Head of the Observatory at Bhillamala
Brahmagupta was born in 598 CE, probably at or near Bhillamala in what is now Rajasthan, then the capital of a kingdom under the Gurjara ruler Vyaghramukha. He became head of the astronomical observatory there, a leading centre of the discipline, and at the age of thirty composed the Brahmasphutasiddhanta, the corrected treatise of Brahma. Around 665 he wrote a second work, the Khandakhadyaka, a practical handbook of astronomical computation that proved popular for centuries and was widely commented upon.
He wrote in verse, like his predecessors, and he wrote combatively. The Brahmasphutasiddhanta contains extended criticism of rival astronomers, Aryabhata prominent among them, on the rotation of the earth, on eclipse theory and on numerical parameters. Some of his objections were sound corrections and some defended orthodoxy against a better idea. He worked within a tradition where astronomy carried religious authority, and he was not shy about invoking it, which makes his willingness to formalise the strange new object of zero all the more striking.
Rules for Fortunes and Debts
Zero had existed as a placeholder and as a mark for an empty column. Brahmagupta made it a number that participates in arithmetic. He set out what happens when zero is added to, subtracted from and multiplied by other quantities, and he did the same for negative numbers, which he described in the commercial language of fortunes and debts: a debt subtracted from nothing is a fortune, the product of two debts is a fortune, and so on. This is the earliest surviving systematic treatment of negative quantities as legitimate results rather than as impossibilities.
Division defeated him. He asserted that zero divided by zero is zero, which is wrong, and left division of a nonzero number by zero unresolved. Bhaskara II, in the twelfth century, proposed that such a quotient was infinite, which is closer to a modern intuition but still not right. The definitive answer, that the operation is undefined, waited for the rigour of nineteenth-century analysis. The mistake is worth remembering: the man who first made zero behave like a number also showed exactly where the difficulty lies.
Cyclic Quadrilaterals and the Road to Baghdad
His geometry contains a jewel. He gave a formula for the area of a cyclic quadrilateral, one whose four vertices lie on a circle, in terms of its four sides, generalising the formula attributed to Heron for triangles; he also gave a rule for its diagonals. He did not state clearly that the result requires the quadrilateral to be cyclic, which caused confusion among later readers. In algebra he studied indeterminate equations of the form now called Pell's, giving a method for generating new solutions from known ones that Euler and Lagrange would later formalise.
Around 770 CE an Indian astronomical text, generally taken to be based on Brahmagupta's work, was brought to the court of the caliph al-Mansur at Baghdad and translated into Arabic as the Sindhind. It carried Indian trigonometry and the decimal place-value system with zero into the Islamic scientific world, where al-Khwarizmi built on it, and from there into Latin Europe. Brahmagupta died around 668. The notation that reached medieval Italy as the new arithmetic was, in substantial part, the working method of his observatory.
