product of half the sums of pairs of opposite sides, strangely enough occurs in a Chinese work of the 6th century as well as in the works of Ahmes, Brahmagupta, Mahāvīra, Boethius and Bede. By Mahāvīra, the idea on which it is based—that the area is a function of the perimeter—is further emphasized. Āryabhata gives an incorrect rule for the volume of a pyramid; incorrect rules for the volume of a sphere are common to Āryabhata, S'rīdhara and Mahāvīra. For cones all the rules assume that . Mahāvīra gives incorrect rules for the circumference and area of an ellipse and so on.
17. Brahmagupta gives a fairly complete set of rules dealing with the cyclic quadrilateral and either he or the mathematician from whom he obtained his material had a definite end in view—the construction of a cyclic quadrilateral with rational elements.—The commentators did not fully appreciate the theorems, some of which are given in the works of Mahāvīra and S'rīdhara; and by the time of Bhāskara they had ceased to be understood. Bhāskara indeed condemns them outright as unsound. "How can a person" he says "neither specifying one of the perpendiculars, nor either of the diagonals, ask the rest? Such a questioner is a blundering devil and still more so is he who answers the question."
Besides the two rules (i) and (ii) already given in paragraph 15, Brahmagupta gives rules corresponding to the formula
(iii) , etc., and
(iv) If and then the quadrilateral is cyclic and has its diagonals at right angles.
This figure is sometimes termed "Brahmagupta's trapezium." From the triangles (3, 4, 5) and (5, 12, 13) a commentator obtains the quadrilateral (39, 60, 52, 25),
