The corpus, every claim citable.

Āryabhaṭīya, Gītikā 1 (499 CE) fixes the Earth's eastward rotations per yuga at 1,582,237,500 against 4,320,000 solar revolutions — implying 1,577,917,500 civil days per yuga and a sidereal day of 86,164.10 seconds (23h 56m 4.10s). The modern value is 86,164.09 seconds: agreement to about 0.01 s. The verse also encodes the rotating-Earth doctrine numerically ("of the Earth eastward") — the parameter later tradition recast as revolutions of the stars.

The Aryabhatiya of Aryabhata · 499

The Tantrasaṅgraha of Nīlakaṇṭha Somayājī (Kerala, c. 1500 CE) states the alternating series π/4 = 1 − 1/3 + 1/5 − 1/7 + … as a verse rule for the circumference of a circle of given diameter, together with a rational end-correction that sharply accelerates convergence. The Kerala school's commentaries attribute the series to Mādhava (c. 1340–1425). Leibniz published the same series in Europe in 1673; Charles Whish first reported the Kerala texts to European scholarship in 1834.

On the Hindu Quadrature of the Circle, and the Infinite Series of the Proportion of the Circumference to the Diameter Exhibited in the Four Sastras · 1500

Bhāskara II's Bījagaṇita (1150 CE) defines the quotient 3/0 as khahara — "termed an infinite quantity" — and states it is unaltered by adding or subtracting finite amounts. A deliberate algebraic definition of division by zero, five centuries before Europe acquired a working infinity symbol (Wallis, 1655). His wider system still misfires — elsewhere a quantity multiplied then divided by zero recovers its finite value — but the definition itself is the first of its kind.

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

Mahāvīra's Ganita-sara-sangraha I.52 (850 CE) gives the first explicit recognition that √(negative) is undefined on reals. The reasoning: every square is non-negative, so no real number squared yields a negative. Cardano (1545) calls these roots "fictitious"; Bombelli (1572) treats them as imaginary; Gauss (1799) gives the complex-number foundation. The impossibility acknowledgement is Mahāvīra's, ~700 years before Cardano.

The Ganita-sara-sangraha of Mahaviracarya · 850

Mahāvīra's Ganita-sara-sangraha VI.218 (850 CE) gives the general algorithmic statement of the nCr formula: write 1..n ascending and n..1 descending in two rows; the product of the top r entries divided by the product of the bottom r is nCr. Pingala (~200 BCE) had the binomial-prosody special case; Pascal's Traité (1654 CE) gives the European systematic form. Mahāvīra's algorithm is the general procedural statement, 800 years before Pascal.

The Ganita-sara-sangraha of Mahaviracarya · 850

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

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 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

Bhaskara II's Bijaganita (1150 CE) gives a complete cyclic algorithm (chakravala) for solving Nx² + 1 = y², the equation Europeans would later call "Pell's equation" after a 17th-c. misattribution. Applied to N=61, the algorithm yields x=226,153,980 and y=1,766,319,049 — the smallest positive-integer solution. Fermat proposed exactly this case to Frenicle and Wallis in 1657 as a challenge problem; Lagrange was the first European to find a general method in 1768, six centuries after Bhaskara.

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

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

The decimal place-value system originated in India and spread in documented stages: epigraphic evidence (including a 605 CE Cambodian inscription) shows coverage "roughly of the size of Europe" by the end of the 6th century; Arab mathematicians adopted it in the 8th century; common European use came only around the 16th — popular almanacs of 1557–96 still print Roman numerals. Datta & Singh (1938) document each stage from primary evidence.

History of Hindu Mathematics — A Source Book · 1938

Pañcasiddhāntikā III.21 (505 CE) states that the summer solstice once turned from the middle of Āśleṣā — "then the ayana was right" — but at present begins from Punarvasu: a shift of about 23°, roughly 1,700 years of equinoctial precession separating the old record from current observation. Hipparchus discovered precession c. 130 BCE; this verse documents the Indian tradition registering the same drift by checking its inherited solstice positions against the sky.

The Panchasiddhantika: The Astronomical Work of Varaha Mihira · 575

Varāhamihira's Pañcasiddhāntikā (505 CE) summarizes and ranks five astronomical schools; the Romaka ("Roman") Siddhānta places in the top three. Its luni-solar yuga of 2,850 years with 1,050 intercalary months is exactly 150 Metonic cycles (19 years, 7 intercalations each), and its epoch is reckoned from sunset at Yavanapura — Alexandria. Greco-Roman astronomy circulated inside the Indian canon, openly named and rated.

The Panchasiddhantika: The Astronomical Work of Varaha Mihira · 575

Sūrya-Siddhānta i.30 fixes the Moon's sidereal revolutions per Age at 57,753,336; i.37 fixes the Age's civil days at 1,577,917,828. The implied sidereal month, 27.321674 days, differs from the modern 27.321662 by about 1.1 seconds — roughly 0.5 parts per million. Babylonian-Greek lunar theory reached comparable precision by other routes; the Siddhānta's whole-number encoding is the Indian tradition's own and stayed in computational use for over a millennium.

Translation of the Surya-Siddhanta · 400

The Yajurveda Saṁhitā (Vājasaneyī xvii.2, c. 1200–800 BCE) lists thirteen decimal denominations — eka (1) through parārdha (10¹²) — each ten times the preceding; the same list recurs in the Taittirīya Saṁhitā. Datta & Singh (1938) contrast this with Greek terminology, which stopped at the myriad (10⁴), and Roman, at mille (10³). Named decuple ranks are a documented Vedic-era feature of Sanskrit, many centuries before written place-value numerals.

History of Hindu Mathematics — A Source Book · 1938

Sūrya-Siddhānta XII.53 (c. 400–500 CE core text, Burgess 1860 translation) states that the Earth is a globe in space with no absolute up or down: every observer takes their own place to be uppermost. Verses 51–52 apply it concretely — dwellers at opposite points of the globe each suppose the other underneath. Greek astronomy established terrestrial sphericity earlier (Aristotle, c. 350 BCE); the Siddhānta's plain statement of the relativity of "up" is among the clearest in any ancient text.

Translation of the Surya-Siddhanta · 400

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

Pingala's Chandahsutra (~200 BCE) gives a four-aphorism recursive algorithm for counting metrical arrangements of n syllables. The rules ("halve; subtract one when odd; multiply by two; square when halved") implement exponentiation-by-squaring — the same recurrence modern computers use to compute 2ⁿ in O(log n) steps. Halayudha's 10th-century commentary makes the recursion explicit. The algorithm predates Leibniz's binary arithmetic (1703) by ~1,900 years.

History of Hindu Mathematics — A Source Book · 1938

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