The corpus, every claim citable.

Aryabhata gives π ≈ 62832/20000 = 3.1416 in Aryabhatiya II.10 (the Ganitapada). Crucially, the Sanskrit word for "approximately" he uses is *āsanna* — "near, approaching but not reaching." This is the earliest explicit acknowledgement in any tradition that π is an irrational constant that can only be approximated, predating Lambert's 1761 formal proof by ~1,262 years.

The Aryabhatiya of Aryabhata · 499

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 I.3 gives the cosmic-cycle structure: 14 Manus × 72 yugas × 4,320,000 years = 4,354,560,000 years per kalpa (~4.35 billion). Modern radiometric Earth age (Patterson 1956): 4.54 billion. ~5% match between a 6th-century Sanskrit recurrence period and 20th-century radioactive-decay measurement. Almost certainly coincidence — Aryabhata was computing planetary cycles, not Earth's age — but striking.

The Aryabhatiya of Aryabhata · 499

Aryabhatiya II.32-33 (499 CE) gives the kuttaka algorithm for solving the linear indeterminate equation ax + by = c in integers, via reciprocal (Euclidean) division of a and b, then working the quotient chain backwards. Same algorithm later named "Chinese Remainder Theorem" via Qin Jiushao (1247 CE) and powers modern RSA key recovery (1977). Aryabhata's motivation was astronomical: computing when planets would all return to a given longitude.

The Aryabhatiya of Aryabhata · 499

Āryabhaṭīya IV (Gola) 5, 499 CE: half of the Earth, the planets, and the asterisms is dark — shadowed by the body itself — and the half turned toward the Sun is light. Applied to the Moon, this is the reflected-sunlight account of moonlight and of lunar phases. The insight has earlier independent precedents (Anaxagoras, c. 450 BCE, in Greece); Aryabhata's formulation embeds it in a quantitative astronomy curriculum used continuously in India for over a millennium.

The Aryabhatiya of Aryabhata · 499

Aryabhatiya I.6 (Dasagitika, 499 CE) gives Earth's axial tilt (the obliquity of the ecliptic) as 24°. Modern measurement is 23.44°. The 0.56° discrepancy isn't a measurement error: Earth's obliquity oscillates between ~22.1° and ~24.5° on a 41,000-year cycle (Milankovitch orbital forcing), and was ~24° several thousand years before Aryabhata wrote — consistent with his value reflecting inherited observational tradition from earlier Indian astronomy.

The Aryabhatiya of Aryabhata · 499

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