{
  "schemaVersion": "1.0",
  "discourse": {
    "id": "euclid-elements-book-viii",
    "name": "Euclid's Elements, Book VIII",
    "subject": "number_theory",
    "variant": "classical",
    "description": "Continued proportions. 27 propositions. Depends on Book VII. Source: David E. Joyce.",
    "structure": {
      "books": 8,
      "propositions": 27,
      "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 VIII Dependency Graph. Programming Framework.",
    "keywords": [
      "Euclid",
      "Elements",
      "Book VIII",
      "continued proportion",
      "plane",
      "solid"
    ]
  },
  "sources": [
    {
      "id": "joyce",
      "type": "digital",
      "authors": "Joyce, David E.",
      "title": "Euclid's Elements, Book VIII",
      "year": "1996",
      "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookVIII/bookVIII.html",
      "notes": "Clark University"
    }
  ],
  "nodes": [
    {
      "id": "BookVII",
      "type": "foundation",
      "label": "Book VII — Number theory",
      "shortLabel": "Book VII",
      "short": "Foundation",
      "book": 7,
      "colorClass": "foundation"
    },
    {
      "id": "Prop1",
      "type": "proposition",
      "label": "Continued proportion, extremes relatively prime: least in ratio",
      "shortLabel": "Prop. VIII.1",
      "short": "Continued proportion, extremes prime",
      "book": 8,
      "number": 1,
      "colorClass": "proposition"
    },
    {
      "id": "Prop2",
      "type": "proposition",
      "label": "To find numbers in continued proportion, least in given ratio",
      "shortLabel": "Prop. VIII.2",
      "short": "Find numbers in continued proportion",
      "book": 8,
      "number": 2,
      "colorClass": "proposition"
    },
    {
      "id": "Prop3",
      "type": "proposition",
      "label": "Least in continued proportion: extremes relatively prime",
      "shortLabel": "Prop. VIII.3",
      "short": "Least: extremes prime",
      "book": 8,
      "number": 3,
      "colorClass": "proposition"
    },
    {
      "id": "Prop4",
      "type": "proposition",
      "label": "Given ratios in least numbers, find least in continued proportion",
      "shortLabel": "Prop. VIII.4",
      "short": "Find continued proportion from ratios",
      "book": 8,
      "number": 4,
      "colorClass": "proposition"
    },
    {
      "id": "Prop5",
      "type": "proposition",
      "label": "Plane numbers have ratio compounded of ratios of sides",
      "shortLabel": "Prop. VIII.5",
      "short": "Plane numbers: compound ratio",
      "book": 8,
      "number": 5,
      "colorClass": "proposition"
    },
    {
      "id": "Prop6",
      "type": "proposition",
      "label": "Continued proportion: if first does not measure second, none measures another",
      "shortLabel": "Prop. VIII.6",
      "short": "First does not measure second",
      "book": 8,
      "number": 6,
      "colorClass": "proposition"
    },
    {
      "id": "Prop7",
      "type": "proposition",
      "label": "Continued proportion: if first measures last, it measures second",
      "shortLabel": "Prop. VIII.7",
      "short": "First measures last",
      "book": 8,
      "number": 7,
      "colorClass": "proposition"
    },
    {
      "id": "Prop8",
      "type": "proposition",
      "label": "Numbers between two in continued proportion correspond to ratios",
      "shortLabel": "Prop. VIII.8",
      "short": "Numbers between in proportion",
      "book": 8,
      "number": 8,
      "colorClass": "proposition"
    },
    {
      "id": "Prop9",
      "type": "proposition",
      "label": "Two relatively prime: numbers between them as between each and unit",
      "shortLabel": "Prop. VIII.9",
      "short": "Relatively prime: numbers to unit",
      "book": 8,
      "number": 9,
      "colorClass": "proposition"
    },
    {
      "id": "Prop10",
      "type": "proposition",
      "label": "Numbers between number and unit correspond to between two numbers",
      "shortLabel": "Prop. VIII.10",
      "short": "Numbers from unit",
      "book": 8,
      "number": 10,
      "colorClass": "proposition"
    },
    {
      "id": "Prop11",
      "type": "proposition",
      "label": "Between two squares one mean proportional; duplicate ratio",
      "shortLabel": "Prop. VIII.11",
      "short": "Mean proportional of squares",
      "book": 8,
      "number": 11,
      "colorClass": "proposition"
    },
    {
      "id": "Prop12",
      "type": "proposition",
      "label": "Between two cubes two mean proportionals; triplicate ratio",
      "shortLabel": "Prop. VIII.12",
      "short": "Two means between cubes",
      "book": 8,
      "number": 12,
      "colorClass": "proposition"
    },
    {
      "id": "Prop13",
      "type": "proposition",
      "label": "Continued proportion: products proportional; products of products",
      "shortLabel": "Prop. VIII.13",
      "short": "Products proportional",
      "book": 8,
      "number": 13,
      "colorClass": "proposition"
    },
    {
      "id": "Prop14",
      "type": "proposition",
      "label": "Square measures square iff side measures side",
      "shortLabel": "Prop. VIII.14",
      "short": "Square measures square",
      "book": 8,
      "number": 14,
      "colorClass": "proposition"
    },
    {
      "id": "Prop15",
      "type": "proposition",
      "label": "Cube measures cube iff side measures side",
      "shortLabel": "Prop. VIII.15",
      "short": "Cube measures cube",
      "book": 8,
      "number": 15,
      "colorClass": "proposition"
    },
    {
      "id": "Prop16",
      "type": "proposition",
      "label": "Square does not measure square iff side does not measure side",
      "shortLabel": "Prop. VIII.16",
      "short": "Square does not measure square",
      "book": 8,
      "number": 16,
      "colorClass": "proposition"
    },
    {
      "id": "Prop17",
      "type": "proposition",
      "label": "Cube does not measure cube iff side does not measure side",
      "shortLabel": "Prop. VIII.17",
      "short": "Cube does not measure cube",
      "book": 8,
      "number": 17,
      "colorClass": "proposition"
    },
    {
      "id": "Prop18",
      "type": "proposition",
      "label": "Between similar plane numbers one mean proportional; duplicate ratio",
      "shortLabel": "Prop. VIII.18",
      "short": "Similar plane: mean proportional",
      "book": 8,
      "number": 18,
      "colorClass": "proposition"
    },
    {
      "id": "Prop19",
      "type": "proposition",
      "label": "Between similar solid numbers two mean proportionals; triplicate ratio",
      "shortLabel": "Prop. VIII.19",
      "short": "Similar solid: two means",
      "book": 8,
      "number": 19,
      "colorClass": "proposition"
    },
    {
      "id": "Prop20",
      "type": "proposition",
      "label": "If one mean between two numbers, they are similar plane",
      "shortLabel": "Prop. VIII.20",
      "short": "One mean: similar plane",
      "book": 8,
      "number": 20,
      "colorClass": "proposition"
    },
    {
      "id": "Prop21",
      "type": "proposition",
      "label": "If two means between two numbers, they are similar solid",
      "shortLabel": "Prop. VIII.21",
      "short": "Two means: similar solid",
      "book": 8,
      "number": 21,
      "colorClass": "proposition"
    },
    {
      "id": "Prop22",
      "type": "proposition",
      "label": "Three in continued proportion, first square: third square",
      "shortLabel": "Prop. VIII.22",
      "short": "Three in proportion: first square",
      "book": 8,
      "number": 22,
      "colorClass": "proposition"
    },
    {
      "id": "Prop23",
      "type": "proposition",
      "label": "Four in continued proportion, first cube: fourth cube",
      "shortLabel": "Prop. VIII.23",
      "short": "Four in proportion: first cube",
      "book": 8,
      "number": 23,
      "colorClass": "proposition"
    },
    {
      "id": "Prop24",
      "type": "proposition",
      "label": "If ratio as square to square and first square, second square",
      "shortLabel": "Prop. VIII.24",
      "short": "Ratio as square to square",
      "book": 8,
      "number": 24,
      "colorClass": "proposition"
    },
    {
      "id": "Prop25",
      "type": "proposition",
      "label": "If ratio as cube to cube and first cube, second cube",
      "shortLabel": "Prop. VIII.25",
      "short": "Ratio as cube to cube",
      "book": 8,
      "number": 25,
      "colorClass": "proposition"
    },
    {
      "id": "Prop26",
      "type": "proposition",
      "label": "Similar plane numbers have ratio of square to square",
      "shortLabel": "Prop. VIII.26",
      "short": "Similar plane: square ratio",
      "book": 8,
      "number": 26,
      "colorClass": "proposition"
    },
    {
      "id": "Prop27",
      "type": "proposition",
      "label": "Similar solid numbers have ratio of cube to cube",
      "shortLabel": "Prop. VIII.27",
      "short": "Similar solid: cube ratio",
      "book": 8,
      "number": 27,
      "colorClass": "proposition"
    }
  ],
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    },
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    },
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    },
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    },
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