The corpus, every claim citable.

Aryabhatiya Golapada IV.37 (499 CE) gives the correct geometric mechanism of both eclipses: the Sun is obscured when the Moon comes between Earth and Sun; the Moon is obscured when it passes into the Earth's shadow. Brahmagupta directly attacked this in Brahmasphuta- siddhanta XI.9 (628 CE), calling Aryabhata's eclipse account "false" and re-affirming the demon Rahu. The geometric reading did not become Indian astronomical mainstream until the Kerala school ~1500 CE.

The Aryabhatiya of Aryabhata · 499

Aryabhatiya Golapada IV.9 uses the boat analogy to argue that the apparent westward motion of stars is an illusion caused by Earth's eastward axial rotation. Aryabhatiya I.1 quantifies it: 1,582,237,500 rotations per yuga ≈ 366.26 sidereal rotations per year, accurate to ~4 parts per million against modern measurement. Predates Copernicus by 1,044 years; contested within Indian astronomy itself (Brahmagupta rejected it in 628 CE).

The Aryabhatiya of Aryabhata · 499

Brahmasphutasiddhanta XVIII.22 (628 CE) gives the four sign-multiplication rules: (−)·(+) = (−), (−)·(−) = (+), (+)·(+) = (+), with the parallel zero rules. The hardest case — negative times negative — was hedged in European mathematics for centuries: Cardano (1545) calls negative roots "fictitious"; Wallis (1685) is the first to argue geometrically for the rule; Hankel (1867) gives the first formal construction. Brahmagupta states all four rules without apology in 628 CE.

Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara · 628

Brahmasphutasiddhanta XVIII.19 (628 CE) gives explicit rules for arithmetic with zero as a number: addition, subtraction, multiplication, square root. Brahmagupta also writes a rule for division by zero — getting it wrong (treats x/0 as finite) but pioneering the question itself. Bhaskara II refined the rule ~500 years later (1150 CE) treating 1/0 as khahara — closer to modern infinity-as-limit.

Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara · 628

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 · 628