{
  "schemaVersion": "1.0",
  "discourse": {
    "id": "euclid-elements-book-xii",
    "name": "Euclid's Elements, Book XII",
    "subject": "measurement",
    "variant": "classical",
    "description": "Measurement of figures: circles, pyramids, cones, cylinders, spheres. 18 propositions. Depends on Books I, V, VI, XI. Source: David E. Joyce.",
    "structure": {
      "books": 12,
      "propositions": 18,
      "foundationTypes": [
        "foundation"
      ]
    }
  },
  "metadata": {
    "created": "2026-03-18",
    "lastUpdated": "2026-03-18",
    "version": "1.0.0",
    "license": "CC BY 4.0",
    "authors": [
      "Welz, G."
    ],
    "methodology": "Programming Framework",
    "citation": "Welz, G. (2026). Euclid's Elements Book XII Dependency Graph. Programming Framework.",
    "keywords": [
      "Euclid",
      "Elements",
      "Book XII",
      "measurement",
      "pyramid",
      "cone",
      "cylinder",
      "sphere"
    ]
  },
  "sources": [
    {
      "id": "joyce",
      "type": "digital",
      "authors": "Joyce, David E.",
      "title": "Euclid's Elements, Book XII",
      "year": "1996",
      "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookXII/bookXII.html",
      "notes": "Clark University"
    }
  ],
  "nodes": [
    {
      "id": "BookI",
      "type": "foundation",
      "label": "Book I — Plane geometry",
      "shortLabel": "Book I",
      "short": "Foundation",
      "book": 1,
      "colorClass": "foundation"
    },
    {
      "id": "BookV",
      "type": "foundation",
      "label": "Book V — Proportions",
      "shortLabel": "Book V",
      "short": "Foundation",
      "book": 5,
      "colorClass": "foundation"
    },
    {
      "id": "BookVI",
      "type": "foundation",
      "label": "Book VI — Similar figures",
      "shortLabel": "Book VI",
      "short": "Foundation",
      "book": 6,
      "colorClass": "foundation"
    },
    {
      "id": "BookXI",
      "type": "foundation",
      "label": "Book XI — Solid geometry",
      "shortLabel": "Book XI",
      "short": "Foundation",
      "book": 11,
      "colorClass": "foundation"
    },
    {
      "id": "Prop1",
      "type": "proposition",
      "label": "Similar polygons in circles: to one another as squares on diameters",
      "shortLabel": "Prop. XII.1",
      "short": "Similar polygons: as squares on diameters",
      "book": 12,
      "number": 1,
      "colorClass": "proposition"
    },
    {
      "id": "Prop2",
      "type": "proposition",
      "label": "Circles are to one another as the squares on their diameters",
      "shortLabel": "Prop. XII.2",
      "short": "Circles: as squares on diameters",
      "book": 12,
      "number": 2,
      "colorClass": "proposition"
    },
    {
      "id": "Prop3",
      "type": "proposition",
      "label": "Pyramid with triangular base: divided into two pyramids, two prisms; prisms greater than half",
      "shortLabel": "Prop. XII.3",
      "short": "Pyramid divided",
      "book": 12,
      "number": 3,
      "colorClass": "proposition"
    },
    {
      "id": "Prop4",
      "type": "proposition",
      "label": "Two pyramids same height, triangular bases, divided: base to base as all prisms",
      "shortLabel": "Prop. XII.4",
      "short": "Pyramids: base as prisms",
      "book": 12,
      "number": 4,
      "colorClass": "proposition"
    },
    {
      "id": "Prop5",
      "type": "proposition",
      "label": "Pyramids same height, triangular bases: to one another as bases",
      "shortLabel": "Prop. XII.5",
      "short": "Pyramids: as bases",
      "book": 12,
      "number": 5,
      "colorClass": "proposition"
    },
    {
      "id": "Prop6",
      "type": "proposition",
      "label": "Pyramids same height, polygonal bases: to one another as bases",
      "shortLabel": "Prop. XII.6",
      "short": "Pyramids polygonal: as bases",
      "book": 12,
      "number": 6,
      "colorClass": "proposition"
    },
    {
      "id": "Prop7",
      "type": "proposition",
      "label": "Prism with triangular base: divided into three equal pyramids",
      "shortLabel": "Prop. XII.7",
      "short": "Prism into three pyramids",
      "book": 12,
      "number": 7,
      "colorClass": "proposition"
    },
    {
      "id": "Prop8",
      "type": "proposition",
      "label": "Similar pyramids triangular bases: in triplicate ratio of corresponding sides",
      "shortLabel": "Prop. XII.8",
      "short": "Similar pyramids: triplicate ratio",
      "book": 12,
      "number": 8,
      "colorClass": "proposition"
    },
    {
      "id": "Prop9",
      "type": "proposition",
      "label": "Equal pyramids triangular bases: bases reciprocally proportional to heights",
      "shortLabel": "Prop. XII.9",
      "short": "Equal pyramids: bases reciprocally proportional",
      "book": 12,
      "number": 9,
      "colorClass": "proposition"
    },
    {
      "id": "Prop10",
      "type": "proposition",
      "label": "Any cone is third part of cylinder same base and equal height",
      "shortLabel": "Prop. XII.10",
      "short": "Cone third of cylinder",
      "book": 12,
      "number": 10,
      "colorClass": "proposition"
    },
    {
      "id": "Prop11",
      "type": "proposition",
      "label": "Cones and cylinders same height: to one another as bases",
      "shortLabel": "Prop. XII.11",
      "short": "Cones, cylinders: as bases",
      "book": 12,
      "number": 11,
      "colorClass": "proposition"
    },
    {
      "id": "Prop12",
      "type": "proposition",
      "label": "Similar cones and cylinders: in triplicate ratio of diameters of bases",
      "shortLabel": "Prop. XII.12",
      "short": "Similar cones, cylinders: triplicate",
      "book": 12,
      "number": 12,
      "colorClass": "proposition"
    },
    {
      "id": "Prop13",
      "type": "proposition",
      "label": "Cylinder cut by plane parallel to opposite: cylinder to cylinder as axis to axis",
      "shortLabel": "Prop. XII.13",
      "short": "Cylinder cut: as axes",
      "book": 12,
      "number": 13,
      "colorClass": "proposition"
    },
    {
      "id": "Prop14",
      "type": "proposition",
      "label": "Cones and cylinders on equal bases: to one another as heights",
      "shortLabel": "Prop. XII.14",
      "short": "Cones, cylinders equal bases: as heights",
      "book": 12,
      "number": 14,
      "colorClass": "proposition"
    },
    {
      "id": "Prop15",
      "type": "proposition",
      "label": "Equal cones and cylinders: bases reciprocally proportional to heights",
      "shortLabel": "Prop. XII.15",
      "short": "Equal cones, cylinders: reciprocally proportional",
      "book": 12,
      "number": 15,
      "colorClass": "proposition"
    },
    {
      "id": "Prop16",
      "type": "proposition",
      "label": "Given two circles same center: inscribe in greater equilateral polygon even sides not touching lesser",
      "shortLabel": "Prop. XII.16",
      "short": "Inscribe polygon in greater circle",
      "book": 12,
      "number": 16,
      "colorClass": "proposition"
    },
    {
      "id": "Prop17",
      "type": "proposition",
      "label": "Given two spheres same center: inscribe in greater polyhedral solid not touching lesser",
      "shortLabel": "Prop. XII.17",
      "short": "Inscribe polyhedron in greater sphere",
      "book": 12,
      "number": 17,
      "colorClass": "proposition"
    },
    {
      "id": "Prop18",
      "type": "proposition",
      "label": "Spheres are to one another in triplicate ratio of their diameters",
      "shortLabel": "Prop. XII.18",
      "short": "Spheres: triplicate ratio",
      "book": 12,
      "number": 18,
      "colorClass": "proposition"
    }
  ],
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