History of Hindu Mathematics: A Source Book v2.c17.s18.1

v2.c17.s18.1

**Rational Right Triangles: Early Solutions.** The earliest Hindu solutions of the equation $x² + y² = z² 1$ are found in the *Sulba*. Baudhâyana (*c.* 800 B.C.), Âpastamba and Kâtyâyana (*c.* 500 B.C.)¹ give a method for the transformation of a rectangle into a square, which is the equivalent of the algebraical identity $mn = (m - (m-n)/2)² - ((m-n)/2)²,$ where *m*, *n* are any two arbitrary numbers. Thus we get $(√mn)² + ((m-n)/2)² = ((m+n)/2)².$ Substituting p², q² for *m*, *n* respectively, in order to eliminate the irrational quantities, we get $p²q² + ((p²-q²)/2)² = ((p²+q²)/2)²,$ which gives a rational solution of (1).