Page:Indian mathematics, Kaye (1915).djvu/29

 {| Mahāvīra gives many examples in which he employs formula (V) of which he terms m and n the 'elements.' From given elements he constructs triangles and from given triangles he finds the elements, e.g., "What are the 'elements' of the right-angled triangle (48, 55, 73)? Answer: 3, 8."
 * | A
 * | B
 * | $$\scriptstyle{\sqrt{A^2+B^2}}$$
 * | Authorities.
 * | i
 * | n
 * | $$\scriptstyle{\frac{n^2-1}{2}}$$
 * | $$\scriptstyle{\frac{n^2+1}{2}}$$
 * | Pythagoras (according to Proclus)
 * | ii
 * | $$\scriptstyle{\sqrt{mn}}$$
 * | $$\scriptstyle{\frac{mn}{2}}$$
 * | $$\scriptstyle{\frac{m+n}{2}}$$
 * | Plato (according to Proclus)
 * | iii
 * | $$\scriptstyle{2mn}$$
 * | $$\scriptstyle{m^2-n^2}$$
 * | $$\scriptstyle{m^2+n^2}$$
 * | Euclid and Diophantus.
 * | iv
 * | $$\scriptstyle{\frac{m(n^2-1)}{n^2+1}}$$
 * | $$\scriptstyle{\frac{2mn}{n^2+1}}$$
 * | m
 * | Diophantus.
 * | v
 * | 2mn
 * | $$\scriptstyle{m^2-n^2}$$
 * | $$\scriptstyle{m^2+n^2}$$
 * | Brahmagupta and Mahāvīra.
 * | vi
 * | $$\scriptstyle{\sqrt{m}}$$
 * | $$\scriptstyle{\frac{1}{2}(\frac{m}{n}-n)}$$
 * | $$\scriptstyle{\frac{1}{2}(\frac{m}{n}+n)}$$
 * | Brahmagupta, Mahāvīra, and Bhāskara.
 * | vii
 * | m
 * | $$\scriptstyle{\frac{2mn}{n^2-1}}$$
 * | $$\scriptstyle{\frac{m(n^2+1)}{n^2-1}}$$
 * | Bhāskara.
 * | viii
 * | $$\scriptstyle{\frac{m(m^2-1)}{n^2+1}}$$
 * | $$\scriptstyle{\frac{2mn}{n^2+1}}$$
 * | m
 * | Bhāskara.
 * | ix
 * | 2lmn
 * | $$\scriptstyle{l(m^2-n^2)}$$
 * | $$\scriptstyle{l(m^2+n^2)}$$
 * | General formula.
 * }
 * | $$\scriptstyle{\frac{2mn}{n^2+1}}$$
 * | m
 * | Bhāskara.
 * | ix
 * | 2lmn
 * | $$\scriptstyle{l(m^2-n^2)}$$
 * | $$\scriptstyle{l(m^2+n^2)}$$
 * | General formula.
 * }
 * }