Brahmagupta's Equations: Solving What Europe Couldn't

How a 7th-century Indian solved problems Europe tackled 1,000 years later

Explore Brahmagupta's systematic methods for solving linear and quadratic equations in his Brahmasphutasiddhanta.

The Verse That Surprised Calcutta

In a high-ceilinged office in Calcutta, sometime in the early years of the nineteenth century, the British East India Company judge Henry Thomas Colebrooke sets down his copy of the Brahmasphutasiddhanta and reads the next sūtra aloud, in Sanskrit, twice.

Colebrooke is in his forties, a senior official of the Sadar Diwani Adalat, and quietly the most accomplished Sanskritist in the British administration. He has spent two decades translating mathematical and legal works from the original. The text in front of him is more than a thousand years old. The room is hot. A fan-puller dozes in the corridor outside. A pile of palm-leaf manuscripts sits beside the inkwell, weighted down with a brass paperweight. Colebrooke is alone with the verse and his pen.

The verse is sūtra 18.44 of the Brahmasphutasiddhanta, composed by an astronomer named Brahmagupta in the year 628 CE. Translated literally into English, it reads: 'To the absolute number multiplied by four times the coefficient of the square, add the square of the coefficient of the middle term. The square root of the same, less the coefficient of the middle term, being divided by twice the coefficient of the square, is the value.'

Colebrooke pauses. He has been taught, like every English schoolboy of his generation, that the general quadratic formula was a triumph of European mathematics. Girolamo Cardano had written it down in his Ars Magna in Milan in 1545. Before Cardano, no one. After Cardano, every algebra textbook in Europe.

Translated into modern symbols, what Brahmagupta is saying is that for the equation ax² + bx = c, the unknown x equals the square root of (4ac + b²), minus b, all divided by 2a. That is, exactly, the quadratic formula. It is general. It works for any positive coefficients. It produces an exact root every time. And it was written nine hundred and seventeen years before Cardano's Ars Magna, on the western edge of the Indian subcontinent, by a thirty-year-old astronomer working in a city named Bhillamala.