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

	A	B	${\displaystyle \scriptstyle {\sqrt {A^{2}+B^{2}}}}$	Authorities.
i	n	${\displaystyle \scriptstyle {\frac {n^{2}-1}{2}}}$	${\displaystyle \scriptstyle {\frac {n^{2}+1}{2}}}$	Pythagoras (according to Proclus)
ii	${\displaystyle \scriptstyle {\sqrt {mn}}}$	${\displaystyle \scriptstyle {\frac {mn}{2}}}$	${\displaystyle \scriptstyle {\frac {m+n}{2}}}$	Plato (according to Proclus)
iii	${\displaystyle \scriptstyle {2mn}}$	${\displaystyle \scriptstyle {m^{2}-n^{2}}}$	${\displaystyle \scriptstyle {m^{2}+n^{2}}}$	Euclid and Diophantus.
iv	${\displaystyle \scriptstyle {\frac {m(n^{2}-1)}{n^{2}+1}}}$	${\displaystyle \scriptstyle {\frac {2mn}{n^{2}+1}}}$	m	Diophantus.
v	2mn	${\displaystyle \scriptstyle {m^{2}-n^{2}}}$	${\displaystyle \scriptstyle {m^{2}+n^{2}}}$	Brahmagupta and Mahāvīra.
vi	${\displaystyle \scriptstyle {\sqrt {m}}}$	${\displaystyle \scriptstyle {{\frac {1}{2}}({\frac {m}{n}}-n)}}$	${\displaystyle \scriptstyle {{\frac {1}{2}}({\frac {m}{n}}+n)}}$	Brahmagupta, Mahāvīra, and Bhāskara.
vii	m	${\displaystyle \scriptstyle {\frac {2mn}{n^{2}-1}}}$	${\displaystyle \scriptstyle {\frac {m(n^{2}+1)}{n^{2}-1}}}$	Bhāskara.
viii	${\displaystyle \scriptstyle {\frac {m(m^{2}-1)}{n^{2}+1}}}$	${\displaystyle \scriptstyle {\frac {2mn}{n^{2}+1}}}$	m	Bhāskara.
ix	2lmn	${\displaystyle \scriptstyle {l(m^{2}-n^{2})}}$	${\displaystyle \scriptstyle {l(m^{2}+n^{2})}}$	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."