01Geometry for the altar
A Vedic yajna needs altars of exact shape and area: square, circle, half-moon, and the great falcon-shaped altar of the Agnichayana built of hundreds of fired bricks. Change the shape and the area must stay the same. The Shulba (‘cord’) Sutras give the rules, using only a rope and pegs. Baudhayana’s states: the diagonal of a rectangle produces by itself both the areas which its length and breadth produce separately. He knew the triples (3, 4, 5), (5, 12, 13), (8, 15, 17) and (12, 35, 37), and he gave ways to turn a square into a circle of nearly the same area and back.
02The square root of two
To double a square, take its diagonal as the new side. To measure that diagonal, Baudhayana gives a rule: increase the side by its third, that third by its own fourth, less the thirty-fourth part of that fourth. In numbers, √2 ≈ 1 + 1/3 + 1/(3·4) − 1/(3·4·34) = 1.4142156…, correct to five decimal places. The true value is 1.4142135… . The same author left a Shrauta Sutra and a Dharma Sutra, and his school remains one of the living sutra traditions followed by Taittiriya brahmins of the south.
“The diagonal of a rectangle produces by itself both the areas which its length and breadth produce separately.”
Baudhayana Shulba Sutra 1.48