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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

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 Gaṇita-sāra-saṅgraha (850 CE) poses the hundred-birds problem: pigeons at 5 for 3 paṇas, sārasas at 7 for 5, swans at 9 for 7, peacocks at 3 for 9 — buy 100 birds for 100 paṇas: two equations, four unknowns, integer solutions required. The same problem-type appears in Zhang Qiujian's Chinese classic (c. 475 CE, priority) and Alcuin's Latin puzzles (c. 800) — a marker problem for the circulation of mathematics across Eurasia.

History of Hindu Mathematics — A Source Book · 1938

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