BRAHMAGUPTA’S CONTRIBUTION
Pythagorean triples
In
chapter twelve of his Brahmasphutasiddhanta, Brahmagupta provides a
formula useful for generating Pythagorean triples:
The height of a mountain multiplied by a given
multiplier is the distance to a city; it is not erased. When it is divided by
the multiplier increased by two it is the leap of one of the two who make the
same journey.
Or, in other words, if d = mx/x + 2, then
a traveller who "leaps" vertically upwards a distance d from the top of a
mountain of height m, and
then travels in a straight line to a city at a horizontal distance mx from the base of the
mountain, travels the same distance as one who descends vertically down the
mountain and then travels along the horizontal to the city. Stated
geometrically, this says that if a right-angled triangle has a base of length a = mx and
altitude of length b = m + d, then
the length, c, of
its hypotenuse is given by c = m(1
+ x) − d. And, indeed,
elementary algebraic manipulation shows that a2 + b2 = c2 whenever d has the value stated.
Also, if m and x are rational, so are d, a, b and c. A Pythagorean triple can
therefore be obtained from a, b and c by multiplying each of
them by the least common multiple of
their denominators.
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