{
  "schemaVersion": "1.0",
  "discourse": {
    "id": "euclid-elements-book-ii",
    "name": "Euclid's Elements, Book II",
    "subject": "geometry",
    "variant": "classical",
    "description": "The 14 propositions of Book II on geometric algebra (rectangles, squares). Props 1-10 are logically independent within Book II; 11-14 depend on 6, 4, 7, 5. All depend on Book I. Source: David E. Joyce, Clark University.",
    "structure": {
      "books": 2,
      "propositions": 14,
      "foundationTypes": [
        "definition"
      ]
    }
  },
  "metadata": {
    "created": "2026-03-15",
    "lastUpdated": "2026-03-15",
    "version": "1.0.0",
    "license": "CC BY 4.0",
    "authors": [
      "Welz, G."
    ],
    "methodology": "Programming Framework",
    "citation": "Welz, G. (2026). Euclid's Elements Book II Dependency Graph. Programming Framework.",
    "keywords": [
      "Euclid",
      "Elements",
      "Book II",
      "geometric algebra",
      "rectangles",
      "squares",
      "golden section",
      "quadrature"
    ]
  },
  "sources": [
    {
      "id": "joyce",
      "type": "digital",
      "authors": "Joyce, David E.",
      "title": "Euclid's Elements, Book II",
      "year": "1996",
      "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookII/bookII.html",
      "notes": "Clark University; dependency table from Logical structure"
    },
    {
      "id": "euclid-heath",
      "type": "primary",
      "authors": "Heath, T.L.",
      "title": "The Thirteen Books of Euclid's Elements",
      "year": "1908",
      "edition": "2nd",
      "publisher": "Cambridge University Press",
      "url": "https://archive.org/details/euclidheath00heatiala",
      "notes": "Standard English translation"
    }
  ],
  "nodes": [
    {
      "id": "BookI",
      "type": "foundation",
      "label": "Book I — Fundamentals of plane geometry",
      "shortLabel": "Book I",
      "short": "Foundation",
      "book": 1,
      "colorClass": "foundation"
    },
    {
      "id": "Def1",
      "type": "definition",
      "label": "Rectangle contained by two straight lines containing the right angle",
      "shortLabel": "Def. II.1",
      "book": 2,
      "number": 1,
      "colorClass": "definition"
    },
    {
      "id": "Def2",
      "type": "definition",
      "label": "Gnomon: parallelogram about diameter with two complements",
      "shortLabel": "Def. II.2",
      "book": 2,
      "number": 2,
      "colorClass": "definition"
    },
    {
      "id": "Prop1",
      "type": "proposition",
      "label": "If one line is cut into segments, rectangle by whole equals sum of rectangles by each segment",
      "shortLabel": "Prop. II.1",
      "short": "Rectangle = sum of rectangles",
      "book": 2,
      "number": 1,
      "colorClass": "proposition"
    },
    {
      "id": "Prop2",
      "type": "proposition",
      "label": "If a line is cut at random, sum of rectangles by whole and each segment equals square on whole",
      "shortLabel": "Prop. II.2",
      "short": "Sum of rectangles = square on whole",
      "book": 2,
      "number": 2,
      "colorClass": "proposition"
    },
    {
      "id": "Prop3",
      "type": "proposition",
      "label": "If a line is cut at random, rectangle by whole and one segment equals rectangle by segments plus square on that segment",
      "shortLabel": "Prop. II.3",
      "short": "Rectangle = rectangle + square",
      "book": 2,
      "number": 3,
      "colorClass": "proposition"
    },
    {
      "id": "Prop4",
      "type": "proposition",
      "label": "If a line is cut at random, square on whole equals squares on segments plus twice rectangle contained by segments",
      "shortLabel": "Prop. II.4",
      "short": "Square on whole = squares + 2×rectangle",
      "book": 2,
      "number": 4,
      "colorClass": "proposition"
    },
    {
      "id": "Prop5",
      "type": "proposition",
      "label": "If a line cut into equal and unequal segments, rectangle by unequal segments plus square on difference equals square on half",
      "shortLabel": "Prop. II.5",
      "short": "Unequal segments: rectangle + square = square on half",
      "book": 2,
      "number": 5,
      "colorClass": "proposition"
    },
    {
      "id": "Prop6",
      "type": "proposition",
      "label": "If a line bisected and added to, rectangle by whole-with-added and added plus square on half equals square on half-plus-added",
      "shortLabel": "Prop. II.6",
      "short": "Bisected + added: rectangle + square = square",
      "book": 2,
      "number": 6,
      "colorClass": "proposition"
    },
    {
      "id": "Prop7",
      "type": "proposition",
      "label": "If a line cut at random, square on whole plus square on one segment equals twice rectangle by whole and segment plus square on remainder",
      "shortLabel": "Prop. II.7",
      "short": "Square on whole + square on segment",
      "book": 2,
      "number": 7,
      "colorClass": "proposition"
    },
    {
      "id": "Prop8",
      "type": "proposition",
      "label": "If a line cut at random, four times rectangle by whole and one segment plus square on remainder equals square on whole-plus-segment",
      "shortLabel": "Prop. II.8",
      "short": "Four times rectangle + square",
      "book": 2,
      "number": 8,
      "colorClass": "proposition"
    },
    {
      "id": "Prop9",
      "type": "proposition",
      "label": "If a line cut into equal and unequal segments, sum of squares on unequal segments is double sum of square on half and square on difference",
      "shortLabel": "Prop. II.9",
      "short": "Unequal segments: sum of squares",
      "book": 2,
      "number": 9,
      "colorClass": "proposition"
    },
    {
      "id": "Prop10",
      "type": "proposition",
      "label": "If a line bisected and added to, square on whole-with-added plus square on added equals double sum of square on half and square on half-plus-added",
      "shortLabel": "Prop. II.10",
      "short": "Bisected + added: sum of squares",
      "book": 2,
      "number": 10,
      "colorClass": "proposition"
    },
    {
      "id": "Prop11",
      "type": "proposition",
      "label": "To cut a given line so that rectangle by whole and one segment equals square on remaining segment",
      "shortLabel": "Prop. II.11",
      "short": "Cut line: rectangle = square (golden section)",
      "book": 2,
      "number": 11,
      "colorClass": "proposition"
    },
    {
      "id": "Prop12",
      "type": "proposition",
      "label": "In obtuse-angled triangles, square on side opposite obtuse angle greater than sum of squares on sides containing it",
      "shortLabel": "Prop. II.12",
      "short": "Obtuse triangle: law of cosines",
      "book": 2,
      "number": 12,
      "colorClass": "proposition"
    },
    {
      "id": "Prop13",
      "type": "proposition",
      "label": "In acute-angled triangles, square on side opposite acute angle less than sum of squares on sides containing it",
      "shortLabel": "Prop. II.13",
      "short": "Acute triangle: law of cosines",
      "book": 2,
      "number": 13,
      "colorClass": "proposition"
    },
    {
      "id": "Prop14",
      "type": "proposition",
      "label": "To construct a square equal to a given rectilinear figure",
      "shortLabel": "Prop. II.14",
      "short": "Construct square = rectilinear figure",
      "book": 2,
      "number": 14,
      "colorClass": "proposition"
    }
  ],
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