{
  "id": "aryabhatiya",
  "domain": "indian-mathematics",
  "type": "Work",
  "title": "Āryabhaṭīya",
  "slug": "aryabhatiya",
  "url": "/mathematics/indian-mathematics/aryabhatiya/",
  "summary": "Sanskrit mathematical and astronomical treatise composed by Āryabhaṭa I in 499 CE. Contains place-value arithmetic, a sine table, an approximation of π to four decimal places, the kuṭṭaka linear-congruence algorithm, and a heliocentric rotation model. One of the most influential scientific texts of the first millennium.",
  "created": "499-01-01",
  "added_to_codex": "2026-05-27",
  "last_verified": "2026-05-27",
  "author": {
    "name": "Āryabhaṭa I",
    "born": "476 CE",
    "born_place": "Kusumapura (Pāṭaliputra), Magadha",
    "died": "c. 550 CE"
  },
  "composition": {
    "year_ce": 499,
    "language": "Sanskrit",
    "verse_form": "āryā metre",
    "total_verses": 121,
    "sections": [
      { "name": "Gītikāpāda", "verses": 13, "subject": "astronomical constants, large numbers" },
      { "name": "Gaṇitapāda", "verses": 33, "subject": "mathematics: arithmetic, geometry, series, algebra" },
      { "name": "Kālakriyāpāda", "verses": 25, "subject": "time reckoning, calendar" },
      { "name": "Golādhyāya", "verses": 50, "subject": "spherical astronomy, celestial mechanics" }
    ]
  },
  "key_results": [
    {
      "result": "π approximation",
      "statement": "π ≈ 62832/20000 = 3.1416",
      "significance": "Accurate to four decimal places; Āryabhaṭa explicitly calls it āsanna (approximate)"
    },
    {
      "result": "Sine table",
      "statement": "24 sine values at 3.75° intervals; first-difference scheme",
      "significance": "Foundational for Islamic and European trigonometry via translation"
    },
    {
      "result": "Kuṭṭaka algorithm",
      "statement": "Systematic solution of linear congruences ax ≡ b (mod c)",
      "significance": "Earliest known general algorithm for linear Diophantine equations"
    },
    {
      "result": "Place-value arithmetic",
      "statement": "Uses Sanskrit alphabetic encoding of a full decimal place-value system",
      "significance": "Evidence of place-value in India by 499 CE at latest"
    },
    {
      "result": "Earth's rotation",
      "statement": "Apparent motion of stars due to Earth rotating on its axis",
      "significance": "Heliocentric-compatible model, centuries before Copernicus"
    }
  ],
  "civilisational_context": {
    "tradition": "Indian (Gupta period)",
    "predecessors": ["Vedic sulbasūtras", "Jain cosmological mathematics", "Babylonian planetary models (via Hellenistic transmission)"],
    "transmission": ["Al-Khwārizmī (Baghdad, c. 820 CE)", "Latin translations via Arabic", "Kerala school commentaries (Nīlakaṇṭha, 1500 CE)"]
  },
  "sources": [
    {
      "tier": 1,
      "citation": "Shukla, K.S. and Sarma, K.V. (1976). Āryabhaṭīya of Āryabhaṭa. Indian National Science Academy, New Delhi.",
      "type": "critical_edition_translation",
      "notes": "Standard critical edition with Sanskrit text, English translation, and mathematical commentary"
    },
    {
      "tier": 1,
      "citation": "Plofker, K. (2009). Mathematics in India. Princeton University Press.",
      "pages": "111–127",
      "type": "scholarly_monograph"
    },
    {
      "tier": 2,
      "citation": "Clark, W.E. (1930). The Āryabhaṭīya of Āryabhaṭa. University of Chicago Press.",
      "type": "translation"
    },
    {
      "tier": 2,
      "citation": "O'Connor, J.J. and Robertson, E.F. (2006). Āryabhaṭa the Elder. MacTutor History of Mathematics Archive.",
      "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Aryabhata_I/",
      "type": "reference_article"
    },
    {
      "tier": 3,
      "citation": "Joseph, G.G. (2011). The Crest of the Peacock: Non-European Roots of Mathematics. 3rd ed. Princeton University Press.",
      "type": "popular_scholarly"
    }
  ],
  "relationships": [
    { "type": "authored_by", "target": "aryabhata-i", "label": "Āryabhaṭa I" },
    { "type": "part_of_tradition", "target": "indian-mathematics-overview", "label": "Indian Mathematical Tradition" },
    { "type": "introduces", "target": "kuttaka", "label": "Kuṭṭaka algorithm" },
    { "type": "introduces", "target": "sine-function", "label": "Sine table (jyā)" },
    { "type": "commentated_by", "target": "aryabhatiyabhasya-bhaskara-i", "label": "Bhāskara I commentary (629 CE)" },
    { "type": "commentated_by", "target": "aryabhatiyabhasya-nilakantha", "label": "Nīlakaṇṭha commentary (c. 1500 CE)" },
    { "type": "influenced", "target": "brahmasphutasiddhanta", "label": "Brāhmasphuṭasiddhānta (Brahmagupta, 628 CE)" },
    { "type": "transmitted_to", "target": "al-khwarizmi", "label": "Al-Khwārizmī, Baghdad" }
  ],
  "entry_status": "live",
  "reading_levels_available": ["curious", "exploring", "deep_dive"],
  "completeness": 1.0,
  "codex_version": "1.0"
}