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

Written while on sabbatical leave at Utrecht University (The Netherlands). The author wishes to thank Prof Dr Jan Hogendijk and the Department of Mathematics for their hospitality.

Abstract

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-Shannī provided a proof of the Indian's claim. In this paper I discuss al-Shannī's derivation and compare it with two other derivations. The first of these is by the Kerala mathematician and astronomer Jyeṣṭhadeva (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.

Notes

Written while on sabbatical leave at Utrecht University (The Netherlands). The author wishes to thank Prof Dr Jan Hogendijk and the Department of Mathematics for their hospitality.

¹ See Chapter 7 of Wreede (2007) for a historical overview. Heron's original derivation (along with a few other ones) is also discussed in studies of Dunham (1990, 118–129, 1999, 130–3, 143–5).

² Nor is there any indication that he might have done so in any of his other works. For a purely mathematical argument that Heron could very well have derived such an area formula from his formula for the scalene trapezoid, see Weissenborn (1879).

³ See for instance Mukhopadhyay and Adhikari (1997).

⁴ On this, see Chapter 5 of Sarasvati Amma (1979). For a tentative reconstruction of Brahmagupta's thought process, see Kichenassamy (2010).

⁵ On the Kerala school (and why this name is a misnomer), see Chapter 7 of Plofker (2009). For a translation of the Malayalam original into English, see Sarma et al. (2008). Jyeṣṭhadeva's proof is also discussed in Sarasvati Amma (1979, 115–6).

⁶ See Suter (1910, 40–2, 70) for a German translation of the relevant section as well as a short commentary.

⁷ Particularly, instead of the classical language of proportions between line segments, I will use fractions and symbols denoting the lengths of line segments. I have also rearranged the order of some of the steps for clarity.

⁸ See also Wreede (2007, 185, 241, passim).

⁹ On this, see Davids (1986, 117), and the literature referenced there.

¹⁰ There is no argument for this claim, but if we add , the centre of the circle and , the reflection of in , to De Graaf's drawing, the relation follows immediately from the similarity of the triangles and . Of course, De Graaf's observation also is an immediate consequence of the (extended) Law of Sines, which was well known by the late seventeenth century.

¹¹ See Tropfke (1923, 188). The history of the use of this identity in the Western world is somewhat murky. It was only elevated to the status of a standard proposition by Legendre, who combined it with Ptolemy's theorem as Proposition XXXII of Book III of Legendre (1794, 92–3).

¹² ‘Eius quidem demonstratio, si analysis in subsidium vocetur, non est difficilis, sed qui eam more apud Geometras recepto adornare sunt conati, maximas experti sunt difficultates, Cl: quondam Naudeaus non parum in hoc genere laboravit, et geminam huius quoque regulae demonstrationem protulit in Misc. Berol. verum utraque non solum maxime est intricata et multitudine linearum in figura ductarum obruta, ut sine summa attentione ne capi qui possit, sed etiam ubique nimis luculenta vestigia analytici calculi offendunt (...).’ See Euler (1750, 57).