From Brahmagupta to Euler: on the formula for the area of a cyclic quadrilateral

A formula for the area of a cyclic quadrilateral in terms of its sides was first stated without proof by the early seventh-century Indian mathematician Brahmagupta. As early as the late tenth century, the Persian mathematician al-Shannı provided a proof of the Indian’s claim. In this paper I discuss al-Shannı’s derivation and compare it with two other derivations. The first of these is by the Kerala mathematician and astronomer Jyesṭ ḥ adeva (sixteenth century), while the second is a forgotten proof by the Dutch mathematical practitioner Abraham de Graaf that was published in 1706. I conclude with a discussion of Euler’s much better known derivation of 1748.