Brahmagupta (588-670), son of Jisnugupta

Brahmagupta also introduced the formula which bears his name:

Brahmagupta's formula

Given a cyclic quadrilateral (one which can be inscribed in a circle) with side lengths a,b,c,d, the area of the quadrilateral is ((s-a)(s-b)(s-c)(s-d))1/2, where s = (a+b+c+d)/2

The proof of this rather elegant statement is quite hideous and involves lots of trigonometry and messy algebra. It is quite probable that Brahmagupta did not prove it, but rather took it as a generalisation of Heron's formula. It is quite easy to see that Heron's formula follows, simply by considering the case where d=0.

Other notable achievements of Brahmagupta are rules for the summing of various series (first n squares, first n cubes). He developed a method of long multiplication very similar to that in common use today and found quite accurate methods for approximating square roots and sines.