangle , AKMD is a squareALGH is a required square

Turning a rectangle into a square by dissection

Consider a rectangle with sides a and b, such that, a < b < 2*a. We show here one way to cut it into three pieces and rearrange them into a square.

• Compute, s = √(a*b). Open a compass to length s, put a pin in corner A of the rectangle, and mark point X on side CD. The distance AX is s. Draw the line AX. Draw another line (using an index card) BY, perpendicular to AX.

• Cut out the rectangle ABDC, cut the line AX, and cut the line YB.

• In order to get a square in slanted position, move the left triangle to the right, and move the top triangle to the bottom right.

References

1. Baudhayana, wiki pedia the free encyclopedia.

2. Shulbha sutras, wiki pedia the free encyclopedia.

3. Squaring the circle, wikipedia the free encyclopedia.

4.Turning a rectangle into a square by dissection

http://sofia.nmsu.edu/~breakingaway/Lessons/R2S/R2S.html