MathematicsT1
Brahmagupta's bhāvanā lemma (Brahmasphutasiddhanta XVIII.64-65,
628 CE; rendered §76-77 in Colebrooke 1817): if (x₁, y₁) solves
x²N + k₁ = y² and (x₂, y₂) solves x²N + k₂ = y², then their
cross-product (x₁y₂ ± x₂y₁, y₁y₂ ± Nx₁x₂) solves x²N + (k₁k₂) = y².
The mathematical foundation Bhāskara II's chakravala (1150 CE)
stands on. Lagrange's 1768 European solution to "Pell's equation"
rediscovers the same composition + iteration structure.
Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara · tr. H. T. Colebrooke, 1817