The Science of the Shulba: A Study in Early Hindu Geometry c19.s1.10

c19.s1.10

The principle underlying these solutions will be easily detected to be that of the reduction of the sides of any rational right triangle in the ratio of the given leg to the corresponding side of it. Thus the sides of a rational right triangle having a given leg a will be (a, aβ/α, aγ/α), where α, β, γ are the sides of any rational right triangle. So that starting with the solution (II), we shall find that all rational right triangles having the leg a will be given by $a, ((p²-q²)/2pq)a, ((p²+q²)/2pq)a.$ But this general solution is not stated anywhere in the *Śulba*. The commentators of Āpastamba, however, give the solution $a, ((m²+2m)/(2m+2))a, ((m²+2m+2)/(2m+2))a.$