The corpus, every claim citable.

Hoernle's 1888 study of the Bakhshali manuscript (a birch-bark mathematical text discovered in 1881 near Peshawar) describes the dot serving two roles: as a placeholder for an unknown quantity (analogous to modern x) AND as a fundamental digit in the decimal place-value system — the zero. The manuscript was carbon-dated by Oxford's Bodleian Library in 2017 to between 224 and 383 CE, making it the earliest extant evidence of place-value zero in any tradition.

On the Bakshali Manuscript · 1888

Aryabhatiya I.2 (Dasagitika, 499 CE) defines an alphabetic numeral system: Sanskrit consonants ka-ma (varga, 25 letters) take values 1-25; ya-ha (avarga, 8) take 30, 40, ... 100; vowels multiply by powers of 100 (a=10⁰, i=10², u=10⁴, ...). This compresses 7-digit astronomical constants into 2-3 Sanskrit syllables, pronounceable in metric verse. Used continuously by Indian astronomers through the Kerala school (~1500 CE).

The Aryabhatiya of Aryabhata · 499

Mahāvīra's Ganita-sara-sangraha VII.21 (850 CE) gives an explicit rule for the area of an ellipse: shorter diameter / 4, multiplied by the circumference (computed from the two diameters in the same verse). Reduces to A = π·a·b with π taken as 3. Accurate to ~10% for moderate eccentricities, ~20% for typical pond-shaped ellipses. First explicit ellipse-area formula in surviving Sanskrit mathematics.

The Ganita-sara-sangraha of Mahaviracarya · 850

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

Aryabhatiya II.22 (499 CE) gives both closed-form summation identities: Σi² for i=1..n equals n(n+1)(2n+1)/6, and Σi³ for i=1..n equals (Σi)² = (n(n+1)/2)². Faulhaber publishes the same identities in 1631 (Academia Algebrae); Bernoulli systematises power-sum formulas in 1713 (Ars Conjectandi) via what became the Bernoulli numbers. The Sanskrit statement predates Faulhaber by 1,132 years.

The Aryabhatiya of Aryabhata · 499

Surya-Siddhanta IV.12-13 (~5th c. CE) gives a computational procedure for the half-duration of a lunar eclipse: the moon's latitude offsets the lunar centre from Earth's shadow; first/last contact occurs when this offset equals (shadow±moon) radii; a right-triangle Pythagorean step plus a relative-motion divide gives the time. NASA's modern eclipse-timing code uses the same triangle structure.

Translation of the Surya-Siddhanta · 400

Bhaskara II's Lilavati §134 (1150 CE) states the Pythagorean theorem: the hypotenuse (Sanskrit karna, here "diagonal") equals the square root of the sum of the squares of the two legs. Not original to Bhaskara — the Indian Sulbasutras (Baudhayana, Apastamba, ~800 BCE) state it geometrically for altar construction, centuries before Pythagoras (~530 BCE). Lilavati §134 is the canonical late-medieval Sanskrit statement of an already-2,000-year-old Indian result.

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

Aryabhatiya II.10 gives 24 first-differences of sines at 3°45' intervals (R = 3,438 minutes). Cumulative sum reconstructs sin(θ) for θ = 3°45'…90°, accurate to ~0.03%. The English word "sine" descends Sanskrit jya → Arabic jaib (translation error: pocket/ bay) → Latin sinus → English "sine." Hipparchus had a chord table 600 years earlier; Aryabhata's is the first half-chord (= modern sine) table in any tradition.

The Aryabhatiya of Aryabhata · 499

Aryabhata's cosmological time-scale: a day of Brahman (kalpa) consists of 14 Manus, each Manu containing 72 yugas — a deep-time framework predating modern geological time scales by over a millennium.

The Aryabhatiya of Aryabhata · 499

Aryabhata (499 CE) gives the diameter of the Earth as 1,050 yojanas (~13,200 km using ~12.5 km/yojana), within ~3% of the modern value of 12,742 km — a striking pre-modern measurement.

The Aryabhatiya of Aryabhata · 499