Sanskrit astronomical and mathematical treatise composed by Brahmagupta in 628 CE. Contains the first systematic rules for arithmetic with zero and negative numbers, the formula for the area of a cyclic quadrilateral, and methods for indeterminate equations. Translated into Arabic c. 771 CE and directly influenced the development of algebra in the Islamic world.
In 628 CE, a 30-year-old mathematician in Ujjain, India sat down and wrote a book that would change arithmetic forever. Brahmagupta's Brāhmasphuṭasiddhānta — "The Correctly Established Doctrine of Brahma" — covered astronomy, but buried inside was something even more important: the first written rules for calculating with zero and negative numbers.
About 140 years after Brahmagupta wrote it, an Arabic translation was made in Baghdad. That translation reached Al-Khwārizmī, who used it in his famous algebra book — the book that gave us the word "algorithm." From there the ideas travelled to Europe via Fibonacci (1202 CE). So when European merchants started using the number 0 in the 13th century, they were using rules that Brahmagupta had written 600 years earlier.
| Author | Brahmagupta (598 – c. 668 CE) |
| Composed | 628 CE, Ujjain, India |
| Language | Sanskrit verse |
| Chapters | 25 |
| Title meaning | "The Correctly Established Doctrine of Brahma" |
Brahmagupta states rules for three types of quantity: dhana (fortune, positive), ṛṇa (debt, negative), and kha (cipher, zero).
For a cyclic quadrilateral with consecutive sides a, b, c, d and semi-perimeter s = (a+b+c+d)/2:
Setting d = 0 gives Heron's formula for triangles. This formula is exact — it requires only the four side lengths, with no reference to angles or diagonals.
Chapter 18 also contains methods for solving integer equations of the form ax + by = c, extending Āryabhaṭa's earlier kuṭṭaka algorithm. Brahmagupta further studied the equation Nx² + 1 = y², which Europeans would later call "Pell's equation."
The BSS survives in multiple Sanskrit manuscripts. The standard modern edition is Dvivedin (1902), who collated available manuscripts. The authoritative English translation of the mathematical chapters is Colebrooke (1817), which translates Chapters 12 and 18 (the core mathematical content). Plofker (2009) provides the current scholarly analysis with Sanskrit verse references, historical context, and comparison with parallel traditions.
The BSS reached Baghdad as part of the translation movement under Caliph Al-Manṣūr (r. 754–775 CE). An Indian delegation presented astronomical manuscripts to the caliph's court c. 771 CE; the BSS was translated into Arabic as the Zij al-Sindhind. This translation was used by Al-Fazārī (d. c. 796 CE) and later by Al-Khwārizmī (c. 780–850 CE). Al-Khwārizmī's Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala (c. 820 CE) draws on this Indian material. The word "algorithm" derives from the Latinisation of Al-Khwārizmī's name.
The proof of Brahmagupta's formula uses the law of cosines applied to two triangles formed by a diagonal, then the identity sin²θ + cos²θ = 1. For a cyclic quadrilateral, opposite angles are supplementary (sum to 180°), which allows the two cosine terms to cancel. The result Area = √[(s−a)(s−b)(s−c)(s−d)] is then obtained by algebraic simplification. No proof survives from Brahmagupta himself; Indian mathematical tradition emphasised results over proofs in the Greek sense. The modern proof appears in most advanced geometry texts.
Brahmagupta's BSS contains explicit criticism of Āryabhaṭa (BSS Ch. 11) on the question of Earth's rotation. Brahmagupta argues (incorrectly) that the Earth cannot rotate because objects would fly off it. This illustrates that Indian mathematics was not a single unified tradition — it was a competitive intellectual field with disagreements and debates, like any living scholarly community.