Brahmagupta’s propositions on the perpendiculars of cyclic quadrilaterals

AbstractWe continue a recent analysis of Propositions XII.21–28 of Brahmagupta’s Brāhma-sphuṭa-siddhānta (India, 628 A.D.), on the area and diagonals of the cyclic quadrilateral, by examining Propositions XII.29–32, that explain how to determine the perpendiculars as well as all the portions of diagonals and perpendiculars. These results include the result nowadays referred to as “Brahmagupta’s theorem” (XII.30–31). Brahmagupta describes both the geometric situation and the key elements of the derivation of his results. We analyze the expression of hypotheses and derivations, using only Brahmagupta’s conceptual framework, that does not include the notion of angle, and uses proportion only in a standard form (XII.25)