RATIONAL GEOMETRICAL FIGURE method of transforming a rectangle into a square, which is equivalent to the algebraic identity : 261 mn-(m-"-") -(""). where m, n, are any two arbitrary numbers. Brahmagupta in connection with the solution of rational triangles says: The square of the optional (ista) side is divided and then diminished by an optional number; half the result is the upright, and that increased by the optional number gives the hypotenuse of a rectangle. m. We shall put these statements of Brahmagupta in the algebraic language thus: If m, n be any two rational numbers, then the sides of a right-angled triangle will be ÷ ( ² − n) + ( + * n ) 12 n This Sanskrit term ista may either mean "given" or "optional". With the former meaning the rule would imply the method of finding rational right angles having a given leg. Brahmagupta was the first to give a solution of the equa- tion x²+²-z² in integers. His solution is m²-n², 2mn, m² +n². m and n being two unequal integers.² Thus if m-7 and n=4 then m²-n²-33, 2mm=56 and m²+n²-65; then the three numbers 33, 56 and 65 bear the rela- tion 33+56-65¹. Mahavira (850 A.D.) also states The difference of the squares (of two elements) is the upright, twice their product is the base and the sum of their squares is the diagonal of a generated rectangle.2 Isosceles Triangles with Integral Sides: The following state- ment of Brahmagupta in this connection is very significant : 1. इष्टस्य भुजस्य कृतिर्मको नेष्टेन तद्दलं कोटिः । आयतचतुरस्रस्य च त्रस्येष्टाधिका कर्णः || 2. GSS. VII. 90¹ -BrSpSi. XII. 35
