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# SHULBA SŪTRAS (VEDĀNGAS)

SHULBA SŪTRAS (VEDĀNGAS) The Shulba Sūtras belong to the Vedāngas, or supplementary texts of the Vedas. Although they are part of the Kalpa Sūtras, which deal with ritual, their importance stems from the constructions they provide for building geometric altars. Their contents, written in the condensed sūtra style, deal with geometrical propositions and problems related to rectilinear figures and their combinations and transformations, squaring the circle, as well as arithmetical and algebraic solutions to these problems. The root shulb means measurement, and the word shulba means a cord, rope, or string.

The extant Shulba Sūtras belong to the schools of the Yajurveda. The most important Shulba texts are the ones by Baudhāyana, Āpastamba, Kātyāyana, and Mānava. They have been generally assigned to the period 800 to 500 b.c. Baudhāyana's text is the oldest, and he is believed to have belonged to the Andhra region. Baudhāyana begins with units of linear measurement and then presents the geometry of rectilinear figures, triangles, and circles, and their transformations from one type to another using differences and combinations of areas. An approximation to the square root of 2 and to p are next given.

Then follow constructions for various kinds of geometric altars in the shapes of the falcon (both rectilinear and with curved wings and extended tail), kite, isosceles triangle, rhombus, chariot wheel with and without spokes, square and circular trough, and tortoise.

In the methods of constructing squares and rectangles, several examples of Pythagorean triples are provided. It is clear from the constructions that both the algebraic and the geometric aspects of the so-called Pythagorean theorem were known. This knowledge precedes its later discovery in Greece. The other theorems in the Shulba include:

The diagonals of a rectangle bisect each other.

The diagonals of a rhombus bisect each other at right angles.

The area of a square formed by joining the middle points of the sides of a square is half of the area of the original one.

A quadrilateral formed by the lines joining the middle points of the sides of a rectangle is a rhombus whose area is half of that of the rectangle.

A parallelogram and rectangle on the same base and within the same parallels have the same area.

If the sum of the squares of two sides of a triangle is equal to the square of the third side, then the triangle is right-angled.

A variety of constructions are listed. Some of the geometric constructions in these texts are based on algebraic solutions of simultaneous equations, both linear and quadratic. It appears that geometric techniques were often used to solve algebraic problems.

The Shulbas are familiar with fractions. Algebraic equations are implicit in many of their rules and operations. For example, the quadratic equation and the indeterminate equation of the first degree are a basis of the solutions presented in the constructions.

The Shulba geometry was used to represent astronomical facts. The altars that were built according to the Shulba rules demonstrated knowledge of the lunar and the solar years.

Subhash Kak