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

	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.

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."