The Science of the Shulba: A Study in Early Hindu Geometry c15.115

c15.115

The rational rectangle (15, 36, 39) perhaps deserves more than a passing notice. It could of course be derived from the rational rectangle (5, 12, 13) by multiplying its sides and diagonal by 3. This relation has, indeed, been expressly admitted by Āpastamba.² But it was probably obtained first independently, thinks Bürk,³ as an instance of a rational rectangle in which the difference between the greater side and the diagonal would be 3. This will probably account for Baudhāyana's enumerating it separately along with (5, 12, 13). The early Hindus, particularly those belonging to the Āpastamba school, appear, however, to have special regard for the rational rectangle (15, 36, 39). So the method of constructing several *vedis* has been described in the *Śulba* of this school with particular reference to this rectangle. For instance, take the case of the *Nirūḍhapaśubandha-vedi*. It is of the shape of an isosceles trapezium whose measurements are known from the ancient scriptures to be: face = the yoke of a cart (= 86 aṅgulis); altitude = the pole (= 188 aṅgulis) and base = the axle of the cart (= 104 aṅgulis).¹ As regards the process of its construction, Āpastamba says: